Fractal nil graded Lie superalgebras

Journal of Algebra 466 (2016) 229–283

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Journal of Algebra www.elsevier.com/locate/jalgebra

Fractal nil graded Lie superalgebras Victor Petrogradsky 1 Department of Mathematics, University of Brasilia, 70910-900 Brasilia, DF, Brazil

a r t i c l e

i n f o

Article history: Received 4 February 2016 Available online 3 August 2016 Communicated by J.T. Stafford MSC: 16P90 16N40 16S32 17B50 17B65 17B66 17B70 Keywords: Restricted Lie algebras p-groups Growth Self-similar algebras Nil-algebras Graded algebras Lie superalgebra Lie algebras of differential operators

1

a b s t r a c t The Grigorchuk and Gupta–Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [31], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [41]. Thus, we have examples of (self-similar) finitely generated restricted Lie algebras with a nil p-mapping. In characteristic zero, similar examples of Lie algebras do not exist by a result of Martinez and Zelmanov [27]. The goal of the present paper is to construct analogues of the Grigorchuk and Gupta–Sidki groups in the world of Lie superalgebras of an arbitrary characteristic. We construct two examples R, Q of finitely generated self-similar Lie superalgebras over a field K of an arbitrary characteristic and study their properties and properties of their associative hulls. In case char K = 2, these examples turn into restricted Lie algebras. The virtue of the present construction is that the Lie superalgebras have clear monomial bases. These Lie superalgebras have slow polynomial growth and are multigraded by multidegree in the generators. The Lie superalgebra R is Z2 -graded, while Q has a multidegree Z3 -gradation and a Z2 -gradation. Both algebras R and Q have similar constructions, computations for R are simpler, but Q enjoys some more specific interesting properties. The Z3 -components of Q lie inside an elliptic paraboloid in space, they are at most one-dimensional, thus, the Z3 -grading of Q is fine. In the Z2 -gradation of Q, all components Qnm , n, m ∈ Z, are infinite dimensional except

E-mail address: [email protected]. The author was partially supported by grants CNPq, FEMAT.

http://dx.doi.org/10.1016/j.jalgebra.2016.06.028 0021-8693/© 2016 Elsevier Inc. All rights reserved.

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for Q00 = {0}. The Z2 -gradation also yields a continuum of different decompositions into a direct sum of two locally nilpotent subalgebras Q = Q+ ⊕ Q− . In both examples, ad a is nilpotent, a being even or odd with respect to the Z2 -grading as Lie superalgebras. This property is an analogue of the periodicity of the Grigorchuk and Gupta–Sidki groups. In particular, Q is a nil finely-graded Lie superalgebra, which shows that an extension of a theorem due to Martinez and Zelmanov [27] for the Lie superalgebras of characteristic zero is not valid. Both Lie superalgebras are self-similar, contain infinitely many copies of themselves, let us also call them fractal due to this property. © 2016 Elsevier Inc. All rights reserved.

1. Introduction: self-similar groups and algebras In this section we provide a survey of related results and give motivations for construction of our examples, in Section 2 we supply basic definitions. Main results are formulated in Section 3. 1.1. Golod–Shafarevich algebras and groups The General Burnside Problem puts the question whether a finitely generated periodic group is finite. The first negative answer was given by Golod and Shafarevich, who proved that, for each prime p, there exists a finitely generated infinite p-group [13,14]. The construction is based on a famous construction of a family of finitely generated infinite dimensional associative nil-algebras [13]. This construction also yields examples of infinite dimensional finitely generated Lie algebras L such that (ad x)n(x,y) (y) = 0, for all x, y ∈ L, the field being arbitrary [15]. The field being of positive characteristic p, one obtains an infinite dimensional finitely generated restricted Lie algebra L such that n(x) the p-mapping is nil, namely, x[p ] = 0, for all x ∈ L. This gives a negative answer to a question of Jacobson whether a finitely generated restricted Lie algebra L is finite dimensional provided that each element x ∈ L is algebraic, i.e. satisfies some p-polynomial fp,x (x) = 0 [20, Ch. 5, ex. 17]. It is known that the construction of Golod yields associative nil-algebras of exponential growth. Using specially chosen relations, Lenagan and Smoktunowicz constructed associative nil-algebras of polynomial growth [26]. On further developments concerning Golod–Shafarevich algebras and groups see [47,11]. A close by spirit but different construction was motivated by respective grouptheoretic results. A restricted Lie algebra G is called large if there is a subalgebra H ⊂ G of finite codimension such that H admits a surjective homomorphism on a nonabelian free restricted Lie algebra. Let K be a perfect at most countable field of positive characteristic. Then there exist infinite-dimensional finitely generated nil restricted Lie algebras over K that are residually finite dimensional and direct limits of large restricted Lie algebras [3].

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1.2. Grigorchuk and Gupta–Sidki groups The construction of Golod is rather undirect, Grigorchuk gave a direct and elegant construction of an infinite 2-group generated by three elements of order 2 [16]. This group was defined as a group of transformations of the interval [0, 1] from which rational points of the form {k/2n | 0 ≤ k ≤ 2n , n ≥ 0} are removed. For each prime p ≥ 3, Gupta and Sidki gave a direct construction of an infinite p-group on two generators, each of order p [19]. This group was constructed as a subgroup of an automorphism group of an infinite regular tree of degree p. The Grigorchuk and Gupta–Sidki groups are counterexamples to the General Burnside Problem. Moreover, they gave answers to important problems in group theory. So, the Grigorchuk group and its further generalizations are first examples of groups of intermediate growth [17], thus answering in negative to a conjecture of Milnor that groups of intermediate growth do not exist. The construction of Gupta–Sidki also yields groups of subexponential growth [12]. The Grigorchuk and Gupta–Sidki groups are self-similar, in particular, they contain infinitely many copies of themselves. Now self-similar, in particular so called branch groups, form a well-established area in group theory, see [18] for further developments. Below we discuss existence of analogues of the Grigorchuk and Gupta–Sidki groups for other algebraic structures. 1.3. Self-similar nil graded associative algebras The study of these groups leads to investigation of group rings and other related associative algebras [43]. In particular, there appeared self-similar associative algebras defined by matrices in a recurrent way [5]. The recurrent matrices were also applied to number theoretical problems [1]. Sidki suggested two examples of self-similar associative matrix algebras [42]. A more general family of self-similar associative algebras was introduced in [34], this family generalizes the second example of Sidki [42], also it yields a realization of a Fibonacci restricted Lie algebras (see below) in terms of self-similar matrices [34]. Another important feature of some associative self-similar algebras A constructed in [34] is that they are sums of two locally nilpotent subalgebras A = A+ ⊕ A− (see similar decompositions (1) below). Recall that an algebra is said locally nilpotent if every finitely generated subalgebra is nilpotent. But the desired analogues of the Grigorchuk and Gupta–Sidki groups should be associative self-similar nil-algebras, in a standard way yielding new examples of finitely generated periodic groups. But such examples are not known yet. On close open problems in theory of infinite dimensional algebras see review [48]. 1.4. Nil restricted Lie algebras Unlike associative algebras, for restricted Lie algebras, there are natural analogues of the Grigorchuk and Gupta–Sidki groups. Namely, over a field of characteristic 2,

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the author constructed an example of an infinite dimensional restricted Lie algebra L generated by two elements, called a Fibonacci restricted Lie algebra [31]. Let char K = 2 and R = K[ti |i ≥ 0]/(tpi |i ≥ 0) a truncated polynomial ring. Put ∂i = ∂t∂ i , i ≥ 0. Define the following two derivations of R: v1 = ∂1 + t0 (∂2 + t1 (∂3 + t2 (∂4 + t3 (∂5 + t4 (∂6 + · · · ))))); v2 =

∂2 + t1 (∂3 + t2 (∂4 + t3 (∂5 + t4 (∂6 + · · · )))).

These two derivations generate a restricted Lie algebra L = Liep (v1 , v2 ) ⊂ Der R and an associative algebra A = Alg(v1 , v2 ) ⊂ End R. By Bergman’s theorem, the Gelfand–Kirillov dimension of an associative algebra cannot belong to the interval (1, 2) [22]. Such a gap for Lie algebras does not exist, the Gelfand–Kirillov dimension of a finitely generated Lie algebra can be an arbitrary number {0} ∪ [1, +∞] [30]. The Fibonacci Lie algebra has slow polynomial growth with Gelfand–Kirillov dimension GKdim L = log √5+1 2 ≈ 1.44 [31]. The restricted Lie algebra L is self-similar. Further 2 properties of the Fibonacci restricted Lie algebra and its generalizations are studied in [33,35]. Probably, the most interesting property of L is that it has a nil p-mapping [31], which is an analog of the periodicity of the Grigorchuk and Gupta–Sidki groups. We do not know whether the associative hull A is a nil-algebra. We have a weaker statement. The algebras L, A, and the augmentation ideal of the restricted enveloping algebra u = ωu(L) are direct sums of two locally nilpotent subalgebras [33]: L = L+ ⊕ L− ,

A = A+ ⊕ A− ,

u = u+ ⊕ u− .

(1)

There are examples of infinite dimensional associative algebras which are direct sums of two locally nilpotent subalgebras [24,9]. Infinite dimensional restricted Lie algebras can have finitely many different decompositions into a direct sum of two locally nilpotent subalgebras [36]. In case of an arbitrary prime characteristic, Shestakov and Zelmanov suggested an example of a finitely generated restricted Lie algebra with a nil p-mapping [41]. That example yields the same decompositions (1) for some primes [25,34]. An example of a p-generated nil restricted Lie algebra L, characteristic p being arbitrary, was studied in [36]. The virtue of that example is that for all primes p we have the same decompositions (1) into direct sums of two locally nilpotent subalgebras. But computations for that example are rather complicated. Observe that only the original example has a clear monomial basis [31,33]. In other examples, elements of a Lie algebra are linear combinations of monomials, working with such linear combinations is sometimes an essential technical difficulty, see e.g. [41,36]. A continuum family of nil restricted Lie algebras of slow growth having good monomial basis is constructed in [37], these algebras are close relatives of the Lie superalgebra R of the present paper.

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Let G be a group and G = G1 ⊇ G2 ⊇ · · · its lower central series. One constructs the related Z-graded Lie algebra LK (G) = ⊕i≥1 Li , where Li = Gi /Gi+1 ⊗Z K, i ≥ 1, and char K = p. A product is given by [ai Gi+1 , bj Gj+1 ] = (ai , bj )Gi+j+1 , where (ai , bj ) = −1 a−1 i bj ai bj stands for the group commutator. A residually p-group G is said of finite width if all factors Gi /Gi+1 are finite groups with uniformly bounded orders. The Grigorchuk group G is of finite width, namely, it is proved that dimF2 Gi /Gi+1 ∈ {1, 2} for i ≥ 2 [40,7]. In particular, the respective Lie algebra L = LK (G) = ⊕i≥1 Li has a linear growth. Bartholdi recently proved that the restricted Lie algebra LF2 (G) is nil while LF4 (G) is not nil [6]. 1.5. Lie algebras over a field of characteristic zero In case of characteristic zero a similar example does not exist by the next result. Informally speaking, there are no natural analogues of the Grigorchuk and Gupta–Sidki groups in the world of Lie algebras of zero characteristic. Theorem 1.1 (Martinez and Zelmanov [27]). Let L = ⊕α∈Γ Lα be a Lie algebra over a field K of zero characteristic graded by an abelian group Γ. Suppose that (1) there exists d > 0 such that dimK Lα ≤ d for all α ∈ Γ, (2) every homogeneous element a ∈ Lα , α ∈ Γ, is ad-nilpotent. Then the Lie algebra L is locally nilpotent. 1.6. Fractal nil graded Lie superalgebras R and Q The goal of the present paper is to construct analogues of the Grigorchuk and Gupta– Sidki groups in the world of Lie superalgebras of an arbitrary characteristic. We construct two Lie superalgebras R, Q, which are also analogues of the Fibonacci restricted Lie algebra and other self-similar Lie algebras mentioned above. In both examples, ad a is nilpotent, a being an even or odd element with respect to Z2 -grading as Lie superalgebras. This property is an analogue of the periodicity of the Grigorchuk and Gupta–Sidki groups. The Lie superalgebra R is Z2 -graded, while Q has a natural fine Z3 -gradation and Z2 -gradation. Constructions of both algebras R and Q look similar, computations for R are simpler, but Q enjoys some specific interesting properties. In particular, the example Q shows that an extension of Theorem 1.1 for Lie superalgebras of zero characteristic is not valid. Both Lie superalgebras are self-similar, contain infinitely many copies of themselves, let us also call them fractal due to this property. A detailed exposition of properties of the Lie superalgebras R and Q is given in Section 3. It is interesting that there are many examples of Z2 -graded simple Lie algebras in characteristic zero with one-dimensional components [21].

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2. Basic definitions: (restricted) Lie superalgebras, growth In this section we supply basic definitions. Superalgebras appear naturally in physics and mathematics [23,44,2]. Let K denote the ground field, S K a linear span of a subset S in a K-vector space. A superalgebra A is a Z2 -graded algebra A = A¯0 ⊕A¯1 . The elements a ∈ Aα are called homogeneous of degree deg a = α, α ∈ Z2 . The elements of A¯0 are called even, those of A¯1 odd. Throughout what follows, if deg a enters an expression, then it is assumed that a is homogeneous of degree deg a ∈ Z2 , and the expression extends to the other elements by linearity. Let A, B be superalgebras, a tensor product A ⊗ B is the superalgebra whose space is the tensor product of the spaces A and B with the induced Z2 -grading and the product defined by (a1 ⊗ b1 )(a2 ⊗ b2 ) = (−1)deg b1 ·deg a2 a1 a2 ⊗ b1 b2 ,

ai ∈ A, bi ∈ B.

An associative superalgebra A is just a Z2 -graded associative algebra A = A¯0 ⊕ A¯1 . A Lie superalgebra is a Z2 -graded algebra L = L¯0 ⊕ L¯1 with an operation [ , ] satisfying the following axioms: • [x, y] = −(−1)deg x·deg y [y, x], (superanticommutativity); • [x, [y, z]] = [[x, y], z] + (−1)deg x·deg y [y, [x, z]], (Jacobi identity). In particular, these conditions imply that L¯0 is a Lie algebra and L¯1 a L¯0 -module. All commutators in the present paper are supercommutators. Long commutators are right-normed: [x, y, z] = [x, [y, z]]. We use standard notation ad x(y) = [x, y], where x, y ∈ L. Assume that A = A¯0 ⊕ A¯1 is an associative superalgebra. Then one defines on the same vector space A a supercommutator [x, y] = xy − (−1)deg x·deg y yx,

x, y ∈ A.

This product turns A into a Lie superalgebra denoted by A(−) . Assume that L is a Lie superalgebra, one defines the universal enveloping algebra U (L) = T (L)/(x ⊗ y − (−1)deg x·deg y y ⊗ x − [x, y] | x, y ∈ L), where T (L) is the tensor algebra of the vector space L. Now, the product in L coincides with the supercommutator in U (L)(−) . A basis of U (L) is given by PBW-theorem [2,44]. In case of small characteristics char K = 2, 3 the axioms of the Lie superalgebra have to be modified, see e.g. [2, section 1.10], [8]. Substituting x = y = z ∈ L¯1 in the Jacobi identity and using anticommutativity, we get 3[z, [z, z]] = 0, z ∈ L¯1 . Thus, in case char K = 3, we have an identity: [z, [z, z]] = 0 for all z ∈ L¯1 . Otherwise we add one more axiom for the Lie superalgebra. • (char K = 3) [z, [z, z]] = 0, z ∈ L¯1 .

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Substituting x = y ∈ L¯1 in the Jacobi identity, we get 2(ad x)2 z = [[x, x], z]. In case char K = 2 we get an identity (ad x)2 z =

1 [[x, x], z], 2

x ∈ L¯1 , z ∈ L.

The present paper deals with superalgebras of the form A(−) , they have squares “for free” (−) [x, x] = 2x2 , x ∈ A¯1 . Thus, the identity above takes the following form, which we shall use below without mention (one directly verifies that it is also valid for char K = 2): (ad x)2 z = [x2 , z],

(−)

x ∈ A¯1 , z ∈ A(−) .

