- Email: [email protected]
The Nowicki conjecture for relatively free algebras
Journal Pre-proof The Nowicki conjecture for relatively free algebras
Lucio Centrone, Sehmus ¸ Fındık
PII:
S0021-8693(20)30054-5
DOI:
https://doi.org/10.1016/j.jalgebra.2020.01.017
Reference:
YJABR 17532
To appear in:
Journal of Algebra
Received date:
23 August 2019
Please cite this article as: L. Centrone, S. ¸ Fındık, The Nowicki conjecture for relatively free algebras, J. Algebra (2020), doi: https://doi.org/10.1016/j.jalgebra.2020.01.017.
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier.
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS LUCIO CENTRONE AND S ¸ EHMUS FINDIK
Abstract. A linear locally nilpotent derivation of the polynomial algebra ock K[Xm ] in m variables over a field K of characteristic 0 is called a Weitzenb¨ derivation. It is well known from the classical theorem of Weitzenb¨ ock that ock derivation δ is finitely genthe algebra of constants K[Xm ]δ of a Weitzenb¨ erated. Assume that δ acts on the polynomial algebra K[X2d ] in 2d variables as follows: δ(x2i ) = x2i−1 , δ(x2i−1 ) = 0, i = 1, . . . , d. The Nowicki conjecture states that the algebra K[X2d ]δ is generated by x1 , x3 . . . . , x2d−1 , and x2i−1 x2j − x2i x2j−1 , 1 ≤ i < j ≤ d. The conjecture was proved by several authors based on different techniques. We apply the same idea to two relatively free algebras of rank 2d. We give the infinite set of generators of the algebra of constants in the the free metabelian associative algebras F2d (A), and finite set of generators in the free algebra F2d (G) in the variety determined by the identities of the infinite dimensional Grassmann algebra.
1. Introduction Let K be a field of characteristic zero, Xm = {x1 , . . . , xm } be a set of variables, and KXm be the vector space with basis Xm . Now consider a non-zero nilpotent linear operator δ of KXm . Certainly δ can be extended to a derivation of the polynomial algebra K[Xm ], the free algebra of rank m in the class of all commutative unitary algebras. Such derivations are called Weitzenb¨ ock due to the classical theorem of Weitzenb¨ock [27], dating back to 1932, which states that the algebra K[Xm ]δ = kerδ of constants of the derivation δ in the algebra K[Xm ] is finitely generated. Obviously, any Weitzenb¨ock derivation δ is locally nilpotent and the linear operator exp(δ) which acts on the vector space KXm is unipotent. Hence the algebra K[Xm ]δ of constants of δ is equal to the algebra of invariants K[Xm ]exp(δ) = {x ∈ K[Xm ] | exp(δ)(x) = x}. The algebra of invariants of exp(δ) is equal to the algebra of invariants K[Xm ]G , where the group G consists of all elements exp(cδ), c ∈ K. As an abstract group the group G is isomorphic to the unitriangular group consisting of all 2 × 2 upper triangular matrices with 1’s on the diagonal. The group G is embedded into the general linear group GLm (K) via the representation corresponding to the Jordan normal form of the linear operator δ acting on KXm . This means that the methods inherited from the classical invariant theory are linked to the study of the algebra 2010 Mathematics Subject Classification. 16R10; 16S15;16W25; 13N15; 13A50. Key words and phrases. Free metabelian associative algebras; algebras of constants; Weitzenb¨ ock derivations, the Nowicki conjecture. The first author is partially supported by FAPESP grant 2018/02108-7 and by CNPq. The research of the second named author was partially supported by C ¸ ukurova University (BAP/10746) for Bilateral Scientific Cooperation between Turkey and Brazil. 1
2
LUCIO CENTRONE, S ¸ EHMUS FINDIK
of constants K[Xm ]δ . One can see computational aspects of algebras of constants and invariant theory in the books by Nowicki [23], Derksen and Kemper [5], and Sturmfels [26]. The Jordan normal form of the linear operator δ consists of Jordan cells with zero diagonals. The Weitzenb¨ ock derivations are in one-to-one correspondence with the partitions of m, up to a linear change of variables. Thus there are essentially finite number of Weitzenb¨ock derivations for a fixed dimension m. In particular, let m = 2d, d ≥ 1, and assume that the Jordan form of δ contains the Jordan cells of size 2 × 2 only, i.e., ⎛ ⎞ 0 1 ··· 0 0 ⎜0 0 · · · 0 0⎟ ⎜ ⎟ ⎜ ⎟ J(δ) = ⎜ ... ... . . . ... ... ⎟ . ⎜ ⎟ ⎝0 0 · · · 0 1⎠ 0 0 ··· 0 0 We assume that δ(x2j ) = x2j−1 , δ(x2j−1 ) = 0, j = 1, . . . , d. Nowicki conjectured in his book [23] (Section 6.9, page 76) that in this case the algebra K[X2d ]δ is generated by x1 , x3 . . . . , x2d−1 , and x2i−1 x2j − x2i x2j−1 , 1 ≤ i < j ≤ d. The conjecture was proved by several authors with different techniques. The first published proof appeared in 2004 by Khoury [17] in his PhD thesis, followed by his paper [18], where he makes use of a computational approach involving Gr¨ obner basis techniques. Derksen and Panyushev applied ideas of classical invariant theory in order to prove the Nowicki conjecture but their proof remained unpublished. Later in 2009, Drensky and Makar-Limanov [11] confirmed the conjecture by an elementary proof from undergraduate algebra, without involving invariant theory. Another simple short proof was given by Kuroda [21] in the same year. Bedratyuk [2] proved the Nowicki conjecture by reducing it to a well known problem of classical invariant theory. There are three more proofs of the conjecture of Nowicki. Two of them are mentioned in the paper by Kuroda [21], and the latest proof is given by Drensky [8]. The following problem arises naturally. Let δ be a Weitzenb¨ock derivation of the free algebra Fm (V) of rank m in a given variety of algebras V (or relatively free algebra of rank m of V), then give an explicit set of generators of the algebra δ (V). In [10] Drensky and Gupta studied Weitzenb¨ock derivations of constants Fm δ acting on Fm (V) proving that if the algebra U T2 (K) of 2 × 2 upper triangular δ (V) matrices over a field K of characteristic zero belongs to a variety V, then Fm δ is not finitely generated whereas if U T2 (K) does not belong to V, then Fm (V) is finitely generated by a result of Drensky in [7]. Recently, Dangovski et al. [3, 4] gave some new results in this direction. They showed that the algebra of constants (Fd (V))δ in the commutator ideal Fd (V) of Fd (V) is a finitely generated module of K[Xd ]δ and K[Ud , Vd ]δ , respectively. Here δ acts on Ud and Vd in the same way as on Xd . When studying varieties of associative unitary algebras over a field of characteristic zero, the polynomial identities of U T2 (K) play a crucial role. It is well known the identities of U T2 (K) follow from the metabelian identity [x1 , x2 ][x3 , x4 ] ≡ 0, so every variety V contains U T2 (K) or satisfies the Engel identity [x2 , x1 , . . . , x1 ] ≡ 0. Another precious tool in the study of varieties is given by the identities of the infinitely generated Grassmann algebra G. In a famous work by Kemer (see [16] or his monograph [15]) the author proves that any variety V of associative algebras
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
3
satisfies the Specht property, i.e, any proper subvariety of V is finitely generated. In particular the identities of V are the same as the identities of the so called Grassmann envelope of a finite dimensional superalgebra. In the description of varieties of (not necessarily associative) algebras and their rate of growth the exponent of an algebra is worthy used. In the next lines we shall recall the definition of the exponent. Let V be a variety, then by a well known multilinearization process, in order to study the polynomial identities of V, it is enough studying its multilinear identities. Hence, if we denote by Pn the vector space of multilinear polynomials of degree n, we denote by cn (V) the dimension of Pn modulo the multilinear identities of V of degree n. In [13], [14] Giambruno and Zaicev proved that if V is a variety of associative algebras, then there exists the limit exp(V) = lim n cn (V) n→∞
and it is a non-negative integer called the PI-exponent of V denoted by exp(V). In the Lie case the existence of the exponent has been proved in the finite dimensional case by Zaicev in [28]. Indeed varieties of exponent one are called polynomial growth varieties, hence varieties having a non-polynomial growth may have exponent at least two. The algebras U T2 (K) and G are also fundamental tools in the study of the growth of varieties of exponent two. In particular in [12] the authors constructed five “minimal” varieties {Ai }i=1,...,5 and they proved that a variety V has exponent two if and only if none of the Ai belongs to V and either G or U T2 (K) belongs to V. At the light of the above discussion, the goal of the paper is giving experimental results supporting the computational aspects of Nowicki conjecture for any 2d-generated relatively free algebra of any variety of associative algebras. In the second section we give the full infinite set of generators of the free metabelian associative algebra of rank 2d, d ≥ 1, whose commutator ideal has a K[X2d ]-bimodule structure. In this case its algebra of constants is a K[U2d , V2d ]δ -module, where the elements uj and vj stand for the associative (left side) and Lie (right side) multiplications, respectively. This allows to use the results from Nowicki conjecture as a counterpart of the work. For this purpose, we worked in the abelian wreath product where the algebra of constants is embedded into, and δ acts in the same way, and finally the results were pulled back to the algebra of constants. In Section 3, we give the explicit form of the set of generators for the algebra of constants in the variety G of associative algebras generated by the infinite dimensional Grassmann algebra. Of course the free metabelian associative algebra coincides with the variety V generated by U T2 (K). As noticed above, both G and V have exponent two although G does not belong to the variety generated by U T2 (K). At the light of the results [7] and [10] the algebra of constants in G is finitely generated not as in the metabelian case. 2. The free metabelian associative algebras We start off our investigation with the relatively free algebra of the associative metabelian algebra. Let K be a field of characteristic zero, P2d be the free unitary = P2d [P2d , P2d ]P2d be its commutator associative algebra of rank 2d over K, P2d ideal generated by all elements of the form [x, y] = xy − yx,
x, y ∈ P2d .