(2)

In case char K = 2, we add more axioms for Lie superalgebras: • [x, x] = 0 for all x ∈ L¯0 ; • there exists a quadratic mapping (formal square): (∗)[2] : L¯1 → L¯0 , x → x[2] , x ∈ L¯1 , that satisfies: (λx)[2] = λ2 x[2] ,

(x + y)[2] = x[2] + [x, y] + y [2] ;

x, y ∈ L¯1 , λ ∈ K;

(By putting y = x, we get [y, y] = 0, y ∈ L¯1 .) • a formal substitute of (2) is fulfilled: (ad x)2 z = [x[2] , z],

x ∈ L¯1 , z ∈ L.

In case p = 2, to get the universal enveloping algebra, we additionally factor out {y ⊗ y − y [2] | y ∈ L¯1 }. Also, one has to modify the definition of the derived series: L(0) = L,

L(i+1) = [L(i) , L(i) ] + y 2 | y ∈ (L(i) )¯1 K , i ≥ 0.

Roughly speaking, a Lie superalgebra in case char K = 2 is the same as a Z2 -graded Lie algebra supplied with a quadratic mapping L¯1 → L¯0 , which is similar to the p-mapping (see below). In all cases, the quadratic mapping will be denoted by x2 , x ∈ L¯1 , it also coincides with the ordinary square in U (L). Let V = V¯0 ⊕ V¯1 be a vector space, we say that it is Z2 -graded. The associative algebra of all endomorphisms of the space End V is an associative superalgebra: End V = End¯0 V ⊕ End¯1 V , where Endα V = {φ ∈ End V | φ(Vβ ) ⊂ Vα+β , β ∈ Z2 }, α ∈ Z2 . Thus, End(−) V is a Lie superalgebra, called the general linear superalgebra gl(V ). Let A = A¯0 ⊕ A¯1 be a Z2 -graded algebra of arbitrary signature. A linear mapping φ ∈ Endβ A, β ∈ Z2 , is a superderivative of degree β if it satisfies φ(a · b) = φ(a) · b + (−1)β·deg(a) a · φ(b),

a, b ∈ A.

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Denote by Derα A ⊂ Endα A the space of all superderivatives of degree α ∈ Z2 . An easy check shows that Der A = Der¯0 A ⊕ Der¯1 A is a subalgebra of the Lie superalgebra End(−) A. In this paper by a derivation we always mean a superderivation. Let L be a Lie algebra over a field K of characteristic p > 0. The Lie algebra L is called a restricted Lie algebra (or Lie p-algebra), if it is additionally supplied with a unary operation x → x[p] , x ∈ L, that satisfies the following axioms [20,45,46]: • (λx)[p] = λp x[p] , for λ ∈ K, x ∈ L; • ad(x[p] ) = (ad x)p , x ∈ L; p−1 • (x + y)[p] = x[p] + y [p] + i=1 si (x, y), x, y ∈ L, where isi (x, y) is the coefficient of ti−1 in the polynomial ad(tx + y)p−1 (x) ∈ L[t]. This notion is motivated by the following observation. Let A be an associative algebra over a field K, char K = p > 0. Then the mapping x → xp , x ∈ A(−) , satisfies these conditions considered in the Lie algebra A(−) . A restricted Lie superalgebra L = L¯0 ⊕ L¯1 is a Lie superalgebra such that the even component L¯0 is a restricted Lie algebra and L¯0 -module L¯1 is restricted, i.e. ad(x[p] )y = (ad x)p y, for all x ∈ L¯0 , y ∈ L¯1 (see e.g. [28,2]). Remark that in case char K = 2, the restricted Lie superalgebras and Z2 -graded restricted Lie algebras are the same objects. (Let L = L¯0 ⊕ L¯1 be a restricted Lie superalgebra, we extend the p-mapping on the whole of algebra by (x + y)[2] = x[2] + y [2] + [x, y], x ∈ L¯0 , y ∈ L¯1 .) Suppose that L is a restricted Lie algebra. Let J be an ideal of the universal enveloping algebra U (L) generated by all elements x[p] − xp , x ∈ L. Then u(L) = U (L)/J is called the restricted enveloping algebra. In this algebra, the formal operation x[p] coincides with the pth power xp for any x ∈ L. One has an analogue of Poincare–Birkhoff–Witt’s theorem describing the basis of the restricted enveloping algebra [20, p. 213]. We recall the notion of growth. Let A be an associative (or Lie) algebra generated by a finite set X. Denote by A(X,n) a subspace of A spanned by all monomials in X of length not exceeding n, n ≥ 0. In case of a Lie superalgebra of char K = 2 we also consider formal squares of odd monomials of length at most n/2. If A is a restricted Lie algebra, k put A(X,n) = [x1 , . . . , xs ]p | xi ∈ X, spk ≤ n K [29]. Similarly, one defines the growth for restricted Lie superalgebras. In either situation, one obtains a growth function: γA (n) = γA (X, n) = dimK A(X,n) ,

n ≥ 0.

Let f, g : N → R+ be two eventually monotone increasing and positive valued functions. Write f (n)  g(n) if and only if there exist two positive constants N , C such that f (n) ≤ g(Cn) for all n ≥ N . Introduce equivalence f (n) ∼ g(n) if and only if f (n)  g(n) and g(n)  f (n). Different generating sets of an algebra yield equivalent growth functions [22]. It is well known that the exponential growth is the highest possible growth for finitely generated Lie and associative algebras. A growth function γA (n) is compared

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with polynomial functions nk , k ∈ R+ , by computing upper and lower Gelfand–Kirillov dimensions [22]: ln γA (n) = inf{α > 0 | γA (n)  nα }; ln n ln γA (n) GKdim A = lim = sup{α > 0 | γA (n)  nα }. ln n n→∞

GKdim A = lim

n→∞

Assume that generators X = {x1 , . . . , xk } are assigned positive weights wt(xi ) = λi , i = 1, . . . , k. Define a weight growth function: γ˜A (n) = dimK xi1 · · · xim | wt(xi1 ) + · · · + wt(xim ) ≤ n, xij ∈ X K ,

n ≥ 0.

Set C1 = min{λi | i = 1, . . . , k}, C2 = max{λi | i = 1, . . . , k}, then γ˜A (C1 n) ≤ γA (n) ≤ γ˜A (C2 n) for n ≥ 1. Thus, we obtain an equivalent growth function γ˜A (n) ∼ γA (n). Therefore, we can use the weight growth function γ˜A (n) in order to compute the Gelfand– Kirillov dimensions. Suppose that L is a Lie (super)algebra and X ⊂ L. By Lie(X) denote the subalgebra of L generated by X, (including application of the quadratic mapping in case char K = 2). Let L be a restricted Lie (super)algebra, by Liep (X) denote a restricted subalgebra of L generated by X. Similarly, assume that X is a subset of an associative algebra A. Write Alg(X) ⊂ A to denote an associative subalgebra (without unit) generated by X. 3. Main results: two examples of Lie superalgebras R, Q and their properties In this section we give two examples of Lie superalgebras and formulate their main properties. Let Λn = Λ(x1 , . . . , xn ) be the Grassmann algebra in n variables. By setting deg xi = 1, i = 1, . . . , n, one has the associative superalgebra Λn = Λ¯0 ⊕ Λ¯1 , called (−) the Grassmann superalgebra. Since the superbracket in Λn is trivial, it is called supercommutative. The relations ∂xi (xj ) = δij , i, j ∈ {1, . . . , n}, define superderivations ∂xi ∈ Der Λn . Any superderivation D ∈ Der Λn can be uniquely written as [23]: D=

n 

Pi ∂xi ,

Pi ∈ Λ n .

i=1

If char K = 0 one obtains a simple finite dimensional Lie superalgebra of Cartan type Wn [23]. For more information on theory of Lie superalgebras we refer the reader to [23,44,2]. Assume that I is a well-ordered set of arbitrary cardinality. Put Z2 = {0, 1}. Let ZI2 = {α : I → Z2 } be a set of functions with finitely many nonzero values. Suppose that α ∈ ZI2 has nonzero values at {i1 , . . . , it } ⊂ I, where i1 < · · · < it , put xα = xi1 xi2 · · · xit and |α| = t. Now {xα | α ∈ ZI2 } is a basis of the Grassmann algebra ΛI = Λ(xi | i ∈ I),

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which is an associative superalgebra ΛI = Λ¯0 ⊕Λ¯1 , all xi , i ∈ I, being odd. Let ∂xi , i ∈ I, denote the superderivatives of ΛI , which are determined by the values ∂xi (xj ) = δij , i, j ∈ I. Consider a space of all formal sums W(ΛI ) =

  α∈ZI2



m(α)

xα

j=1

   λα,ij ∂xij  λα,ij ∈ K, ij ∈ I .

It is essential that the sum at each xα , α ∈ ZI2 , is finite. This construction is similar to so called Lie algebra of special derivations, see [38,39,32]. It is similarly verified that the product of elements of W(ΛI ) is well defined and W(ΛI ) acts on ΛI by superderivations. Our superalgebras will be constructed as subalgebras in W(ΛI ) ⊂ Der ΛI . Our goal is to study the following finitely generated (restricted) Lie superalgebras. We consider Grassmann superalgebras Λ in infinitely many variables. We write our elements as infinite sums, they belong to the Lie superalgebra W(Λ) ⊂ Der Λ. We consider that the algebra Λ is naturally embedded into End Λ by sending each element to the left multiplication. We also shall consider respective finitely generated associative subalgebras of End Λ. Example 1. Let Λ = Λ[xi , yi | i ≥ 0] and consider the following odd elements in W(Λ): ai = ∂xi + yi xi (∂xi+1 + yi+1 xi+1 (∂xi+2 + yi+2 xi+2 (∂xi+3 + · · · ))), bi = ∂yi + xi yi (∂yi+1 + xi+1 yi+1 (∂yi+2 + xi+2 yi+2 (∂yi+3 + · · · ))),

i ≥ 0.

(3)

We define a Lie superalgebra R = Lie(a0 , b0 ) ⊂ W(Λ) and its associative hull A = Alg(a0 , b0 ) ⊂ End Λ. In case char K = 2, assume that the quadratic mapping on odd elements is just the square of the respective operator in End Λ. We call {ai , bi | i ≥ 0} the pivot elements. We establish the following main properties of R = Lie(a0 , b0 ) and its associative hull A = Alg(a0 , b0 ). (i) Section 4 yields basic relations of R and shows that R is self-similar. (ii) R has a monomials’ basis consisting of standard monomials of three types (char K = 2, Theorem 5.1). In case char K = 2, a basis of R is given by monomials of the first and second type and squares of the pivot elements (Corollary 5.2). (iii) In case char K = p > 0, we also consider a restricted Lie superalgebra Liep (a0 , b0 ), which in case p = 2 coincides with the restricted Lie algebra also denoted by Liep (a0 , b0 ), the bases coincide with the base of R. (iv) We introduce two weight functions wt, swt, which are additive on products of monomials (Section 6). (v) R is Z2 -graded by multidegree in the generators (Lemma 7.1). We introduce two coordinate systems on plane: multidegree coordinates (X1 , X2 ), and weight coordinates (Z1 , Z2 ) (Section 7).

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(vi) We find bounds on weights of monomials of R and A (Section 8). Monomials of R are in a region of plane bounded by two logarithmic curves (Theorem 8.6, Fig. 1). A similar bound holds for A. (vii) GKdim R = GKdim R = log3 4 ≈ 1.26 (Theorem 8.5). (viii) GKdim A = GKdim A = 2 log3 4 ≈ 2.52 (Theorem 8.8). (ix) We describe generating functions of R (Section 9) and observe a fractal structure of its homogeneous components (Fig. 1). (x) R = R¯0 ⊕R¯1 is a nil graded Lie superalgebra (Theorem 10.1). We have a triangular decomposition into a direct sum of three locally nilpotent subalgebras R = R+ ⊕ R0 ⊕ R− (Corollary 10.2). Example 2. Let Λ = Λ[xi , yi , zi | i ≥ 0] and consider the following odd elements in W(Λ): ai = ∂xi + yi xi (∂xi+1 + yi+1 xi+1 (∂xi+2 + yi+2 xi+2 (∂xi+3 + · · · ))), bi = ∂yi + zi yi (∂yi+1 + zi+1 yi+1 (∂yi+2 + zi+2 yi+2 (∂yi+3 + · · · ))),

i ≥ 0.

(4)

ci = ∂zi + xi zi (∂zi+1 + xi+1 zi+1 (∂zi+2 + xi+2 zi+2 (∂zi+3 + · · · ))), We define a Lie superalgebra Q = Lie(a0 , b0 , c0 ) ⊂ W(Λ) and its associative hull A = Alg(a0 , b0 , c0 ) ⊂ End Λ. In case char K = 2, assume that the quadratic mapping on odd elements is just the square of the respective operator in End Λ. We also call {ai , bi , ci | i ≥ 0} the pivot elements. We establish the following main properties of Q = Lie(a0 , b0 , c0 ) and A = Alg(a0 , b0 , c0 ): (i) Section 11 yields basic relations of Q and shows that Q is self-similar. (ii) Q has a monomials’ basis consisting of standard monomials of three types (char K = 2, Theorem 12.1). In case char K = 2, a basis is given by monomials of the first and second type and squares of the pivot elements (Corollary 12.2). (iii) In case char K = p > 0, we also consider a restricted Lie superalgebra Liep (a0 , b0 , c0 ), which in case p = 2 coincides with the restricted Lie algebra also denoted by Liep (a0 , b0 , c0 ), the bases coincide with the base of Q. (iv) We introduce three weight functions wt, swt, swt, additive on products of monomials (Section 13). (v) Q is Z3 -graded by multidegree in the generators (Lemma 14.1). We introduce three coordinate systems in space: multidegree coordinates (X1 , X2 , X3 ), weight coordinates (Z1 , Z2 , Z3 ) and orthogonal coordinates (Y1 , Y2 , Y3 ) (Section 14). (vi) We establish bounds on weights of monomials of Q and A (Section 15). Monomials of Q are in a region of space bounded by an elliptic paraboloid (Theorem 15.6 and Fig. 2). A similar bound holds for A. A situation when homogeneous components of

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(vii) (viii) (ix)

(x) (xi)

(xii)

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a graded restricted Lie algebra are in a region of space bounded by a paraboloid-like surface was observed before [36]. GKdim Q = GKdim Q = log3 8 ≈ 1.89 (Theorem 15.5). GKdim A = GKdim A = 2 log3 8 (Theorem 15.9). We describe generating functions of Q (Section 16). The initial coefficients show an interesting pattern: the coefficients are at most one and the diagonal coefficients are zero. These observations are true in general, see below. Q = Q¯0 ⊕ Q¯1 is also a nil graded Lie superalgebra (Theorem 16.2). The following properties are specific to Q: The components of the Z3 -grading of Q are at most one-dimensional (Theorem 17.1), so the Z3 -grading of Q is fine. The diagonal components Qnnn , n ≥ 0, are empty (Lemma 17.2). Asymptotically, a density of the standard monomials of Q inside the elliptic paraboloid is zero (Lemma 15.7). Q has a Z2 -grading such that all components Qnm , n, m ∈ Z, are infinite dimensional except for Q00 = {0} (Theorem 18.1), a part of this grading is shown on Fig. 3. There exists a continuum of different decompositions into a direct sum of two locally nilpotent subalgebras: Q = Q+ ⊕ Q− (Theorem 18.2). Q shows that an extension of Theorem 1.1 (Martinez and Zelmanov [27]) for Lie superalgebras of characteristic zero is not valid.

4. Example 1: Lie superalgebra R, relations In this section we start to study the algebras of Example 1. Let Λ = Λ[xi , yi |i ≥ 0] be the Grassmann algebra in infinitely many variables. The Grassmann letters and superderivatives {xi , yi , ∂xi , ∂yi |i ≥ 0} are odd elements of End Λ. As a rule, they anticommute, e.g.: xi xj = −xj xi ,

∂xi ∂xj = −∂xj ∂xi ,

xi yj = −yj xi ;

∂xi xj = −xj ∂xi , i = j; x2i = yi2 = (∂xi )2 = (∂yi )2 = 0,

i ≥ 0.

The nontrivial relations are: ∂xi xi + xi ∂xi = 1,

∂yi yi + yi ∂yi = 1,

i ≥ 0.

For convenience, we present the pivot elements (3) recursively ai = ∂xi + yi xi ai+1 , bi = ∂yi + xi yi bi+1 ;

i ≥ 0.

The pivot elements act on Grassmann variables as follows:

(5)

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an (yk ) = 0, ⎧ ⎪ ⎪ ⎨0, an (xk ) =

241

n, k ≥ 0; n > k; (6)

1, n = k; ⎪ ⎪ ⎩y x y n n n+1 xn+1 · · · yk−1 xk−1 , n < k.