4
LUCIO CENTRONE, S ¸ EHMUS FINDIK
2 Let us consider the quotient algebra F2d = P2d /(P2d ) . The algebra F2d is the free algebra of rank 2d in the variety of all associative algebras satisfying the polynomial identity [x, y][z, t] ≡ 0. Let the free associative metabelian algebra F2d be generated by X2d = {x1 , . . . , x2d }. We assume that all Lie commutators are left normed; i.e.,
[x, y, z] = [[x, y], z], The commutator ideal xξ11
F2d
x, y, z ∈ F2d .
of F2d has the following basis (see e.g. [1, 6])
· · · xξ2d2d [xi , xj , xj1 , . . . , xjm ],
ξi ≥ 0,
i > j ≤ j1 ≤ · · · ≤ jm ≤ 2d.
As a consequence of the metabelian identity in F2d we have the following identity:
xiπ(1) · · · xiπ(m) [xi , xj , xjσ(1) , . . . , xjσ(n) ] ≡ xi1 · · · xim [xi , xj , xj1 , . . . , xjn ], for any permutation π ∈ Sm and σ ∈ Sn . Thus the commutator ideal F2d can be “seen” as a module of polynomial algebra from both sides via the associative (left side) and Lie (right side) multiplication. We recall now some of the results and constructions given in [4]. Let U2d = {u1 , . . . , u2d } and V2d = {v1 , . . . , v2d } be two sets of commuting variables and let as follows. If f ∈ F2d , then K[U2d , V2d ] be the polynomial algebra acting on F2d
f ui = xi f,
f vi = [f, xi ],
i = 1, . . . , 2d.
This action defines a K[Ud , Vd ]-module structure on the vector space F2d . We are going to construct a wreath product which is the same as the one used in [4]. It is a particular case of the construction of Lewin [22] given in [10] and is similar to the construction of Shmel’kin [25] in the case of free metabelian Lie algebras as appeared in [3]. = {v1 , . . . , v2d } be sets of commuting variables Let Y2d = {y1 , . . . , y2d } and V2d ]and let A2d = {a1 , . . . , a2d } in 2d variables. Now let M2d be the free K[U2d , V2d module generated by A2d equipped with with trivial multiplication M2d · M2d = 0. We endow M2d with a structure of a free K[Y2d ]-bimodule structure via the action
yj ai = ai uj ,
ai yj = ai vj ,
i, j = 1, . . . , 2d.
The wreath product W2d = K[Y2d ] M2d is an algebra satisfying the metabelian identity. As well as in [22] F2d can be embedded into W2d . In fact we have the following result. Proposition 2.1. The mapping ε : xj → yj + aj , j = 1, . . . , 2d, extends to an embedding ε of F2d into W2d . We identify vi = vi − ui , i = 1, . . . , 2d, and get (1)
ε(xi1 · · · xim [xi , xj , xj1 , . . . , xjn ]) = (ai vj − aj vi )vj1 · · · vjn ui1 · · · uim
Thus we may assume that M2d is a free K[U2d , V2d ]-module. Clearly the commuta is embedded into M2d , too. An element ai f (U2d , V2d ) ∈ M2d is an tor ideal F2d if and only if vi f (U2d , V2d ) = 0, as a consequence image of some element from F2d of (1). Let δ be the Weitzenb¨ock derivation of F2d acting on the variables U2d , V2d , as well as explained in the Introduction on X2d . By [4] we know that the vector space δ of the constants of δ in the K[U2d , V2d ]-module M2d is a K[U2d , V2d ]δ -module. M2d The following results are particular cases of [7] (Proposition 3) and [4]. δ δ Theorem 2.2. The vector spaces (F2d ) and M2d are finitely generated K[U2d , V2d ]δ modules.
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
5
In the sequel we shall write down an explicit set of generators for the algebra of constants K[U2d , V2d ]δ as a consequence of the Nowicki conjecture [23] (proved in [17, 18, 11, 2, 21]), and one of the results by Drensky and Makar-Limanov [11]. Let the Weitzenb¨ock derivation δ act on X2d , A2d , U2d , V2d by the rule δ(x2i ) = x2i−1 , δ(x2i−1 ) = 0,
δ(a2i ) = a2i−1 , δ(a2i−1 ) = 0,
δ(u2i ) = u2i−1 , δ(u2i−1 ) = 0, δ(v2i ) = v2i−1 , δ(v2i−1 ) = 0, for i = 1, . . . , d. Then the algebra of constants K[U2d , V2d ]δ is generated by u1 , u3 , . . . , u2d−1 , v1 , v3 , . . . , v2d−1 and the determinants
u αpq = u2p−1 u2q − u2p u2q−1 =
2p−1 u2q−1
v βpq = v2p−1 v2q − v2p v2q−1 =
2p−1 v2q−1
u γpq = u2p−1 v2q − u2p v2q−1 =
2p−1 v2q−1 with the following defining relations (2)
u2p
, 1 ≤ p < q ≤ d, u2q
v2p
, 1 ≤ p < q ≤ d, v2q
u2p
, p, q = 1, . . . , d, v2q
u2i−1 αjk − u2j−1 αik + u2k−1 αij = 0,
1 ≤ i < j < k ≤ d,
(3)
u2i−1 γjk − u2j−1 γik + v2k−1 αij = 0, 1 ≤ i < j ≤ d, 1 ≤ k ≤ d,
(4)
u2i−1 βjk − v2j−1 γik + v2k−1 γij = 0, 1 ≤ i ≤ d, 1 ≤ j < k ≤ d,
(5)
v2i−1 βjk − v2j−1 βik + v2k−1 βij = 0,
1 ≤ i < j < k ≤ d,
(6)
αij αkl − αik αjl + αil αjk = 0, 1 ≤ i < j < k < l ≤ d,
(7)
αij γkl − αik γjl + γil αjk = 0, 1 ≤ i < j < k ≤ d, 1 ≤ l ≤ d,
(8)
αij βkl − γik γjl + γil γjk = 0, 1 ≤ i < j ≤ d, 1 ≤ k < l ≤ d,
(9)
γij βkl − γik βjl + γil βjk = 0, 1 ≤ i ≤ d, 1 ≤ j < k < l ≤ d,
(10)
βij βkl − βik βjl + βil βjk = 0,
1 ≤ i < j < k < l ≤ d.
The vector space K[U2d , V2d ]δ has a canonical linear basis consisting of the elements of the form (11) v2i1 −1 · · · v2im −1 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1 such that among the generators βpq , γp q , and αp q there is no intersection, and no one covers v2ik −1 or u2jl −1 . Note that each βpq , γp q , and αp q is identified with the open interval (p+d, q + d), (p , q + d), and (p , q ), respectively, on the real line. The generators intersect each other if the corresponding open intervals have a nonempty intersection and are not contained in each other. On the other hand the generators v2i−1 and u2j−1 are identified with the points i + d and j, respectively. We say also that a generator among βpq , γp q , or αp q covers v2i−1 or u2j−1 if the corresponding open interval covers the corresponding point.
6
LUCIO CENTRONE, S ¸ EHMUS FINDIK
The pairs of indices are ordered in the following way: p1 ≤ · · · ≤ pr and if ; pξ = pξ+1 , then qξ ≤ qξ+1 ; p1 ≤ · · · ≤ ps and if pμ = pμ+1 , then qμ ≤ qμ+1 p1 ≤ · · · ≤ pt and if pσ = pσ+1 , then qσ ≤ qσ+1 ; and i1 ≤ · · · ≤ im , j1 ≤ · · · ≤ jn . it is sufficient to In order to detect the constants in the commutator ideal F2d δ δ δ work in the K[U2d , V2d ] -submodule C2d of M2d , which is generated by a1 , a3 , . . . , a2d−1 and the determinants wpq = a2p−1 v2q − a2q v2p−1
a =
2p−1 v2p−1
a2q
, v2q
p, q = 1, . . . , d.
and spanned as a vector space on the elements of the form (12) a2i0 −1 v2i1 −1 · · · v2im −1 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1 (13) wp0 q0 v2i1 −1 · · · v2im −1 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1 δ for each i0 , p0 , q0 = 1 . . . , d. We also have the following relations in the algebra C2d as a consequence of [11].