The action of bn is similar. Recall that we consider the Lie superalgebra R = Lie(a0 , b0 ) = R¯0 ⊕ R¯1 ⊂ W(Λ) ⊂ Der Λ and its associative hull, the associative superalgebra A = Alg(a0 , b0 ) ⊂ End Λ. Define the shift mappings τ : Λ → Λ and τ : W(Λ) → W(Λ) by τ (xi ) = xi+1 , τ (yi ) = yi+1 ,

τ (∂xi ) = ∂xi+1 , τ (∂yi ) = ∂yi+1 ,

i ≥ 0.

Consider also the permuting mappings θ : Λ → Λ and θ : W(Λ) → W(Λ): θ(xi ) = yi , θ(yi ) = xi ,

θ(∂xi ) = ∂yi , θ(∂yi ) = ∂xi ,

i ≥ 0.

Clearly, τ is an endomorphism and θ an isomorphism, θ2 = 1. Lemma 4.1. We have a sequence of even elements: ci+1 = [ai , bi ] = [bi , ai ] = xi ai+1 + yi bi+1 ∈ R¯0 ,

i ≥ 0.

(7)

Remark that by our construction we have: τ (ai ) = ai+1 , τ (bi ) = bi+1 , τ (ci ) = ci+1 ,

θ(ai ) = bi , θ(bi ) = ai , θ(ci ) = ci ,

i ≥ 0.

Below by degree of relations we mean the degree in the set {ai , bi }, i being fixed. Thus, relation (7) has degree 2. We group relations with those obtained by the action of θ. Lemma 4.2. We have the following relations of degree 2 and 3, i ≥ 0: (1) (2) (3) (4) (5) (6) (7)

[a2i , bi ] = [ai , ai , bi ] = [ai , ci+1 ] = ai+1 ; [b2i , ai ] = [bi , bi , ai ] = [bi , ci+1 ] = bi+1 ; a2i = −yi ai+1 ; b2i = −xi bi+1 ; [ai , ai ] = 2a2i = −2yi ai+1 ; [bi , bi ] = 2b2i = −2xi bi+1 ; [ai , ai , ai ] = [bi , bi , bi ] = 0.

Proof. We use (2), (7), (5) and that x2i = yi2 = 0: [ai , ai , bi ] = [ai , ci+1 ] = [∂xi + yi xi ai+1 , xi ai+1 + yi bi+1 ] = [∂xi , xi ]ai+1 = ai+1 .

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One has the identity (a + b)2 = a2 + [a, b] + b2 , where a, b are odd elements of an associative superalgebra. We get a2i = (∂xi + yi xi ai+1 )2 = [∂xi , yi xi ai+1 ] = −yi [∂xi , xi ]ai+1 = −yi ai+1 .

2

The notion of self-similarity plays important role in group theory [18]. The Fibonacci Lie algebra is self-similar [33]. A general definition was given by Bartholdi [6]. A Lie algebra L is called self-similar if it affords a homomorphism ψ : L → Der R  R ⊗ L, where R is a commutative algebra and Der R its Lie algebra of derivations. This definition easily extends to superalgebras, we consider R to be a supercommutative associative superalgebra. Consider Lie superalgebras generated by two elements: Li = Lie(ai , bi ), i ≥ 0, so L0 = R. Corollary 4.3. We have (1) (2) (3) (4) (5)

ai , bi ∈ R, i ≥ 0; τ i : R → Li is an isomorphism for any i ≥ 0; R is infinite dimensional; θ : R → R is an automorphism; there exists a self-similarity embedding: R → ∂x0 , ∂y0 K  Λ[x0 , y0 ] ⊗ τ (R).

Lemma 4.4. We have the following relations of degree 4, i ≥ 0: (1) (2) (3) (4) (5)

[bi , ai , ai , bi ] = [bi , ai+1 ] = xi yi ci+2 ; [ai , bi , bi , ai ] = [ai , bi+1 ] = −xi yi ci+2 ; [ai , ai , ai , bi ] = [ai , ai+1 ] = 2xi yi yi+1 ai+2 ; [bi , bi , bi , ai ] = [bi , bi+1 ] = −2xi yi xi+1 bi+2 ; [[ai , bi ], [ai , bi ]] = [ci , ci ] = 0.

Proof. Recall that by Lemma 4.2, [ai , ai , bi ] = ai+1 , [bi , bi , ai ] = bi+1 . Let us check claims (1), (3). [bi , ai+1 ] = [∂yi + xi yi bi+1 , ai+1 ] = xi yi [bi+1 , ai+1 ] = xi yi ci+2 ; [ai , ai+1 ] = [∂xi + yi xi ai+1 , ai+1 ] = yi xi [ai+1 , ai+1 ] = 2xi yi yi+1 ai+2 . Relations (2), (4) follow by application of θ. For the last relation recall that ci ∈ R¯0 .

2

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˜ 5. Monomial bases of R and A Let us introduce a notation widely used below. By rn denote a tail monomial: rn = xξ00 y0η0 · · · xξnn ynηn ∈ Λ,

ξi , ηi ∈ {0, 1}; n ≥ 0.

(8)

For n < 0 we assume that rn = 1. If needed, other monomials of type (8) will be denoted by rn , r˜n , etc. Theorem 5.1. Let char K = 2. A basis of the Lie superalgebra R = Lie(a0 , b0 ) is given by the following standard monomials of three types (where rn denote all tail monomials (8)) (1) monomials of the first type: {rn−2 an , rn−2 bn | n ≥ 0}; (2) monomials of the second type: {rn−2 cn | n ≥ 1}; (3) monomials of the third type: {rn−2 yn−1 an , rn−2 xn−1 bn | n ≥ 1} (in case n = 1 we have {y0 a1 , x0 b1 }). Proof. Let us call n the length, an , bn , cn the heads, rn−2 the tails, and the remaining products the necks of the respective monomials. A) We check by induction on length n that all these monomials belong to R. By Lemma 4.1 and Lemma 4.2, we have {a0 , b0 , c1 , a1 , b1 } ⊂ R. In case char K = 2 by Lemma 4.2, we get {y0 a1 , x0 b1 } ⊂ R. So, we get the desired monomials of all three types of lengths n = 0, 1. This is the base of induction. An integer n ≥ 1 being fixed, assume that all monomials of the first type of lengths i = 0, . . . , n belong to R. Take a monomial w = rn−2 an ∈ R. Using Lemma 4.4 and Lemma 4.2, R contains the following elements w = [bn−1 , rn−2 an ] = ±rn−2 [bn−1 , an ] = ±rn−2 xn−1 yn−1 cn+1 , [an , w ] = ±rn−2 xn−1 yn−1 [an , cn+1 ] = ±rn−2 xn−1 yn−1 an+1 , [bn , w ] = ±rn−2 xn−1 yn−1 [bn , cn+1 ] = ±rn−2 xn−1 yn−1 bn+1 . Now we multiply all the elements above by an−1 or bn−1 and delete the letters xn−1 or yn−1 , or both. Thus, we can obtain all desired monomials of the first and second type of length n + 1.

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244

Assume that char K = 2. Take an arbitrary w = rn−2 an ∈ R, using Lemma 4.4, we get w = [an−1 , rn−2 an ] = ±2rn−2 xn−1 yn−1 yn an+1 . We multiply this element by an−1 or bn−1 and delete letters xn−1 or yn−1 , or both. So, we obtain all monomials of the third type of length n + 1. B) Now let us prove that a product of the standard monomials is expressed via the standard monomials. 1) Consider products of monomials of the same length. Until the end of the proof, we use a somewhat different notation, we write monomials of the first and third type in the same way: rn−1 an , rn−1 bn , and present monomials of the second type as cn =  rn−1 an + rn−1 bn . Thus, we get products of the respective heads [an , bn ], [an , an ], [bn , bn ], given by Lemma 4.1 and Lemma 4.2. We obtain standard monomials:    [rn−1 an , rn−1 bn ] = ±rn−1 [an , bn ] = ±rn−1 cn+1 ,    [rn−1 an , rn−1 an ] = ±rn−1 [an , an ] = ±2rn−1 yn an+1 ,  [rn−1 bn , rn−1 bn ]

=

 ±rn−1 [bn , bn ]

=

(9)

 ±2rn−1 xn bn+1 .

2) Consider a product of two standard monomials of different lengths n < m. We present a monomial of shorter length n as follows:  rn−1 an = rn−1 ∂xn + yn xn (∂xn+1 + yn+1 xn+1 (· · · + ym−2 xm−2 (∂xm−1 + ym−1 xm−1 am ) · · · )) =

m−1 

(10)

r¯j−1 ∂xj + r¯m−2 ym−1 xm−1 am .

j=n

We present rn−1 bn similarly. Using cn = xn−1 an + yn−1 bn , we present monomials of the second type as a respective linear combination. We multiply a monomial (10) by a standard monomial of a bigger length m of the first or third type, and proceed similar to (9):  [rn−1 an , rm−1 bm ] =

m−1 

  r¯j−1 ∂xj (rm−1 )bm ± r¯m−2 ym−1 xm−1 rm−1 [am , bm ]

j=n

=

m−2 

   r¯j−1 ∂xj (rm−1 )bm + r¯m−2 ∂xm−1 (rm−1 )bm ± rm−1 cm+1 .

j=n

(11)

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The derivatives ∂xj , j = n, . . . , m − 2 do not change a possible neck in position m − 1  of rm−1 , the next derivative ∂xm−1 can only delete this neck. Thus, (11) yields standard monomials. Consider the case that the longest monomial has the same head  letter: rm−1 am . The arguments remain valid, where the last term in (11) has the factor [am , am ] = −2ym am+1 . Consider the case that a standard monomial of a bigger length m is of the second type. We use presentation (10):  [rn−1 an , rm−2 cm ]

=

m−2 

  r¯j−1 ∂xj (rm−2 )cm ± r¯m−2 rm−2 ∂xm−1 (cm )

j=n  ± r¯m−2 rm−2 [ym−1 xm−1 am , xm−1 am + ym−1 bm ] 

m−2  (j)  ±rm−2 cm ± rm−2 am . =

2

j=n

Corollary 5.2. Let char K = 2. A basis of the Lie superalgebra R = Lie(a0 , b0 ) as well as a basis of the restricted Lie algebra Liep (a0 , b0 ) is given by the standard monomials of the first and second type and squares of the pivot elements: {yn−1 an , xn−1 bn | n ≥ 1}. Proof. Let L = L¯0 ⊕ L¯1 ⊂ W(Λ) be the subalgebra generated by a0 , b0 using only the commutator bracket. By proof of Theorem, its basis {vj | j ∈ J} consists of all monomials of the first and second types. In order to obtain the restricted Lie algebra [pn ] Liep (a0 , b0 ) = Liep (L) it is sufficient to add all powers: {vj | n ≥ 1, j ∈ J}. These powers are trivial except squares of the pivot elements (see Lemma 4.2) and one more case: c2m = (xm−1 am + ym−1 bm )2 = xm−1 ym−1 [am , bm ] = xm−1 ym−1 cm+1 ,

m ≥ 1,

these are monomials of the second type. Consider the case of the Lie superalgebra Lie(a0 , b0 ). We need to add the squares of a basis of the odd component L¯1 . Again, we add the same squares of the pivot elements. 2 Corollary 5.3. Let char K = p ≥ 3. The restricted Lie superalgebra Liep (a0 , b0 ) coincides with the Lie superalgebra R = Lie(a0 , b0 ), both have the same bases. Proof. We need to add pk -powers of a basis of the even component of the Lie superalgebra, a check shows that we add nothing. 2 ˜ ⊃ R spanned by all monomials that We consider a little bigger Lie superalgebra R contain at most one variable xn−1 or yn−1 , the head letter being an or bn . Namely, consider

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246

ξ

η

ξ

η

n−1 n−1 n−1 n−1 {rn−2 xn−1 yn−1 an , rn−2 xn−1 yn−1 bn | ξn−1 , ηn−1 ∈ {0, 1}, ξn−1 + ηn−1 ≤ 1, n ≥ 0},

(12) where rn are tail monomials. We refer to these elements as quasi-standard monomials of length n. Corollary 5.4. The Lie superalgebra R = Lie(a0 , b0 ) is contained in a bigger (restricted) ˜ ⊂ W(Λ) which basis is given by quasi-standard monomials (12). Lie superalgebra R Proof. Computations of Theorem show that a product of quasi-standard monomials is expressed via quasi-standard monomials. 2 We consider an associative subalgebra generated by quasi-standard monomials. ˜ ⊂ End(Λ). Then Theorem 5.5. Let A˜ = Alg(R) (1) a basis of A˜ is given by monomials ξ

η

n−1 n−1 αn βn 0 β0 {rn−2 xn−1 yn−1 an bn · · · aα 0 b0 | ξn−1 , ηn−1 , αi , βi ∈ {0, 1},

ξn−1 + ηn−1 ≤ αn + βn > 0, n ≥ 0}, where rn−2 are tail monomials (8). (2) A˜ contains A = Alg(a0 , b0 ); (3) monomials with ξn−1 = ηn−1 = 0 are linearly independent and belong to A. Proof. The desired products having at most one Grassmann variable in position n − 1 are easily obtained as products of quasi-standard monomials. Consider the described product having both Grassmann variables in position n − 1. Then αn = βn = 1. We have rn−2 xn−1 an · yn−1 bn = −rn−2 xn−1 yn−1 an bn . Thus, the described monomials belong to A˜ as well. The linear independence follows by the same arguments as [33, proof of Theorem 4.1]. Let us check that A˜ is spanned by these monomials. First, we work with products of quasi-standard monomials, we reorder such products using PBW-like arguments, where a total order is fixed obeying to the length n of quasi-standard monomials. In this process, not only powers of odd quasi-standard monomials but also products of quasi-standard monomials of the same length with the same head letter disappear:    (rn−1 an ) · (rn−1 an ) = ±rn−1 a2n = ±rn−1 yn an+1 .

As a result, we get products of quasi-standard monomials, each given head aj , bj appearing at most once, written in the length-decreasing order. For example, we obtain:   rn−1 an · rn−1 bn · rn−2 an−1 · rn−2 bn−1 · · · a0 · b0 ,

(13)

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where there is at least one quasi-standard monomial with the head letter an or bn , while the remaining quasi-standard monomials of lengths n − 1, . . . , 0 are optional. We observe the following senior part property of the product (13) implied by construction of quasi-standard monomials. The product containing both senior head letters an , bn can contain both smaller variables xn−1 , yn−1 . But the product with exactly one of the senior head letters an , bn can contain at most one of {xn−1 , yn−1 }. Second, we move all Grassmann letters in (13) to the left. Let xi (or yi ) be a Grassmann variable in a quasi-standard monomial w = rj−1 aj , then i < j. The quasi-standard monomials before w in (13) have heads ak , bk with k ≥ j > i. Hence, all Grassmann variables xi , yi , i = 0, . . . , n − 1 appearing in (13) super-commute with the preceding heads (see (6)), and the migration of Grassmann variables is a supercommutation only without additional commutators. In particular, the process does not change the senior letters an, bn and existing variables xn−1 , yn−1 . Therefore, the senior part property remains valid, yielding the required monomials of the theorem. The second claim is trivial. By Theorem 5.1, we have rn−2 an , rn−2 bn ∈ R ⊂ A. It follows that monomials specified in the third claim belong to A as well. 2 6. Two weight functions on R Assume that the superderivatives and variables have the weights as follows: wt(∂xi ) = − wt(xi ) = αi ∈ C,

i ≥ 0.

wt(∂yi ) = − wt(yi ) = βi ∈ C,

We want all terms in (5) be homogeneous. Thus, we get equalities: αi = −αi − βi + αi+1 ,

i ≥ 0.

βi = −αi − βi + βi+1 , Thus, we get a recurrence relation:



 αi+1 αi =A , βi+1 βi

i ≥ 0,

where A =

Lemma 6.1. The matrix A has the following properties (1) the eigenvalues are λ = 3,

μ = 1;

(2) the respective eigenvectors are

 1 v1 = , 1

v2 =

1 −1

 ;

2 1 1 2

 .

(14)

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(3) An =

1 2

3n + 1 3n − 1

3n − 1 3n + 1

 n ≥ 0.

,

Consider a set of sequences W = {(αi , βi | αi , βi ∈ C, i ≥ 0, satisfy (14))}. Clearly, this space is two-dimensional. Let us choose a convenient basis in this space. Namely, according to the eigenvalues and eigenvectors we introduce two weight functions. Corollary 6.2. We have the following weight functions: (1) wt(∂xn ) = wt(∂yn ) = wt(an ) = wt(bn ) = 3n , n ≥ 0 (the weight function); (2) swt(∂xn ) = swt(an ) = 1, swt(∂yn ) = swt(bn ) = −1, n ≥ 0 (the superweight function); (3) Wt(a) = (wt(a), swt(a)) ( vector weight function, here a is one of an , bn ).

Proof. For example, we start with

αn βn

α0 β0



 1 = v1 = and obtain 1



= An v1 = 3n v1 =

3n 3n

 n ≥ 0.

,

2

By monomials we call any monomials in letters {xi , yi , ∂xi , ∂yi , ai , bi , ci | i ≥ 0}, belonging to the associative algebra End Λ. Lemma 6.3. Let a, b be any monomials. Then (1) the weight functions are well defined on monomials; (2) the weight functions are additive on products of monomials, e.g.: Wt(a · b) = Wt(a) + Wt(b). Proof. It follows by our construction of weights, see also [33] and [36]. 2 Corollary 6.4. The vector weight function has the values: Wt(an ) = (3n , 1), Wt(bn ) = (3n , −1),

n ≥ 0; n ≥ 0;

Wt(cn ) = (2 · 3n−1 , 0),

(15)

n ≥ 1.

Proof. Follows by definition (Corollary 6.2) and observation that Wt(cn ) = Wt([an−1 , bn−1 ]) = Wt(an−1 ) + Wt(bn−1 ). 2

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7. Z 2 -gradation of R and two coordinate systems Lemma 7.1. The algebras R = Lie(a0 , b0 ) and A = Alg(a0 , b0 ) are Z2 -graded by multidegree in {a0 , b0 }: A=

⊕

n1 ,n2 ≥0

An 1 n 2 ,

R=

⊕

n1 ,n2 ≥0

Rn1 n2 .

Proof. By (15), we have two vectors Wt(a0 ) = (1, 1), Wt(b0 ) = (1, −1) ∈ R2 .

(16)

They are linearly independent and form a weight lattice: Γ = Z Wt(a0 ) ⊕ Z Wt(b0 ) ⊂ R2 . Let An1 n2 ⊂ A be a span of all monomials of degrees n1 , n2 in the generators a0 , b0 , respectively, where n1 , n2 ≥ 0. By Lemma 6.3, for any v ∈ An1 n2 we have Wt(v) = n1 Wt(a0 ) + n2 Wt(b0 ).

(17)

Since vectors (16) are linearly independent we conclude that elements belonging to different components An1 n2 are linearly independent. 2 Let v ∈ An,m , then swt(v) = n − m = 0 iff n = m. We collect such monomials and call ⊕n≥0 Ann a diagonal of the Z2 -gradation of A. Corollary 7.2. (1) The algebras R, A, and u = u(R) allow triangular decomposition into direct sums of three subalgebras: R = R+ ⊕ R0 ⊕ R− ,

A = A+ ⊕ A0 ⊕ A− ,

u = u+ ⊕ u0 ⊕ u− ,

where the respective components are spanned by monomials of positive, zero, and negative superweight. (2) The zero component coincides with the diagonal: R0 = ⊕n≥0 Rnn . Proof. Follows by additivity of the superweight function swt(v).

2

Consider a monomial 0 = v ∈ An1 n2 , n1 , n2 ≥ 0, we introduce a multidegree vector and put it on plane identifying with standard coordinates (X1 , X2 ) ∈ R2 . Denote Gr(v) = (n1 , n2 ) ∈ Z2 ⊂ R2 .

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From (16), (17) we get the following relations with the weight functions: wt(v) = n1 + n2 ;

(18)

swt(v) = n1 − n2 .

Now let (X1 , X2 ) be the standard coordinates of an arbitrary point in R2 , which we also refer to as the multidegree coordinates. We introduce weight coordinates (Z1 , Z2 ) in R2 :  Z1 = X1 + X2 ; (19) Z2 = X1 − X2 ;  X1 = (Z1 + Z2 )/2; (20) X2 = (Z1 − Z2 )/2. Lemma 7.3. Let v ∈ A be a monomial, Gr(v) = (X1 , X2 ) its multidegree, and (Z1 , Z2 ) its weight coordinates introduced above. The latter coincide with the vector weight function: (Z1 , Z2 ) = Wt(v) = (wt(v), swt(v)). Proof. Follows from (18) and (19). 2 Lemma 7.4. The multidegrees of the pivot elements an , bn and cn are as follows: (1) 1 n (3 + 1, 3n − 1), 2 1 Gr(bn ) = (3n − 1, 3n + 1), 2

Gr(an ) =

Gr(cn ) = (3n−1 , 3n−1 ),

n ≥ 0; n ≥ 0;

n ≥ 1.

(2) Sets of points for pivot elements {an |n ≥ 0}, {bn |n ≥ 0}, {cn |n ≥ 0} belong to three lines of plane. The equations in the standard and the weight coordinates are as follows, respectively: an :

X1 − X2 = 1;

Z2 = 1;

bn :

X1 − X2 = −1;

Z2 = −1;

cn :

X1 − X2 = 0;

Z2 = 0.

Proof. The weights of an , bn , cn are known (15), the standard (multidegree) coordinates are obtained by (20). 2

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Throughout what follows, by using Wt(v) or Gr(v), where v ∈ A, we assume that v is homogeneous with respect to the Z2 -gradation of A. 8. Bounds on weights and growth of R The goal of this section is to compute the growth and describe position of points on plane that correspond to homogeneous elements of R and A. Let us find bounds on weights of standard monomials of R. Lemma 8.1. We have the following bounds on weights of the standard monomials of R of three types (applying θ to an we get the head bn ) of length n ≥ 1: 2 · 3n−1 + 1 ≤ wt(rn−2 an ) ≤ 3n ; 3n−1 + 1 ≤ wt(rn−2 cn ) ≤ 2 · 3n−1 ; 3n−1 + 1 ≤ wt(rn−2 yn−1 an ) ≤ 2 · 3n−1 . ηm Proof. Consider a tail rm = xξ00 y0η0 · · · xξmm ym , we get bounds on its weight:

0 ≥ wt(rm ) ≥ −2(1 + · · · + 3m−1 + 3m ) = −2

3m+1 − 1 = −3m+1 + 1, 3−1

m ≥ 0.

(21)

Formally, these inequalities are valid also for m = −1. Using this bound, we evaluate weights of monomials of the three different types as follows 3n ≥ wt(rn−2 an ) ≥ −3n−1 + 1 + 3n = 2 · 3n−1 + 1; 2 · 3n−1 ≥ wt(rn−2 cn ) ≥ −3n−1 + 1 + 2 · 3n−1 = 3n−1 + 1; 2 · 3n−1 = −3n−1 + 3n ≥ wt(rn−2 yn−1 an ) ≥ −3n−1 + 1 − 3n−1 + 3n = 3n−1 + 1.

2

Corollary 8.2. Let w be a quasi-standard monomial. Then wt(w) ≥ 1. Proof. Let w be of length n ≥ 1, the estimate is the same as that for the standard monomials of the third type. For monomials of length n = 0, i.e. a0 , b0 , the bound is trivial. 2 Lemma 8.3. We have estimates on superweights of the standard monomials of R of length n ≥ 1: −(n − 2) ≤ swt(rn−2 an ) ≤ n, −n ≤ swt(rn−2 bn ) ≤ n − 2, −(n − 1) ≤ swt(rn−2 cn ) ≤ n − 1, −(n − 3) ≤ swt(rn−2 yn−1 an ) ≤ n + 1, −n − 1 ≤ swt(rn−2 xn−1 bn ) ≤ n − 3.

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Proof. Recall that swt(an ) = 1, swt(bn ) = −1, swt(cn ) = 0, n ≥ 1. Consider a tail ξ η ηm rm = xξ00 y0η0 · · · xξmm ym , m ≥ 0. Observe that swt(xj j yj j ) ∈ {−1, 0, 1}, j = 0, . . . , m. We get −m − 1 ≤ swt(rm ) ≤ m + 1,

m ≥ −1.

2

Now we give rough general estimates. Lemma 8.4. Let w ∈ R be a standard monomial of length n ≥ 0. Then 3n−1 < wt(w) ≤ 3n ,

| swt(w)| ≤ n + 1.

Proof. The previous lemmas exclude only a0 , b0 , for which the claim is clear. 2 Theorem 8.5. Let R = Lie(a0 , b0 ). Then GKdim R = GKdim R = log3 4 ≈ 1.26. Proof. Let a natural number m be fixed and set n = [log3 m]. Consider standard monomiξn−2 ηn−2 als of the first type with the head an and of length exactly n: w = xξ00 y0η0 · · · xn−2 yn−2 an , n where ξi , ηi ∈ {0, 1}. By Lemma 8.4, wt(w) ≤ 3 ≤ m. The number of such monomials yields a lower bound on the growth: γ˜R (m) ≥ 22(n−1) ≥ 4log3 m−2 =

1 log3 4 m . 16

Let a natural number m be fixed, set n = [log3 (m)] + 1, so m < 3n . By Lemma 8.4, every standard monomial of length at least n + 1 has the weight greater than 3n > m. Thus, the number of standard monomials w with wt(w) ≤ m is evaluated by the number of all standard monomials of length at most n. Let j = 1, . . . , n be a length of a standard monomial, there are 22(j−1) different tails rj−2 and 5 different types of monomials (see a list in Lemma 8.3). We get an upper bound on the growth: γ˜R (m) ≤ 2 + 5

n  j=1

22(j−1) < 2 + 5

20 4n 20 ≤ 2 + 4log3 m = 2 + mlog3 4 . 4−1 3 3

2

Theorem 8.6. Let w ∈ R = Lie(a0 , b0 ) be a standard monomial. The respective point of plane is bounded by two logarithmic curves in terms of the weight coordinates Wt(w) = (Z1 , Z2 ): |Z2 | < log3 Z1 + 2. Proof. Assume that w has length n ≥ 0. By Lemma 8.4, we have bounds on weights Z1 = wt(w) > 3n−1 ,

|Z2 | = | swt(w)| ≤ n + 1.

It follows that |Z2 | ≤ n + 1 < log3 Z1 + 2. 2

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˜ Recall that A˜ is spanned by Now we skip to study the associative algebras A ⊂ A. monomials: ξ

η

n−1 n−1 αn βn 0 β0 w = rn−2 xn−1 yn−1 an bn · · · aα 0 b0 ,

(22)

under the restrictions of Theorem 5.5. Such a monomial will be said of length n. Lemma 8.7. Let w be a monomial (22) of A˜ described in Theorem 5.5, n ≥ 0. Then 3n−1 < wt(w) < 3n+1 ,

| swt(w)| ≤ 2n + 1.

Proof. We evaluate weights of different parts of a monomial (22) applying estimate (21) and using additional restrictions specified in Theorem 5.5: −3n−1 < wt(rn−2 ) ≤ 0; ξ

η

α

β

n−1 n−1 αn βn yn−1 an bn ) ≤ 2 · 3n ; 2 · 3n−1 ≤ wt(xn−1 n−1 n−1 n−1 0 β0 bn−1 · · · aα + · · · + 30 ) < 3n . 0 ≤ wt(an−1 0 b0 ) ≤ 2(3

i βi Summing these inequalities, we get the first estimates. Recall that swt(aα i bi ) and ξi ηi swt(xi yi ), i ≥ 0, take values in the set {−1, 0, 1}. This observation yields the second estimate. 2

Theorem 8.8. Let A = Alg(a0 , b0 ). Then GKdim A = GKdim A = 2 log3 4 ≈ 2.52. Proof. Let a natural number m be fixed and set n = [log3 m] − 1. Consider monomials w of type (22) such that ξn−1 = ηn−1 = 0, having the senior part one of three types: an bn , an , or bn . By Theorem 5.5, they are linearly independent and belong to A. By Lemma 8.7, wt(w) < 3n+1 ≤ m. The number of such monomials gives a lower bound on the growth: γ˜A (m) ≥ 3 · 22(n−1)+2n ≥ 3 · 24 log3 m−10 = 3 · 2−10 mlog3 16 . Let m ≥ 1 be fixed, set n = [log3 m] +1, then m < 3n . By the lower bound of Lemma 8.7, every monomial (22) of length at least n + 1 has the weight greater than 3n > m. Thus, the number of monomials w ∈ A˜ of type (22) with wt(w) ≤ m is evaluated by the number of all such monomials of length at most n. These monomials have the senior part aj bj , aj , or bj , where 0 ≤ j ≤ n, and their number is evaluated as: γ˜A (m) ≤ γ˜A˜ (m) ≤ 3

n  j=0

24j < 3

24 log3 m+8 28 24(n+1) ≤ = mlog3 16 . 16 − 1 5 5

2

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Theorem 8.9. Let w be a monomial of A = Alg(a0 , b0 ). The respective point of plane is bounded by two logarithmic curves in terms of the weight coordinates Wt(w) = (Z1 , Z2 ): |Z2 | < 2 log3 Z1 + 3. Proof. Let w be a monomial (22) of length n. We have bounds on weights by Lemma 8.7: |Z2 | = | swt(w)| ≤ 2n + 1.

Z1 = wt(w) > 3n−1 ,

It follows that |Z2 | ≤ 2n + 1 < 2 log3 Z1 + 3.

2

9. Generating functions of R Let A =

⊕

n,m∈Z

Anm be a Z2 -graded algebra, one has an induced Z-gradation: A =

⊕n An , where An =

⊕

m+k=n

Amk . Define respective generating functions:

H(A, t1 , t2 ) =



dim Anm tn1 tm 2 ;

n,m

H(A, t) =



dim An tn = H(A, t, t).

n

Similarly, define generating functions for graded sets and spaces. Lemma 9.1. Let R = Lie(a0 , b0 ), where char K = 2. The generating function H(R, t1 , t2 ) satisfies a functional relation: −1 2 2 H(R, t1 , t2 ) = (1 + t−1 1 )(1 + t2 )H(R, t1 t2 , t1 t2 ) − t1 t2 .

Proof. Denote by T the set of all standard monomials, by Tn the standard monomials of length n, n ≥ 0. Recall that T0 = {a0 , b0 }, T1 = {a1 , b1 , c1 , y0 a1 , x0 b1 }. Using Lemma 4.2, we have H(T0 , t1 , t2 ) = t1 + t2 ; H(T1 , t1 , t2 ) = t1 t2 + t21 + t22 + t21 t2 + t1 t22 . We have H(R, t1 , t2 ) = H(T, t1 , t2 ). Due to the structure of the standard monomials of all three types (see Theorem 5.1) we obtain bijections between sets of monomials: Tn+1 = {1, x0 , y0 , x0 y0 } · τ (Tn ), T \ (T1 ∪ T0 ) = {1, x0 , y0 , x0 y0 } · τ (T \ T0 ). By Lemma 7.4,

n ≥ 1;

(23) (24)

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Gr(x0 ) = − Gr(a0 ) = (−1, 0),

H({x0 }, t1 , t2 ) = t−1 1 ;

Gr(y0 ) = − Gr(b0 ) = (0, −1),

H({y0 }, t1 , t2 ) = t−1 2 ;

255

−1 −1 −1 −1 −1 H({1, x0 , y0 , x0 y0 }, t1 , t2 ) = 1 + t−1 1 + t2 + t1 t2 = (1 + t1 )(1 + t2 ).

Using (24), we compute the generating function H(R, t1 , t2 ) = H(T0 ∪ T1 , t1 , t2 ) + (1 +

t−1 1 )(1

+

t−1 2 )

 · H(R, t1 , t2 ) − H(T0 , t1 , t2 )  

t1 :=t21 t2 , t2 :=t1 t22

= t1 + t2 + t1 t2 + t21 + t22 + t21 t2 + t1 t22 −1 2 2 2 2 + (1 + t−1 1 )(1 + t2 )(H(R, t1 t2 , t1 t2 ) − t1 t2 − t1 t2 ) −1 2 2 = (1 + t−1 1 )(1 + t2 )H(R, t1 t2 , t1 t2 ) − t1 t2 .

2

Corollary 9.2. Let char K = 2, R = Lie(a0 , b0 ), and Tn the set of standard monomials of length n. Then −1 2 2 H(Tn+1 , t1 , t2 ) = (1 + t−1 1 )(1 + t2 )H(Tn , t1 t2 , t1 t2 ),

Proof. Follows by the proof and (23).

n ≥ 1.