(14) (15)
a2i−1 βjk − wik v2j−1 + wij v2j−1 = 0, wij βkl − wik βjl + wil βjk = 0,
1 ≤ i ≤ d, 1 ≤ j < k ≤ d,
1 ≤ i ≤ d, 1 ≤ j < k < l ≤ d.
δ
δ We denote by L the K[U2d , V2d ] -submodule of C2d generated by the following elements
(16)
wii ,
1 ≤ i ≤ d, 1 ≤ i < j ≤ d,
(17)
wij + wji ,
(18)
a2i−1 v2j−1 − a2j−1 v2i−1 ,
1 ≤ i < j ≤ d,
(19)
a2i−1 βpq − wpq v2i−1 , 1 ≤ i ≤ d, 1 ≤ p < q ≤ d,
(20)
wij βpq − wpq βij , 1 ≤ i < j ≤ d, 1 ≤ p < q ≤ d,
(21)
a2i−1 βjk − a2j−1 βik + a2k−1 βij ,
1 ≤ i < j < k ≤ d,
(22)
wkl γij − wjl γik + wjk γil , 1 ≤ i ≤ d, 1 ≤ j < k < l ≤ d.
(23)
wjk u2i−1 − a2j−1 γik + a2k−1 γij , 1 ≤ i ≤ d, 1 ≤ j < k ≤ d.
One can observe that the generating elements (16)-(23) of L are the images of some of the free associative algebra F2d . elements in the commutator ideal F2d δ /L is spanned on the following elements. Lemma 2.3. The quotient space C2d (24) a2i0 −1 v2i1 −1 · · · v2im −1 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1
(25)
wp0 q0 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1
such that when we replace a2i0 −1 and wp0 q0 by v2i0 −1 and βp0 q0 , respectively, we obtain the basis elements of the algebra K[U2d , V2d ]δ .
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
7
Proof. We start by showing that the spanning elements of the form (13) reduce to the form (25). By (16) and (17) we assume p0 < q0 . Then (22) ensures that βp0 q0 does not intersect with each γpξ qξ . Now by the relation (15) we assume βp0 q0 does not intersect with each βpμ qμ . By definition, βp0 q0 does not intersect with each αpσ qσ and it does not cover each u2ic −1 . By (20) we can fix the order among βp0 q0 , βp1 q1 , . . . , βpr qr . Now if m ≥ 1 in (13), by (19) we may replace wp0 q0 v2im −1 by a2im −1 βp0 q0 . Hence we may assume m = 0, and the expression (13) reduces to (25) with the desired conditions. Let us consider the spanning element of the form (12). We have to check whether we obtain a basis element of K[U2d , V2d ]δ after replacing a2i0 −1 by v2i0 −1 . If v2i0 −1 is covered by some γpξ qξ , then by (23) we replace a2i0 −1 γpξ qξ by wiqξ u2pξ −1 + a2qξ −1 γpξ i . The elements v2qξ −1 is not covered by γpξ i . Additionally we have wiqξ u2pξ −1 v2i1 −1 · · · v2im −1 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1 which is not of the form (25). In this expression, if m = 0, then applying the argument above we reduce it to (25). If m ≥ 1, we apply (19) and get a2im −1 v2i1 −1 · · · v2im−1 −1 βiqξ βp1 q1 · · · βpr qr γp1 q1 · · · γps qs × ×αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1 u2pξ −1 which has a number of v2ic −1 ’s and γp q ’s strictly less than in the previous expression. Hence this process terminates when there is no more γpξ qξ covering v2i0 −1 . Now by (21), we may assume v2i0 −1 is not covered by each βpξ qξ . Then by (18) we are able to fix the order among v2i0 −1 , v2i1 −1 , . . . , v2im −1 . Finally the order among the generators of constants of K[U2d , V2d ]δ is fixed by the defining relations (2)-(10). Now we are in the position to prove the main result of this section. δ Theorem 2.4. The K[U2d , V2d ]δ -submodule L of C2d consists of all commutator δ δ elements in C2d ; i.e., the images in (F2d ) ⊂ F2d .