2

Now, using computer calculations, we have the following. Corollary 9.3. Let char K = 2, R = Lie(a0 , b0 ). (1) The generating functions are: H(R, t1 , t2 ) = t1 + t2 + t21 + t1 t2 + t22 + t21 t2 + t1 t22 + t31 t2 + t21 t22 + t1 t32 + t41 t2 + 2t31 t22 + 2t21 t32 + t1 t42 + t41 t22 + t31 t32 + t21 t42 + t41 t32 + t31 t42 + t51 t32 + 2t41 t42 + t31 t52 + t51 t42 + t41 t52 + . . . ; H(R, t) = 2t + 3t2 + 2t3 + 3t4 + 6t5 + 3t6 + 2t7 + 4t8 + 2t9 + 3t10 + 6t11 + 3t12 + 6t13 + 12t14 . . . ; (2) The diagonal R0 (the zero component of the triangular decomposition in Corollary 7.2) is nontrivial: 10 H(R0 , t1 , t2 ) = t1 t2 + t21 t22 + t31 t32 + 2t41 t42 + t51 t52 + t61 t62 + 4t71 t72 + t81 t82 + t91 t92 + 2t10 1 t2 11 12 12 13 13 14 14 15 15 16 16 17 17 + 2t11 1 t2 + 2t1 t2 + 2t1 t2 + t1 t2 + t1 t2 + 4t1 t2 + t1 t2 + . . .

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Fig. 1. Z2 -components of the Lie superalgebra R, char K = 2.

10. Graded nillity of R The following result shows that the Lie superalgebra R is an analogue of the Grigorchuk and Gupta–Sidki groups. Namely, we prove that R is nil Z2 -graded. The question whether ad a is nil for non-homogeneous elements a ∈ R is open. Theorem 10.1. Let R = Lie(a0 , b0 ) = R¯0 ⊕ R¯1 . For any a ∈ Rn¯ , n ¯ ∈ {¯0, ¯1}, the operator ad(a) is nilpotent. Proof. Let a ∈ R¯1 , then (ad a)2 = ad(a2 ), where a2 = 12 [a, a] ∈ R¯0 (see (2), in case char K = 2 we use the square mapping a2 , which is added to the axioms of the Lie superalgebra). Thus, the proof is reduced to the case of an even element a ∈ R¯0 . Observe that the even standard monomials contain at least one Grassmann variable (Theorem 5.1). We prove a more general statement. Fix a number N > 0. Let V be a set of all quasi-standard monomials (see (12)) of length at most N , each containing at least one Grassmann letter. We are going to prove that the associative algebra Alg(V ) ⊂ End Λ is nilpotent.

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Let w be an M -fold non-zero product of vi ’s, vi ∈ V . Since by Corollary 8.2 wt(vi ) ≥ 1, we get wt(w) ≥ M . We transform the product as follows. We move to the left and order Grassmann variables, while not changing an order of the heads and doing nothing with their mutual products. We only tackle the following interaction. A Grassmann variable xj (or yj ) can disappear by commuting with an appropriate head ak (or bk ) with k ≤ j. There are two options illustrated by the following examples (see (6)): x3 b5 · y5 a7 = −x3 y5 · b5 a7 + x3 · a7 ;

(25)

x3 b4 · y5 a7 = −x3 y5 · b4 a7 + x3 x4 y4 · a7 .

As a result, the product is written as a linear combination of monomials of the form: u = xi1 · · · xin1 yj1 · · · yjn2 ·   

· · · akl · · · bsl · · · ,   

il , jl < N,

kl , sl ≤ N.

(26)

n different Grassmann letters m heads with original order

Since xil , yjl ∈ {x0 , y0 , . . . , xN −1 , yN −1 } and the product is non-zero, the number of different Grassmann letters above satisfies n ≤ 2N . The number of Grassmann letters in the original product w is greater or equal to the number of heads. This property is kept by transformations (25), hence n ≥ m. We evaluate weight of the resulting monomial (26): M ≤ wt(w) = wt(u) ≤ m wt aN ≤ m3N ≤ n3N ≤ 2N 3N . Therefore, Alg(V )N1 = 0, where N1 = 2N 3N + 1. Now let a ∈ R¯0 be a linear combination of standard monomials of length at most N . By statement above, aN1 = 0. Now standard arguments show that (ad a)2N1 −1 = 0. Theorem is proved. 2 Corollary 10.2. All three components of the triangular decomposition (Corollary 7.2) of the Lie superalgebra R = R+ ⊕ R0 ⊕ R− are locally nilpotent subalgebras. Proof. Recall that the superweights of the pivot elements {ai , bi | i ≥ 0}, are nonzero (Corollary 6.2). So, basis monomials spanning R0 contain at least one variable. By observation made in the proof of Theorem, R0 is locally nilpotent. The homogeneous components of R are bounded by two logarithmic curves (Theorem 8.6, see Fig. 1), and R+ , R− are locally nilpotent by the same geometric arguments as in [33]. 2 11. Example 2: Lie superalgebra Q, relations From now on we skip to Example 2, arguments are similar to those of Example 1. Let Λ = Λ[xi , yi , zi | i ≥ 0] be the Grassmann algebra in infinitely many variables. The Grassmann variables and respective superderivatives {xi , yi , zi , ∂xi , ∂yi , ∂zi | i ≥ 0} are odd elements of End Λ. They anticommute except for nontrivial relations:

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258

∂xi xi + xi ∂xi = 1,

∂yi yi + yi ∂yi = 1,

∂zi zi + zi ∂zi = 1;

x2i = yi2 = zi2 = (∂xi )2 = (∂yi )2 = (∂zi )2 = 0,

i ≥ 0.

We present the pivot elements of Example 2 (4) recursively: ai = ∂xi + yi xi ai+1 , bi = ∂yi + zi yi bi+1 ,

i ≥ 0.

(27)

ci = ∂zi + xi zi ci+1 , Recall that we consider the Lie superalgebra Q = Lie(a0 , b0 , c0 ) ⊂ W(Λ) ⊂ Der Λ and the associative algebra A = Alg(a0 , b0 , c0 ) ⊂ End Λ. Define shift mappings τ : Λ → Λ and τ : W(Λ) → W(Λ) by τ (xi ) = xi+1 , τ (yi ) = yi+1 , τ (zi ) = zi+1 , τ (∂xi ) = ∂xi+1 , τ (∂yi ) = ∂yi+1 , τ (∂zi ) = ∂zi+1 ,

i ≥ 0.

Also, consider permuting mappings θ : Λ → Λ and θ : W(Λ) → W(Λ): θ(xi ) = yi , θ(yi ) = zi , θ(zi ) = xi ,

θ(∂xi ) = ∂yi , θ(∂yi ) = ∂zi , θ(∂zi ) = ∂xi ,

i ≥ 0.

Remark that τ (ai ) = ai+1 , τ (bi ) = bi+1 , τ (ci ) = ci+1 ,

θ(ai ) = bi , θ(bi ) = ci , θ(ci ) = ai ,

i ≥ 0.

Clearly, τ is an endomorphism and θ an isomorphism of order 3. Below by degree of relations we mean the degree in the set {ai , bi , ci }, i being fixed. We group relations with their conjugates under θ, θ2 . Lemma 11.1. We have the following relations of degree 2, i ≥ 0: (1) (2) (3) (4) (5) (6) (7) (8) (9)

[ai , bi ] = xi ai+1 , [bi , ci ] = yi bi+1 , [ci , ai ] = zi ci+1 , a2i = −yi ai+1 , b2i = −zi bi+1 , c2i = −xi ci+1 , [ai , ai ] = 2a2i = −2yi ai+1 ; [bi , bi ] = 2b2i = −2zi bi+1 ; [ci , ci ] = 2c2i = −2xi ci+1 .

Proof. We proceed as in Lemma 4.1 and Lemma 4.2.

2

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Lemma 11.2. All relations of degree 3, i ≥ 0: (1) (2) (3) (4) (5) (6) (7) (8)

[ai , ai , bi ] = [a2i , bi ] = ai+1 , [bi , bi , ci ] = [b2i , ci ] = bi+1 , [ci , ci , ai ] = [c2i , ai ] = ci+1 , [ai , ai , ci ] = [a2i , ci ] = xi yi zi zi+1 ci+2 , [bi , bi , ai ] = [b2i , ai ] = xi yi zi xi+1 ai+2 , [ci , ci , bi ] = [c2i , bi ] = xi yi zi yi+1 bi+2 , [ai , bi , ci ] = [bi , ci , ai ] = [ci , ai , bi ] = 0, [ai , ai , ai ] = [bi , bi , bi ] = [ci , ci , ci ] = 0.

Proof. Let us check claims 1 and 4: [ai , ai , bi ] = [ai , xi ai+1 ] = [∂xi + yi xi ai+1 , xi+1 ai+1 ] = ai+1 ; [ai , ai , ci ] = [ai , zi ci+1 ] = [∂xi + yi xi ai+1 , zi ci+1 ] = −yi xi zi [ai+1 , ci+1 ] = xi yi zi zi+1 ci+2 .

2

Consider Lie superalgebras Li = Lie(ai , bi , ci ), i ≥ 0, so L0 = Q. Corollary 11.3. We have (1) (2) (3) (4) (5)

ai , bi , ci ∈ Q for all i ≥ 0; τ i : Q → Li is an isomorphism for any i ≥ 1; Q is infinite dimensional; θ : Q → Q is an automorphism of order 3; there exists a natural self-similarity embedding: Q → ∂x0 , ∂y0 , ∂z0 K  Λ[x0 , y0 , z0 ] ⊗ τ (Q).

All nonzero relations of degree 3 given by Lemma 11.2 are θ-conjugates of those of items 1) and 4). To get relations of degree 4 we multiply those formulas by ai , bi , ci and consider θ-conjugates. Lemma 11.4. Relations of degree 4 are: (1) (2) (3) (4) (5) (6)

[ai , ai , ai , bi ] = [ai , ai+1 ] = 2xi yi yi+1 ai+2 , [bi , ai , ai , bi ] = [bi , ai+1 ] = −yi zi xi+1 ai+2 , [ci , ai , ai , bi ] = [ci , ai+1 ] = xi zi zi+1 ci+2 , [ai , ai , ai , ci ] = yi zi zi+1 ci+2 , [bi , ai , ai , ci ] = −xi zi zi+1 ci+2 , [ci , ai , ai , ci ] = xi yi zi+1 ci+2 ,

and their θ-conjugates as well, where i ≥ 0.

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Proof. Using Claim 1) of Lemma 11.2, we get: [ai , ai+1 ] = [∂xi + yi xi ai+1 , ai+1 ] = yi xi [ai+1 , ai+1 ] = 2xi yi yi+1 ai+2 ; [bi , ai+1 ] = [∂yi + zi yi bi+1 , ai+1 ] = zi yi [bi+1 , ai+1 ] = −yi zi xi+1 ai+2 ; [ci , ai+1 ] = [∂zi + xi zi ci+1 , ai+1 ] = xi zi [ci+1 , ai+1 ] = xi zi zi+1 ci+2 . Using Claim 4) of Lemma 11.2, we get: [ai , ai , ai , ci ] = [∂xi + yi xi ai+1 , xi yi zi zi+1 ci+2 ] = yi zi zi+1 ci+2 ; [bi , ai , ai , ci ] = [∂yi + zi yi bi+1 , xi yi zi zi+1 ci+2 ] = −xi zi zi+1 ci+2 ; [ci , ai , ai , ci ] = [∂zi + xi zi ci+1 , xi yi zi zi+1 ci+2 ] = xi yi zi+1 ci+2 .

2

˜ 12. Monomial bases of Q and A As above, by rn denote a tail monomial: rn = xξ00 y0η0 z0ζ0 · · · xξnn ynηn znζn ∈ Λ,

ξi , ηi , ζi ∈ {0, 1}; n ≥ 0.

(28)

If n < 0, we consider that rn = 1. Another monomials of type (28) will be denoted by rn , r˜n , etc. Theorem 12.1. Let char K = 2. A basis of the Lie superalgebra Q = Lie(a0 , b0 , c0 ) is given by the following standard monomials of three types (where rn denote tail monomials (28)) (1) monomials of the first type: {rn−2 an , rn−2 bn , rn−2 cn | n ≥ 0}; (2) monomials of the second type: {rn−2 xn−1 an , rn−2 yn−1 bn , rn−2 zn−1 cn | n ≥ 1}; (3) monomials of the third type: β {rn−3 xα n−2 yn−2 yn−1 an , β α rn−3 yn−2 zn−2 zn−1 bn , α rn−3 zn−2 xβn−2 xn−1 cn | α, β ∈ {0, 1}, n ≥ 1}

(in case n = 1 we have {y0 a1 , z0 b1 , x0 c1 }).

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Proof. Let us call n the length, an , bn , cn the heads, rn−2 (or rn−3 ) the tails, and the remaining products the necks of the respective monomials. A) We check by induction on the length n that all these monomials belong to Q. By Lemma 11.2 and Lemma 11.1, we get {a0 , b0 , c0 , a1 , b1 , c1 } ⊂ Q, {x0 a1 , y0 b1 , z0 c1 } ⊂ Q, and {y0 a1 , z0 b1 , x0 c1 } ⊂ Q. Thus, we have monomials of all three types for n = 0, 1. This is the base of induction. Let a number n ≥ 1 be fixed and assume that all monomials of the first type of length i = 0, . . . , n belong to Q. Then we take w = rn−3 an−1 ∈ Q and use Lemma 11.2: w = [bn−1 , bn−1 , w] = rn−3 [bn−1 , bn−1 , an−1 ] = rn−3 xn−1 yn−1 zn−1 xn an+1 .

(29)

Observe that we can delete the letters {xn−1 , yn−1 , zn−1 , xn } in (29) in any combination, for example: [an−1 , w ] = ±rn−3 [∂xn−1 + yn−1 xn−1 an , xn−1 yn−1 zn−1 xn an+1 ] = ±rn−3 yn−1 zn−1 xn an+1 ; [an , w ] = ±rn−3 [∂xn + yn xn an+1 , xn−1 yn−1 zn−1 xn an+1 ] = ±rn−3 xn−1 yn−1 zn−1 an+1 . This argument yields all desired monomials of the first and second type of length n + 1. We start with a monomial of the first type w = rn−2 an ∈ Q and use the first relation of Lemma 11.4: [an−1 , w] = ±rn−2 [an−1 , an ] = ±2rn−2 xn−1 yn−1 yn an+1 . As above, we can delete any of the letters xn−1 , yn−1 , thus getting all the desired monomials of the third type of length n + 1. B) Now let us prove that a product of the standard monomials is expressed via the standard monomials. 1) Consider products of monomials of the same length. Assume that the heads are different letters, e.g. “a” and “c”. We write monomials of all three types uniformly:  v = rm−1 am , and w = rm−1 cm . Then   [v, w] = ±rm−1 rm−1 [am , cm ] = ±rm−1 zm cm+1 ,

we get a monomial of the second type. Assume that the heads are the same letters, e.g. both are “a”. Take two monomials of  the first type v = rm−2 am , and w = rm−2 am . Then   [v, w] = ±rm−2 rm−2 [am , am ] = ±2rm−2 ym am+1 .

Consider a product of monomials of the same length of the second and third type:

(30)

V. Petrogradsky / Journal of Algebra 466 (2016) 229–283

262

  [rm−2 xm−1 am , rm−2 ym−1 am ] = ±rm−2 xm−1 ym−1 [am , am ]  = ±2rm−2 xm−1 ym−1 ym am+1 .

(31)

Similarly, all products of monomials of the same length with the same head yield monomials of the third type. 2) Consider monomials of different lengths. Assume that the heads are different letters, e.g. “a” and “c”. We present monomials of all three types uniformly: v = rn−1 an , and  w = rm−1 cm . Assume that n < m, we have a presentation  rn−1 an = rn−1 ∂xn + yn xn (∂xn+1 + yn+1 xn+1 (· · · + ym−2 xm−2 (∂xm−1 + ym−1 xm−1 am ) · · · )) =

m−1 

(32)

r¯j−1 ∂xj + r¯m−2 ym−1 xm−1 am .

j=n

We use this presentation  [rn−1 an , rm−1 cm ] =

m−1 

  r¯j−1 ∂xj (rm−1 )cm ± r¯m−2 ym−1 xm−1 rm−1 [am , cm ]

j=n

=

m−2 

   r¯j−1 ∂xj (rm−1 )cm + r¯m−2 ∂xm−1 (rm−1 )cm ± rm−1 zm cm+1 .

j=n

(33)  The last term is of the second type. The factors ∂xi (rm−1 ), i = n, . . . , m − 2, keep the  letter xm−1 or zm−1 of the neck (rm−1 can have at most one of those), and do not add new Grassmann letters with index m − 2. Thus, the summation in (33) yields standard  monomials of the same type as rm−1 cm . The middle monomial in (33) is non-zero only in  the case ∂xm−1 (rm−1 ) = 0 which implies that the only letter xm−1 of the neck disappears, we get a monomial of the first type. Now consider monomials with the same head letter of different lengths. Take two  monomials v = rn−1 an , and w = rm−1 am such that n < m. We use presentation (32)

[v, w] =

m−1 

  r¯j−1 ∂xj (rm−1 )am + r¯m−2 [ym−1 xm−1 am , rm−1 am ].

j=n

We treat the summation as above. Consider the commutator of the last term, it is nonzero  only in the case rm−1 am is of the first type, similar to (31), we get a monomial of the third type. 2 The following corollaries are proved as above.

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263

Corollary 12.2. Let char K = 2. A basis of the Lie superalgebra Q = Lie(a0 , b0 , c0 ) as well as a basis of the restricted Lie algebra Liep (a0 , b0 , c0 ) is given by standard monomials of the first and second type and squares of the pivot elements: {yn−1 an , zn−1 bn , xn−1 cn | n ≥ 1}. Corollary 12.3. Let char K = p ≥ 3. The restricted Lie superalgebra Liep (a0 , b0 , c0 ) coincides with the Lie superalgebra Q = Lie(a0 , b0 , c0 ), both have the same bases. Define monomials ξ

η

ζ

n−1 n−1 n−1 {rn−2 xn−1 yn−1 zn−1 an | ξn−1 , ηn−1 , ζn−1 ∈ {0, 1}, ξn−1 + ηn−1 + ζn−1 ≤ 1, n ≥ 0},

(34) and their θ-conjugates. As above, we refer to them as quasi-standard monomials. Corollary 12.4. The algebra Q = Lie(a0 , b0 , c0 ) is contained in a bigger (restricted) Lie ˜ ⊂ W(Λ), which basis is given by quasi-standard monomials (34). superalgebra Q We consider an associative subalgebra generated by quasi-standard monomials. ˜ ⊂ End(Λ). Then Theorem 12.5. Let A˜ = Alg(Q) (1) a basis of A˜ is given by monomials ξ

η

ζ

n−1 n−1 n−1 αn βn γn 0 β0 γ0 {rn−2 xn−1 yn−1 zn−1 an bn cn · · · aα 0 b0 c0 | ξn−1 , ηn−1 , ζn−1 , αi , βi , γi ∈ {0, 1};

ξn−1 + ηn−1 + ζn−1 ≤ αn + βn + γn > 0; n ≥ 0}; (the inequality above we call the senior part property); (2) A˜ contains A = Alg(a0 , b0 , c0 ); (3) monomials with ξn−1 = ηn−1 = ζn−1 = 0 are linearly independent and belong to A. Proof. We proceed as in Theorem 5.5.

2

13. Three weight functions on Q Assume that the superderivatives and Grassmann letters have the weights as follows: wt(∂xi ) = − wt(xi ) = αi ∈ C, wt(∂yi ) = − wt(yi ) = βi ∈ C, wt(∂zi ) = − wt(zi ) = γi ∈ C,

i ≥ 0.

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264

In order to define weight for the elements an , bn , cn , n ≥ 0, we want all terms in (27) be homogeneous. Thus, we get equalities: αi = −αi − βi + αi+1 , βi = −βi − γi + βi+1 , γi = −αi − γi + γi+1 ,

i ≥ 0.

Thus, we get a recurrence relation 

αi+1 βi+1 γi+1



 =A

αi βi γi



 i ≥ 0;

,

where

A=

2 0 1

1 2 0

0 1 2

 .

(35)

Below i ∈ C, i2 = −1, z¯ denotes the complex conjugation of z ∈ C. We shall use the following constants: √ −1 + 3i = e2πi/3 , = 2 √ μ = 2 +  = 3eπi/6 ,

√ −1 − 3i = e−2πi/3 ; ¯ = 2 √ μ ¯ = 2 + ¯ = 3e−πi/6 ,

λ = 3.

In the next lemma we use sequences of numbers:  πn , := 3 + μ + μ ¯ =3 +2 cos 6 

√ πn 2π n n n n n n Θ2 := 3 + μ + ¯ μ + , ¯ = 3 + 2 3 cos 6 3 

√ πn 2π − ; μn = 3n + 2 3n cos Θn3 := 3n + ¯μn + ¯ 6 3 Θn1

n

n

n

√

n

3n

n ≥ 0.

Lemma 13.1. The matrix A (35) has the following properties: (1) λ = 3, μ, μ ¯ are the eigenvalues, where the respective eigenvectors are as follows:   1 v1 = 1 , 1

  1 v2 =  , ¯

  1 v3 = ¯ ; 

(2) the transition matrix C to the basis {v1 , v2 , v3 } and its inverse are:  C=

1 1 1

1 1  ¯ ¯ 

 ,

C −1

1 = 3



1 1 1

1 1 ¯   ¯

 ;

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265

(3)

A

−1

1 = 9



−2 4 1

4 1 −2

1 −2 4

 ;

(4) 1 A = 3 n



Θn1 Θn2 Θn3

Θn3 Θn1 Θn2

Θn2 Θn3 Θn1

 ,

n ≥ 0.

Proof. The first three claims are checked by multiplication of matrices. The last claim is checked by induction. 2 According to three eigenvalues and eigenvectors we introduce three weight functions. Corollary 13.2. For all n ≥ 0 let wt(an ) = wt(bn ) = wt(cn ) = 3n ( a weight function); swt(an ) = μn , swt(bn ) = μn , swt(cn ) = ¯μn (a superweight function); swt(an ) = μ ¯n , swt(bn ) = ¯ μ ¯n , swt(cn ) = ¯ μn (a conjugated superweight function); Wt(a) = (wt(a), swt(a), swt(a)) (a vector weight function, were a is one of an , bn , cn ); (5) the space of weight functions is three-dimensional and {wt, swt, swt} is its basis. (1) (2) (3) (4)

Proof. We proceed as in the proof of Corollary 6.2. 2 By monomials we call any monomials in letters {xi , yi , zi , ∂xi , ∂yi , ∂zi , ai , bi , ci | i ≥ 0}, belonging to End Λ. Lemma 13.3. Let a, b be any monomials. Then (1) the weight functions are well defined on monomials; (2) the weight functions are additive on products of monomials, e.g.: Wt(a · b) = Wt(a) + Wt(b). (3) swt(a) = swt(a) is the complex conjugation. Proof. It follows by our construction of weights, see also [33] and [36]. 2 Corollary 13.4. The vector weight function has the following values on the pivot elements:

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Wt(an ) = (3n , μn , μ ¯n ), Wt(bn ) = (3n , μn , ¯ μ ¯n ), n ≥ 0.

Wt(cn ) = (3n , ¯μn , ¯ μn ), 14. Z3 -gradation of Q and three coordinate systems

Lemma 14.1. The Lie superalgebra Q = Lie(a0 , b0 , c0 ) and its associative hull A = Alg(a0 , b0 , c0 ) are Z3 -graded by multidegree in the generators {a0 , b0 , c0 }: A=

⊕

n1 ,n2 ,n3 ≥0

An1 ,n2 ,n3 ,

Q=

⊕

n1 ,n2 ,n3 ≥0

Qn1 ,n2 ,n3 .

Proof. By Corollary 13.4, we get three vectors Wt(a0 ) = (1, 1, 1), Wt(b0 ) = (1, , ¯), Wt(c0 ) = (1, ¯, ) ∈ C3 .

(36)

Let An1 ,n2 ,n3 be spanned by all monomials of A of degrees n1 , n2 , n3 in the generators a0 , b0 , c0 , respectively, where n1 , n2 , n3 ≥ 0. For any v ∈ An1 ,n2 ,n3 we apply Lemma 13.3 Wt(v) = n1 Wt(a0 ) + n2 Wt(b0 ) + n3 Wt(c0 ).

(37)

Since the vectors (36) are linearly independent we conclude that elements belonging to different components An1 ,n2 ,n3 are linearly independent. 2 We put relations (36) and (37) in the matrix form:  T

Wt (v) =

wt(v) swt(v) swt(v)



 =

1 1 1 1   1  



n1 n2 n3

 .

(38)

Consider a monomial 0 = v ∈ An1 ,n2 ,n3 , n1 , n2 , n3 ≥ 0, we introduce a multidegree vector and put it in space identifying with the standard coordinates (X1 , X2 , X3 ) ∈ R3 . Gr(v) := (n1 , n2 , n3 ) ∈ Z3 ⊂ R3 ⊂ C3 (we warn that in [33] the function Gr(∗) was denoted as Wt(∗)). Now let (X1 , X2 , X3 ) be the standard coordinates of an arbitrary point in C3 , which we also refer to as the multidegree coordinates. We introduce the weight coordinates (Z1 , Z2 , Z3 ) in C3 by the relations: 

Z1 Z2 Z3



 =

1 1 1

1  

1  



X1 X2 X3

We use the inverse matrix for C given in Lemma 13.1:

 .

(39)

V. Petrogradsky / Journal of Algebra 466 (2016) 229–283



X1 X2 X3



1 = 3



1 1 1

1  

1  



Z1 Z2 Z3

267

 .

(40)

Lemma 14.2. Let v ∈ A be a monomial, Gr(v) = (X1 , X2 , X3 ) its multidegree, and (Z1 , Z2 , Z3 ) its weight coordinates introduced above. They coincide with the vector weight function: (Z1 , Z2 , Z3 ) = Wt(v) = (wt(v), swt(v), swt(v)). Proof. Follows from (38) and (39).

2

Let (X1 , X2 , X3 ) be the standard coordinates in R3 ⊂ C3 . Using (39), we introduce real orthogonal coordinates as follows: Z1 X1 + X2 + X3 √ Y1 = √ = ; 3 3 √ 2 2X1 − X2 − X3 √ Y2 = √ Re(Z2 ) = ; 3 6 √ X2 − X3 2 √ √ Y3 = Im(Z2 ) = . 3 2

(41)

The transition matrix (41) is orthogonal, thus the inverse transition is given by the transpose matrix:  ⎞    ⎛ √ 1/√3 2/3 0√ X1 Y1 √ X2 = ⎝ 1/√3 −1/√6 1/ √2 ⎠ Y2 . (42) X3 Y3 1/ 3 −1/ 6 −1/ 2 For any monomial v ∈ A, we refer to these coordinates as the orthogonal coordinates Ort(v) = (Y1 , Y2 , Y3 ). Two last equations (41) imply that

 2 X12 + X22 + X32 − X1 X2 − X2 X3 − X1 X3 . (43) Y22 + Y32 = 3 Thus, given a point of space P = (X1 , X2 , X3 ) ∈ R3 , we have three different coordinates related by (39), (40), (41), and (42), which we write Gr(P ) = (X1 , X2 , X3 ), Ort(P ) = (Y1 , Y2 , Y3 ), and Wt(P ) = (Z1 , Z2 , Z3 ) ∈ C3 . Lemma 14.3. Using notations of Lemma 13.1, the multidegrees of the pivot elements an , bn , cn are as follows: 1 n n n (Θ , Θ , Θ ), 3 1 3 2 1 Gr(bn ) = (Θn2 , Θn1 , Θn3 ), 3 1 n n n Gr(cn ) = (Θ3 , Θ2 , Θ1 ), 3

Gr(an ) =

n ≥ 0.

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Proof. Any initial data (α0 , β0 , γ0 ) ∈ C3 yields a weight function given by the recurrence relation (35). In particular, by setting α0 = 1, β0 = γ0 = 0, we get a weight function 

αn βn γn



  1 0 , =A 0 n

n ≥ 0.

(44)

By additivity, this tuple yields the degrees of an , bn , cn in the letter a0 . On the other hand, this is the first column of An . By similar arguments, we conclude that the rows of An yield the multidegrees of the elements an , bn , cn in {a0 , b0 , c0 }. 2 Lemma 14.4. (1) The orthogonal coordinates of the pivot elements an , bn , cn , n ≥ 0, are:

 πn √  πn  √ 1 ; Ort(an ) = √ 3n , 2·3n cos , 2·3n sin 6 6 3

 πn 2π √  πn 2π  √ 1 n n n √ 3 , 2·3 cos Ort(bn ) = + + ; , 2·3 sin 6 3 6 3 3

 πn 2π √  πn 2π  √ 1 − − . Ort(cn ) = √ 3n , 2·3n cos , 2·3n sin 6 3 6 3 3 (2) The respective points of space for an , bn , cn , n ≥ 0, belong to an elliptic paraboloid (a smaller one on Fig. 2), which equation in orthogonal and standard coordinates is as follows: √  3 Y1 = Y22 + Y32 ; 2 X1 + X2 + X3 = X12 + X22 + X32 − X1 X2 − X2 X3 − X1 X3 . Proof. The formulas for the orthogonal coordinates follow from the expression of Y ’s via Z’s given by (41) and values of the weight coordinates (Corollary 13.4). Now the equation of the elliptic paraboloid in orthogonal coordinates is directly checked. The equation in standard coordinates is obtained using (41) and (43). 2 Denote by Π the plane X1 + X2 + X3 = 0 in the standard coordinates of R3 (equivalently, Y1 = 0). Consider a point in space given in orthogonal coordinates Y¯ = (Y1 , Y2 , Y3 ), let prΠ (Y¯ ) := (Y2 , Y3 ) be its orthogonal projection on Π (this projection is parallel to line Y2 = Y3 = 0, or X1 = X2 = X3 using the standard coordinates, see (41)). For convenience, let us identify (Y2 , Y3 ) ∈ Π with Y2 + iY3 ∈ C2 . Now we can describe a geometrical meaning of the superweight function. Lemma 14.5. Let v ∈ A, Ort(v) = (Y1 , Y2 , Y3 ) ∈ R3 . The orthogonal projection on Π coincides with the superweight function up to scalar:

V. Petrogradsky / Journal of Algebra 466 (2016) 229–283

prΠ (v) = (Y2 , Y3 ) = Y2 + iY3 =

269

 2/3 swt(v).

Proof. Recall that swt(v) = Z2 . The result follows by relations between Y ’s and Z’s given by two last formulas (41). 2 We call A0 =

 n

Annn a diagonal of the Z3 -gradation, it is also described as follows.

Lemma 14.6. Let v ∈ AX1 X2 X3 and Ort(v) = (Y1 , Y2 , Y3 ). The following conditions are equivalent: (1) (2) (3) (4)

X1 = X2 = X3 ; swt(v) = 0; prΠ (v) = 0; Y2 = Y3 = 0.

Proof. By Corollary 13.2, we have three vectors on plane swt(a0 ) = 1,

swt(b0 ) =  = e2πi/3 ,

swt(c0 ) = ¯ = e−2πi/3 ,

that yield their linear combination: swt(v) = X1 swt(a0 ) + X2 swt(b0 ) + X3 swt(c0 ) = X1 + X2 e2πi/3 + X3 e−2πi/3 . ¯ i = Xi − X, i = 1, 2, 3, assume that X = X3 . Then Put X = min{X1 , X2 , X3 }, and X the condition swt(v) = 0 implies that ¯1 + X ¯ 2 e2πi/3 + X ¯ 3 e−2πi/3 + X(1 + e2πi/3 + e−2πi/3 ) = X ¯1 + X ¯ 2 e2πi/3 = 0, swt(v) = X ¯1 = X ¯ 2 = 0, and X1 = X2 = X3 . We showed that the first two conditions are hence X equivalent.  Since swt(v) = 3/2(Y2 + iY3 ), these conditions are also equivalent to Y2 = Y3 = 0. 2 We shall need a technical fact describing an action of the endomorphism τ on homogeneous elements. Lemma 14.7. Consider 0 = v ∈ An1 n2 n3 , having standard coordinates Gr(v) = (n1 , n2 , n3 ), n1 , n2 , n3 ≥ 0. Let its image under endomorphism τ have coordinates Gr(τ (v)) = (m1 , m2 , m3 ). Then 

m1 m2 m3



 =

2n1 + n3 n1 + 2n2 n2 + 2n3



 = AT ·

n1 n2 n3

 .

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Proof. By assumption, v is a linear combination of products with n1 , n2 , n3 factors a0 , b0 , c0 , respectively. By the first three relations of Lemma 11.2, Gr(a1 ) = (2, 1, 0), Gr(b1 ) = (0, 2, 1, ), Gr(c1 ) = (1, 0, 2). Since τ (a0 ) = a1 , τ (b0 ) = b1 , τ (c0 ) = c1 , we get Gr(τ (v)) = n1 Gr(a1 ) + n2 Gr(b1 ) + n3 Gr(c1 ) = (2n1 + n3 , n1 + 2n2 , n2 + 2n3 ) = (n1 , n2 , n3 ) · A.

2

Now we describe the action of the endomorphism τ on homogeneous elements in terms of orthogonal coordinates and the projection on Π. Lemma 14.8. Let 0 = v ∈ A, Ort(v) = (Y1 , Y2 , Y3 ), and Ort(τ (v)) = (Y1 , Y2 , Y3 ). Then √ Y1 = 3Y1 and τ acts on projections onto the plane Π as a dilation 3 times and rotation by π/6 around the center: prΠ (τ (v)) =

√

3eπi/6 prΠ (v),

swt(τ (v)) =

√ πi/6 3e swt(v).

Proof. By additivity of the weight functions, it is sufficient to check the action of the endomorphism τ on the generators. Consider e.g. v = a0 , we use Corollary 13.2: Y1 = wt(τ (a0 )) = wt(a1 ) = 3 = 3 wt(a0 ) = 3Y1 ; √ swt(τ (a0 )) = swt(a1 ) = μ = 3eπi/6 swt(a0 ).

2

15. Bounds on weights, growth, and elliptic paraboloid for Q First, let us find bounds on weights of standard monomials. Lemma 15.1. We have bounds on weights of standard monomials of Q = Lie(a0 , b0 , c0 ) (bounds for the heads bn , cn are the same): 3n /2 < wt(rn−2 an ) ≤ 3n ,

n ≥ 0;

3n−1 /2 < wt(rn−2 xn−1 an ) ≤ 2 · 3n−1 ,

n ≥ 1;

β n−1 , 3n−1 · 5/6 < wt(rn−3 xα n−2 yn−2 yn−1 an ) ≤ 2 · 3

n ≥ 1.

Proof. The upper bound of the first inequality is evident. The upper bounds of the remaining inequalities follow from wt(xn−1 an ) = 3n − 3n−1 = 2 · 3n−1 . To prove the lower bounds we use an estimate for the tail (28):

0 ≥ wt(rn−2 ) ≥ −

n−2  j=0

3 · 3j > −

3n 3n−1 =− . 1 − 1/3 2

2

(45)

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271

Corollary 15.2. Let w be a standard monomial of Q = Lie(a0 , b0 , c0 ) of length n ≥ 0. Then 3n /6 < wt(w) ≤ 3n . Corollary 15.3. Let w be a quasi-standard monomial of length n ≥ 1. Then wt(w) ≥ 2. Proof. The lower estimate of Corollary 15.2 applies to quasi-standard monomials as well. In case n ≥ 3 the result follows. The cases n = 1, 2 are checked directly. The lowest values are given by the following examples: wt(x0 a1 ) = 3 − 1 = 2, and wt(x0 y0 z0 x1 a2 ) = 9 − 3 − 3 = 3. 2 Lemma 15.4. Let w be a standard monomial of Q of length n. We have bounds on the superweights: √ | swt(w)| = | swt(w)| < θ 3n , n ≥ 0, where θ has values 1.8, 1.4, and 2, respectively, for three types of monomials. Proof. Let ξj , ηj , ζj ∈ {0, 1}, where j ≥ 0, then by definition of the superweight function (claim 2 of Corollary 13.2): ξ

η

ζ

swt(xj j yj j zj j ) = −μj (ξj + ηj + ζj ). By observing respective regular hexagonal, we get the following possible values for the factor above: {ξj + ηj e2πi/3 + ζj e−2πi/3 | ξj , ηj , ζj ∈ {0, 1}} = {ekπi/3 | k = 0, . . . , 5} ∪ {0}.

(46)

Thus, √    swt(xξj y ηj z ζj ) ≤ |μ|j = ( 3)j , j j j

j = 0, . . . , n − 2.

(47)

Consider standard monomials listed in Lemma 15.1. Using (47), we evaluate superweights of the tails: √ √ n−2  √ ( 3)n−1 3 + 3√ n j | swt(rn−2 )| ≤ ( 3) < √ 3 . = 6 3−1 j=0 Next, we evaluate superweights of the necks and heads for three types of monomials using Corollary 13.2: √ | swt(an )| = |μn | = ( 3)n ; √ | swt(xn−1 an )| = | − μn−1 + μn | = |μn−1 (μ − 1)| = |μn−1 eπi/3 | = ( 3)n−1 ; √ | swt(yn−1 an )| = | − μn−1 + μn | = |μn−1 (μ − )| = |2μn−1 | = 2( 3)n−1 .

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Finally, we get total estimates for our three types of monomials: √ √ √ 3 + 3√ n √ n 9 + 3√ n | swt(rn−2 an )| ≤ 3 + 3 = 3 < 1.8 3n ; 6 6 √ √ √ √ 3 + 3√ n 3 + 3 3√ n 3√ n | swt(rn−2 xn−1 an )| ≤ 3 + 3 = 3 < 1.4 3n ; 6 3 6 √ √ √ √ 3 + 3√ n 2 3√ n 3 + 5 3√ n | swt(rn−2 yn−1 an )| ≤ 3 + 3 = 3 < 2 3n . 6 3 6

2

Theorem 15.5. Let Q = Lie(a0 , b0 , c0 ). Then GKdim Q = GKdim Q = log3 8 ≈ 1.89. Proof. Let a natural number m be fixed, set n = [log3 m]. Consider standard monomials of the first type having the head an , of length exactly n: w = xξ00 y0η0 z0ζ0 · · · ξn−2 ηn−2 ζn−2 yn−2 zn−2 an . By Corollary 15.2, wt(w) ≤ 3n ≤ m. Then number of such monomixn−2 als gives a lower bound on the growth: γ˜Q (m) ≥ 23(n−1) ≥ 8log3 m−2 =

1 log3 8 m . 64

Let a natural number m be fixed, set n = [log3 (6m)], then m < 3n+1 /6. By Corollary 15.2, all standard monomials of length at least n + 1 have weights greater than 3n+1 /6 > m. Thus, a number of the standard monomials w with wt(w) ≤ m is evaluated by a number of all standard monomials of length at most n. We count the number of the standard monomials of the first, second, and third type, of length j = 0, 1, . . . , n. We get an upper bound on the growth:

γ˜Q (m) ≤ 3 1 +

n 

2

3(j−1)

+

j=1

≤ C0 8

log3 (6m)

n 

2

3(j−1)

j=1

≤ C1 m

log3 8

+1+

n 

 2

2+3(j−2)

≤ C0 23n

j=2

.

2

Theorem 15.6. Let standard monomials w ∈ Q = Lie(a0 , b0 , c0 ) be drawn in R3 using the multidegree coordinates Gr(w) = (X1 , X2 , X3 ) ∈ Z3 . The respective points are inside an elliptic paraboloid which equations in standard and orthogonal coordinates Ort(w) = (Y1 , Y2 , Y3 ) are as follows:  1 2 X + X22 + X32 − X1 X2 − X2 X3 − X1 X3 ; 15 1  1  Y1 > √ Y22 + Y32 . 10 3

X1 + X2 + X3 >

Proof. Let w be a standard monomial of length n. We have bounds on weights by Corollary 15.2 and Lemma 15.4 Z1 = wt(w) > σ3n ,

√ |Z2 | = | swt(w)| < θ 3n ,

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Fig. 2. Standard monomials of Q (first type – red, second type – green, third type – blue, pivot elements are marked by arrows) and two elliptic paraboloids cut by plane. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

where the constants σ and θ depend on the type. We apply these bounds and transition formulas (41)

2 Z1 σ  2 σ3n σ |Z2 | Y1 = √ > √ > √ =√ Re (Z2 ) + Im2 (Z2 ) θ 3 3 3 3θ2 3σ 1 = √ (Y22 + Y32 ) = √ (Y22 + Y32 ), 2 2 3θ C 3

C=

2θ2 . 3σ

For the first type σ = 1/2, θ = 1.8 we get C = 4.32; for the second type σ = 1/6, θ = 1.4 and C = 7.84; for the third type, σ = 5/18, θ = 2 and C = 9.6 < 10. Thus, the formula in the orthogonal coordinates is established. Now the formula in the standard coordinates follows by (41) and (43). 2 It seems that the standard monomials are contained inside a smaller paraboloid, such as one shown on Fig. 2. But a proof of this fact requires more heavy computations. The next result shows that, asymptotically, a probability that an integer point inside the elliptic paraboloid corresponds to a standard monomial is zero.

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Lemma 15.7. Fix m > 0 and let g(m) be a number of all points u ∈ Z3 inside the elliptic paraboloid with wt(u) ≤ m. Then lim

m→+∞

γ˜Q (m) = 0. g(m)

Proof. By Theorem 15.6, the standard monomials w ∈ Q with wt(w) ≤ m are represented by integer points (in the standard coordinates) inside a cut of an elliptic paraboloid:   Y1 = C Y22 + Y32 ,

Y1 ≤ m,

where C is a constant. We choose a parallelepiped that fits properly inside the elliptic paraboloid: P = {(Y1 , Y2 , Y3 ) | m/2 + 2 ≤ Y1 ≤ m − 2,   |Y2 | < m/(4C) − 2, |Y3 | < m/(4C) − 2}. Consider a lattice point (X1 , X2 , X3 ) ∈ Z3 , it is a center of a unitary cube with vertices (X1 ± 1/2, X2 ± 1/2, X3 ± 1/2) and volume 1. Consider the cubes that intersect P , then √ they cover P . Since we stepped by 2 which is less than their diagonals 3, the cubes lie inside our cut of the elliptic paraboloid. Therefore, g(m) ≥ Volume(cubes intersecting P ) ≥ Volume(P ) ≈ C1 m2 ,

m → ∞.

On the other hand, by proof of Theorem 15.5, we have an estimate for the nominator: γ˜Q (m) ≤ C2 mlog3 8 . The result follows. 2 ˜ Recall that A˜ is spanned by monomials: Now let us study associative algebras A ⊂ A. ξ

η

ζ

n−1 n−1 n−1 αn βn γn 0 β0 γ0 w = rn−2 xn−1 yn−1 zn−1 an bn cn · · · aα 0 b0 c0 ,

(48)

which satisfy restrictions of Theorem 12.5. Such a monomial is said of length n. Lemma 15.8. Let w be a monomial (48) of A˜ described in Theorem 12.5, n ≥ 0. We have bounds √ 3n+2 3n−1 < wt(w) < , | swt(w)| ≤ 4 3n . 2 2 n Proof. We have an upper bound wt(w) ≤ 3 j=0 3j < 3n+2 /2. By the senor part property of Theorem 12.5, ξ

η

ζ

n−1 n−1 n−1 αn βn γn wt(xn−1 yn−1 zn−1 an bn cn ) ≥ wt(xn−1 an ) = 2 · 3n−1 .

We use (45) and get the lower bound on the weight.

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Similar to (47) we also have √    swt(aαj bβj cγj ) ≤ |μ|j = ( 3)j , j j j

j = 0, . . . , n.

Thus, | swt(w)| ≤ 2

n−1 √

3j +

√

3n <

j=0

√

√  2 √ √ 3n √ + 1 = 3n (2 + 3) < 4 3n . 3−1

2

Theorem 15.9. Let A = Alg(a0 , b0 , c0 ). Then GKdim A = GKdim A = 2 log3 8. Proof. Let a natural number m be fixed and set n = [log3 (2m)] − 2. Consider monomials w of type (48) such that ξn−1 = ηn−1 = ζn−1 = 0. By Theorem 12.5, they are linearly independent and belong to A. By Lemma 15.8, wt(w) < 3n+2 /2 ≤ m. There are 7 types n βn γn of the senior part aα n bn cn , the remaining factors are arbitrary and give a lower bound: γ˜A (m) ≥ 7 · 23(n−1)+3n ≥ 7 · 26 log3 (2m)−21 = C0 mlog3 64 . Let m ≥ 1 be fixed, set n = [log3 (2m)] + 1, then m < 3n /2. By the lower bound of Lemma 15.8, every monomial (48) of length at least n + 1 has weight greater than 3n /2 > m. Thus, the number of monomials w ∈ A˜ of type (48) with wt(w) ≤ m is evaluated by the number of all such monomials of length at most n. There are 7 types of the senior part, the remaining factors for a monomial of length j are arbitrary and give an upper bound: γ˜A (m) ≤ 7

n 

26j < 7

j=0

7 6 log3 (2m)+12 26(n+1) ≤ 2 = C1 mlog3 64 . 64 − 1 63

2

Theorem 15.10. Lattice points in R3 for monomials of A = Alg(a0 , b0 , c0 ) are inside an elliptic paraboloid which equations in standard and orthogonal coordinates are as follows:  1 2 X + X22 + X32 − X1 X2 − X2 X3 − X1 X3 ; 96 1  1  Y1 > √ Y22 + Y32 . 64 3

X1 + X2 + X3 >

Proof. Let w be a monomial (48) of length n. By Lemma 15.8, we have bounds √ Z1 = wt(w) > 3n /6, |Z2 | = | swt(w)| < 4 3n . We apply these bounds and transition formulas (41)

2 Z1 1  1 3n 1 |Z2 | Y1 = √ > √ > √ = √ Re2 (Z2 ) + Im2 (Z2 ) = √ (Y22 + Y32 ). 4 3 6 3 6 3 96 3 64 3 The formula in the standard coordinates follows by (41) and (43). 2

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16. Generating functions, nillity of Q In this section we describe the generating function of Q. The computation of its initial coefficients showed an interesting pattern: the coefficients are at most one and the diagonal coefficients are zero. These observations are true in general, we show below that the components of Z3 -grading of Q are at most one-dimensional (Theorem 17.1), so Z3 -grading of Q is fine. Also, the diagonal components Qnnn , n ≥ 0, are empty (Lemma 17.2). Let A = ⊕n,m,k Anmk be a Z3 -graded algebra, one has a induced Z-gradation: A = ⊕ Anmk . Define respective generating functions: ⊕n An , where Al = n+m+k=l

H(A, t1 , t2 , t3 ) =



k dim Anmk tn1 tm 2 t3 ;

n,m,k

H(A, t) =



dim An tn = H(A, t, t, t).

n

By Tnk denote the set of all standard monomials of length n, n ≥ 0, and type k ∈ {1, 2, 3}, described in Theorem 12.1. Lemma 16.1. Let char K = 2, Q = Lie(a0 , b0 , c0 ), and Tnk the set of standard monomials of length n ≥ 1 and type k ∈ {1, 2, 3}, except T13 . Then (1) we have a bijection: k k 0 β0 γ0 Tn+1 = {xα 0 y0 z0 | α0 , β0 , γ0 ∈ {0, 1}} · τ (Tn );

(2) we have a recurrence relation between generating functions: −1 −1 k k 2 2 2 H(Tn+1 , t1 , t2 , t3 ) = (1 + t−1 1 )(1 + t2 )(1 + t3 )H(Tn , t1 t2 , t2 t3 , t3 t1 ).

Proof. The first claim follows by structure of the standard monomials. To prove the second claim we proceed as in proof of Theorem 9.1 or its Corollary 9.2. 2 Computer calculations yield the following series. H(Q, t1 , t2 , t3 ) = t1 + t2 + t3 + t21 + t1 t2 + t1 t3 + t22 + t2 t3 + t23 + t21 t2 + t21 t3 + t1 t22 + t1 t23 + t22 t3 + t2 t23 + t31 t2 + t31 t3 + t21 t22 + t21 t2 t3 + t21 t23 + t1 t32 + t1 t22 t3 + t1 t2 t23 + t1 t33 + t32 t3 + t22 t23 + t2 t33 + t41 t2 + t31 t22 + t31 t2 t3 + t31 t23 + t21 t32 + t21 t22 t3 + t21 t2 t23 + t21 t33 + t1 t32 t3 + t1 t22 t23 + t1 t2 t33 + t1 t43 + t42 t3 + t32 t23 + t22 t33 + t41 t22 + t41 t2 t3 + t31 t32 + t31 t2 t23 + t31 t33 + t21 t32 t3 + t21 t43 + t1 t42 t3 + t1 t22 t33 + t1 t2 t43 + t42 t23 + t32 t33 + . . . ;

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H(Q, t) = 3(t + 2t2 + 2t3 + 4t4 + 5t5 + 4t6 + 6t7 + 6t8 + 6t9 + 12t10 + 12t11 + 9t12 + 15t13 + 15t14 + 9t15 + 12t16 + 12t17 + 10t18 + 18t19 + 18t20 + 12t21 + 18t22 + 18t23 + 12t24 + 18t25 + 18t26 + 18t27 + 36t28 + 36t29 + 24t30 + 36t31 + · · · ). Theorem 16.2. Let Q = Lie(a0 , b0 , c0 ) = Q¯0 ⊕ Q¯1 . For any a ∈ Qn¯ , n ¯ ∈ {¯0, ¯1}, the operator ad(a) is nilpotent. Proof. The same as that of Theorem 10.1. 2 17. Properties of Z 3 -grading of Q We start with an observation on the Z3 -grading of Q. The computed above coefficients of H(Q, t1 , t2 , t3 ) are 0 and 1. Below, we establish this fact for the whole series. In particular, Z3 -grading of Q is fine, i.e. it cannot be split by taking a bigger grading group (see a definition in [4,10]). Theorem 17.1. The components of the multidegree Z3 -grading of Q = Lie(a0 , b0 , c0 ) are at most one-dimensional. Proof. By way of contradiction assume that u, v are different standard monomials and Gr(u) = Gr(v) = (X1 , X2 , X3 ). Assume that u has length 0. If v has also length zero, then u, v ∈ {a0 , b0 , c0 } and u = v. Let v have a nonzero length, by Corollary 15.3, wt(v) ≥ 2, a contradiction with wt(u) = 1. Thus, we can assume that u, v have nonzero length and our example is minimal, namely, the minimum of lengths of u, v is minimal among such examples. The first claim of Lemma 16.1 yields bijections, we need a weaker observation, by structure of the standard monomials (Theorem 12.1) we have 0 β0 γ0 u = xα u), 0 y0 z0 τ (˜

α

β γ

α0 , β0 , γ0 , α0 , β0 , γ0 ∈ {0, 1},

v = x0 0 y0 0 z0 0 τ (˜ v ),

where u ˜, v˜ are standard monomials, which lengths are equal to lengths of u, v minus one, respectively. u) = (n1 , n2 , n3 ) and Gr(˜ v ) = (m1 , m2 , m3 ). Using Lemma 14.7, we get a Let Gr(˜ relation for the multidegree coordinates:  A · T

We use Lemma 13.1

n1 n2 n3



 −

α0 β0 γ0



 =A · T

m1 m2 m3



 −

α0 β0 γ0



 =

X1 X2 X3

 .

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278



n1 − m1 n2 − m2 n3 − m3



 T −1

= (A )

·

α0 − α0 β0 − β0 γ0 − γ0



1 = 9



4 −2 1

1 −2 4 1 −2 4

  ·

δ1 δ2 δ3

 ,

where δi ∈ {−1, 0, 1} for all i = 1, 2, 3. One directly checks that such nonzero vector (δ1 , δ2 , δ3 ) cannot yield a nonzero integer vector on the left hand side. Therefore, δ1 = δ2 = δ3 = 0, and by the formula above, we have (α0 , β0 , γ0 ) = (α0 , β0 , γ0 ) and (n1 , n2 , n3 ) = (m1 , m2 , m3 ), i.e., Gr(˜ u) = Gr(˜ v ). By minimality of the example, we have u ˜ = v˜. Hence, u = v, this contradiction proves the result. 2 The series above has trivial diagonal coefficients, this is true in general. We show that the diagonal (see Lemma 14.6) Q0 = ⊕n≥0 Qnnn of the Z3 -grading is trivial. Lemma 17.2. Let Q = Lie(a0 , b0 , c0 ). Then dim Qnnn = 0, n ≥ 0. Proof. Easy check shows that the standard monomials of lengths 0, 1 do not belong to the diagonal (the multidegrees of the monomials of length 1 follow from relations of Lemma 11.1, where i = 0). Let w be a standard monomial with w ∈ QN N N , we assume that w has a minimal length among monomials belonging to the diagonal. Then w is of length at least two. By structure of the standard monomials (Theorem 12.1) we have 0 β0 γ0 w = xα ˜ 0 y0 z0 τ (w),

α0 , β0 , γ0 ∈ {0, 1},

where w ˜ is a standard monomial of smaller length. Let Gr(w) ˜ = (n1 , n2 , n3 ) ∈ Z3 . Applying Lemma 14.7,  A · T

n1 n2 n3



 −

α0 β0 γ0



 T

= Gr (w) =

N N N

 ,

α0 , β0 , γ0 ∈ {0, 1}.

We are going to prove that this is impossible provided that some n1 , n2 , n3 are different. Due to the cyclic symmetry assume that n3 = min{n1 , n2 , n3 }. Put m1 = n1 − n3 , m2 = n2 − n3 , m3 = 0. Then m1 , m2 ≥ 0, using (35) we get:  A · T

m1 m2 0



 −

α0 β0 γ0



 =

N N N

where N  = N − 3n3 . Now use the formula for A (35): 2m1 − α0 = N  , m1 + 2m2 − β0 = N  , m 2 − γ0 = N  .

 ,

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Subtracting the third equality from the second one and using that α0, β0 , γ0 ∈ {0, 1}, we get: 0 = |(m1 + 2m2 − β0 ) − (m2 − γ0 )| ≥ m1 + m2 − |β0 − γ0 | ≥ m1 + m2 − 1. Therefore, m1 + m2 ≤ 1. Consider the case: m1 = 1, m2 = 0. A difference of the first and third equalities yields a contradiction: 0 = |(2m1 − α0 ) − (m2 − γ0 )| = |2 − (α0 − γ0 )| ≥ 2 − |α0 − γ0 | ≥ 1. Consider the case: m1 = 0, m2 = 1. A difference of the second and first equalities yields a contradiction: 0 = |(m1 + 2m2 − β0 ) − (2m1 − α0 )| = |2 − (β0 − α0 )| ≥ 2 − |β0 − α0 | ≥ 1. Consider the remaining case: m1 = m2 = 0. Hence, n1 = n2 = n3 , and w ˜ belongs to the diagonal, a contradiction with minimality of w. 2 Observe that, in contrast to the result above, the associative hull A = Alg(a0 , b0 , c0 ) has a nontrivial diagonal, because it contains the following elements κ

κ

κ

δ

δ

δ

n−2 n−2 n−2 n−1 n−1 n−1 A0 ⊃ xκ0 0 y0κ0 z0κ0 · · · xn−2 yn−2 zn−2 an bn cn an−1 bn−1 cn−1

· · · aδ00 bδ00 cδ00 | κj , δj ∈ {0, 1} K . 18. Z 2 -grading of Q We have the multidegree vectors of the generators: Gr(a0 ) = (1, 0, 0), Gr(b0 ) = (0, 1, 0), Gr(c0 ) = (0, 0, 1), they generate a lattice: Γ = Gr(a0 ), Gr(b0 ), Gr(c0 ) Z = {(n1 , n2 , n3 ) | ni ∈ Z} = Z3 ⊂ R3 . Consider a projection of Γ on plane Π (see its definition before Lemma 14.5, where we also identify Π with C). For convenience, instead of the projection, we shall use  swt(P ) = 3/2 prΠ (P ), P ∈ R3 (see Lemma 14.5). We get a lattice generated by images of the generators: swt(a0 ) = 1,

swt(b0 ) =  = e2πi/3 ,

swt(c0 ) = ¯ = e−2πi/3 .

¯ = swt(Γ) = 1, , ¯ Z ⊂ C = Π, isomorphic to Z2 (a basis is given by We obtain a lattice Γ ¯ are naturally represented by points of a triangular any pair of these vectors). Points of Γ ¯ is marked there by z2 c3 , this means that z2 c3 is grid (see Fig. 3). For example, 3 ∈ Γ projected on that point, namely, swt(z2 c3 ) = 3.

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Fig. 3. Components of projection of Q on plane Π: Y1 = 0; Z2 -grading of Q.

By additivity of the function swt(∗), the projection yields a Z2 -gradation of Q and A. We have the following interesting property. Theorem 18.1. The Lie superalgebra Q = Lie(a0 , b0 , c0 ) is Z2 -graded so that dim Qnm =

 0, ∞,

(n, m) = (0, 0); (n, m) = (0, 0).

Proof. Triviality of the zero component follows from Lemma 17.2 and Lemma 14.6. We shall prove that any nonzero component contains infinitely many standard monomials of the second type. Let v be a standard monomial of the second type. (Below we are dealing with standard monomials of the second type only.) By structure of such monomials (Theorem 12.1), all elements 0 β0 γ0 {xα 0 y0 z0 τ (v) | α0 , β0 , γ0 ∈ {0, 1}}

(49)

are again standard monomials of the second type. Consider the induced action of τ on ¯ its image is spanned by images of the generators: lattice Γ, ¯ = swt(τ (a0 )), swt(τ (b0 )), swt(τ (c0 )) Z = swt(a1 ), swt(b1 ), swt(c1 ) Z . τ (Γ) ¯ are marked by thick points. By Lemma 14.8, τ acts on projecOn Fig. 3, points of τ (Γ) √ tions on plane Π as a multiplication by λ = 3eπi/6 .

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As we remarked above (see (46)), 0 β0 γ0 swt({xα 0 y0 z0 | α0 , β0 , γ0 ∈ {0, 1}})

= −{α0 + β0 e2πi/3 + γ0 e−2πi/3 | α0 , β0 , γ0 ∈ {0, 1}} = {ekπi/3 | k = 0, . . . , 5} ∪ {0}. Thus, the first factor in (49) yields a step of length one (or none) on the triangular grid (see Fig. 3). As a base of induction, let us prove that all six points on a circle of radius one have infinite dimensional components. Put w0 = x0 a1 , wn+1 = y0 τ (wn ), n ≥ 0, they belong to the same component. Indeed, τ (x0 a1 ) = x1 a2 and the factor swt(y0 ) = − swt(b0 ) returns back to swt(x0 a1 ). These movements are shown by dashed arrows on the picture. We obtain an infinite dimensional component: swt(wn ) = swt(x0 a1 ), n ≥ 0.  Put wn+1 = x0 y0 τ (wn ), n ≥ 0, yielding another infinite dimensional component:  swt(wn+1 ) = swt(τ (wn )) + swt(x0 ) + swt(y0 ) = swt(x1 a2 ) − swt(a0 ) − swt(b0 ) = swt(b0 ) for all n ≥ 0. Thus, the components of x0 a1 and b0 are infinite dimensional. Applying the automorphisms θ, θ2 , we conclude that the components corresponding to all six lattice points on the unitary circle are infinite dimensional. ¯ ⊂ Γ ¯ are drawn Let us make a geometric observation. Points of the sublattice τ (Γ) ¯ \ τ (Γ) ¯ thick on the picture. Observe the following property: any non-thick point P ∈ Γ has three adjacent thick points S1 , S2 , S3 ∈ τ¯(Γ) with |P − Si | = 1, i = 1, 2, 3, situated in vertices of an equilateral triangle, P being the center of the triangle (see a shaded triangle on the picture as an example). Moreover, let a radius R ≥ 1 be fixed and assume that |P | ≤ R, then at least one vertex satisfies the same condition |Si | ≤ R. Indeed, let {h ∈ C | |h| = R} ∩ {h ∈ C | |h − P | = 1} = K1 ∪ K2 , where K1 = K2 is possible. (If the intersection is empty then all S1 , S2 , S3 will do.) Consider an angle K1 P K2 containing O, geometric observations show that K1 P K2 ≥ 2π/3. (This angle is shown on the picture, where we also have R = 3.) (The extreme case is as follows: R = 1, |P | = 1, then we get the angle 2π/3, one can find this situation on the picture. Otherwise the angle is bigger. For example, if we keep the point |P | = 1 and increase R then the angle increases.) Therefore, the angle contains at least one desired point Si (see the shaded triangle on the picture). ¯ Now we are ready to prove an inductive step. We assume that all lattice points P ∈ Γ with |P | ∈ [1, R], where R ≥ 1, have infinite dimensional components and prove that all √ lattice points with |P  | ∈ [1, 3R] also have infinite dimensional components. Indeed, √ ¯ |S| ≤ 3R, have infinite the endomorphism τ yields that all thick points S ∈ τ (Γ), √ ¯ \ τ (Γ), ¯ |P | ≤ 3R. By the dimensional components. Take a non-thick point: P ∈ Γ √ ¯ |S| ≤ 3R, |P −S| = 1. geometric observation above, there exists a thick point: S ∈ τ (Γ), Now we choose an appropriate first factor in (49) that makes the step on the triangular grid, in order to move the infinite dimensional component of S into the point P . The inductive step is proved. 2

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Theorem 18.2. Let Q = Lie(a0 , b0 , c0 ). (1) There exists a decomposition into a direct sum of two locally nilpotent subalgebras: Q = Q+ ⊕ Q− ; (2) There is a continuum of such different decompositions. Proof. Pick any line on plane Π passing through zero and not intersecting any point of the triangular grid except zero. Let Q+ , Q− be sums of homogeneous components that lie on different sides of the line. Since, Q00 = {0}, we get a direct decomposition Q = Q+ ⊕Q− . A position of a homogeneous monomial v on Π is given by the superweight function swt(v), additivity of which implies that Q+ and Q− are subalgebras. The line also yields a plane in space R3 passing through the diagonal line X1 = X2 = X3 . This plane splits the elliptic hyperboloid (Theorem 15.6) into two halves. Now the same geometric arguments as in [36] prove that Q+ , Q− are locally nilpotent. Points of the triangular grid yield a countable set of directions that the line should avoid. There is a continuum of remaining directions. Different lines yield different decompositions because directions to the grid points are dense among all directions. 2 References [1] R. Bacher, Determinants related to Dirichlet characters modulo 2, 4 and 8 of binomial coefficients and the algebra of recurrence matrices, Internat. J. Algebra Comput. 18 (3) (2008) 535–566. [2] Yu.A. Bahturin, A.A. Mikhalev, V.M. Petrogradsky, M.V. Zaicev, Infinite Dimensional Lie Superalgebras, de Gruyter Exp. Math., vol. 7, de Gruyter, Berlin, 1992. [3] Yu. Bahturin, A. Olshanskii, Large restricted Lie algebras, J. Algebra 310 (1) (2007) 413–427. [4] Yu.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2) (2001) 677–698. [5] L. Bartholdi, Branch rings, thinned rings, tree enveloping rings, Israel J. Math. 154 (2006) 93–139. [6] L. Bartholdi, Self-similar Lie algebras, J. Eur. Math. Soc. (JEMS) 17 (12) (2015) 3113–3151. [7] L. Bartholdi, R.I. Grigorchuk, Lie methods in growth of groups and groups of finite width. Computational and geometric aspects of modern algebra, in: London Math. Soc. Lecture Note Ser., vol. 275, Cambridge Univ. Press, Cambridge, 2000, pp. 1–27. [8] S. Bouarroudj, P. Grozman, D. Leites, Classification of finite dimensional modular Lie superalgebras with indecomposable Cartan matrix, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009) 060. [9] V. Drensky, L. Hammoudi, Combinatorics of words and semigroup algebras which are sums of locally nilpotent subalgebras, Canad. Math. Bull. 47 (3) (2004) 343–353. [10] A. Elduque, Fine gradings on simple classical Lie algebras, J. Algebra 324 (12) (2010) 3532–3571. [11] M. Ershov, Golod–Shafarevich groups: a survey, Internat. J. Algebra Comput. 22 (5) (2012) 1230001. [12] J. Fabrykowski, N. Gupta, On groups with sub-exponential growth functions, J. Indian Math. Soc. (N.S.) 49 (3–4) (1985) 249–256. [13] E.S. Golod, On nil-algebras and finitely approximable p-groups, Amer. Math. Soc. Transl. Ser. 2 48 (1965) 103–106; translation from Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 273–276. [14] E.S. Golod, I.R. Shafarevich, On the class field tower, Amer. Math. Soc. Transl. Ser. 2 48 (1965) 91–102; translation from Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 261–272. [15] E.S. Golod, On some problems of Burnside type, Amer. Math. Soc. Transl. Ser. 2 84 (1969) 83–88; translation from Tr. Mezdunar. Kongr. Mat. Moskva 1966 (1968) 284–289. [16] R.I. Grigorchuk, On the Burnside problem for periodic groups, Funktsional. Anal. i Prilozhen. 14 (1) (1980) 53–54. [17] R.I. Grigorchuk, Degrees of growth of finitely generated groups, and the theory of invariant means, Math. USSR, Izv. 25 (1985) 259–300; translation from Izv. Akad. Nauk SSSR Ser. Mat. 48 (5) (1984) 939–985.

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