Proof. Let Bpq = βp1 q1 · · · βpr qr , Gp q = γp1 q1 · · · γps qs , Ap q = αp1 q1 · · · αpt qt , Vi = v2i1 −1 · · · v2im −1 , Uj = u2j1 −1 · · · u2jn −1 , and let σ pqp q p q j wp0 q0 Bpq Gp q Ap q Uj π ipqp q p q j a2i0 −1 Vi Bpq Gp q Ap q Uj + be a commutator element. By the metabelian identity we get π ipqp q p q j v2i0 −1 Vi Bpq Gp q Ap q Uj + σ pqp q p q j βp0 q0 Bpq Gp q Ap q Uj = 0. On the other hand the two sums are linearly independent basis elements of the algebra of constants K[U2d , V2d ]δ . Then by Lemma 2.3 we have π ipqp q p q j = pqp q p q j = 0, and σ π ipqp q p q j a2i0 −1 Vi Bpq Gp q Ap q Uj + σ pqp q p q j wp0 q0 Bpq Gp q Ap q Uj = 0,
because the constants π ipqp q p basis elements.
q j
and σ pqp q p
q j
are uniquely determined by the
8
LUCIO CENTRONE, S ¸ EHMUS FINDIK
δ As a consequence of the previous result we get the full list of generators of (F2d ) as a K[U2d , V2d ]-module. δ Corollary 2.5. The K[U2d , V2d ]δ -module (F2d ) is generated by the following polynomials:
g1 (i) = [x2i−1 , x2i ], g2 (i, j) = [x2i−1 , x2j−1 ],
1 ≤ i ≤ d, 1 ≤ i < j ≤ d,
g3 (i, j) = [x2i−1 , x2j ] + [x2j−1 , x2i ],
1 ≤ i < j ≤ d,
g4 (i, p, q) = [x2i−1 , x2p−1 , x2q ] − [x2i−1 , x2p , x2q−1 ], 1 ≤ i ≤ d, 1 ≤ p < q ≤ d, g5 (i, j, k) = [x2i−1 , x2j−1 , x2k ] − [x2i−1 , x2k−1 , x2j ] + [x2j−1 , x2k−1 , x2i ], 1 ≤ i < j < k ≤ d, g6 (i, j, p, q) = [x2i−1 , x2p−1 , x2j , x2q ] + [x2i , x2p , x2j−1 , x2q−1 ] −[x2i−1 , x2p , x2j , x2q−1 ] − [x2i , x2p−1 , x2j−1 , x2q ], 1 ≤ i < j ≤ d, 1 ≤ p < q ≤ d, g7 (i, j, k, l) = x2i [x2j−1 , x2k−1 , x2l ] + x2i−1 [x2j , x2k , x2l−1 ] −x2i [x2j−1 , x2k , x2l−1 ] − x2i−1 [x2j , x2k−1 , x2l ] 1 ≤ i ≤ d, 1 ≤ j < k < l ≤ d, g8 (i, j, k) = x2i [x2j−1 , x2k−1 ] − x2i−1 [x2j , x2k−1 ], 1 ≤ i ≤ d, 1 ≤ j < k ≤ d. We have to add for the generating set of the whole algebra (F2d )δ the constants x2i−1 and x2i−1 x2j −x2i x2j−1 , 1 ≤ i < j ≤ d, which are needed for the generation of the factor algebra of (F2d )δ modulo the commutator ideal of F2d . These generators are the ones lifted from the algebra K[X2d ]δ of constants of the polynomial algebra to the algebra (F2d )δ by the fact stated in Corollary 4.3 of the paper [10] by Drensky and Gupta. The following result gives an infinite generating set of the subalgebra of constants (F2d )δ of the free metabelian associative algebra F2d as an algebra. Corollary 2.6. The algebra (F2d )δ of the constants is generated by x1 , x3 , . . . , x2d−1 , x2i−1 x2j − x2i x2j−1 , g 1 f1 , . . . , g 8 f8 , where 1 ≤ i < j ≤ d, f1 , . . . , f8 ∈ K[U2d , V2d ]δ , and gi s are as in Corollary 2.5.
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
9
3. The variety G generated by the Grassmann algebra The variety G consists of all associative unitary algebras satisfying the polynomial identity [z1 , z2 , z3 ] ≡ 0. We shall set F2l := F2l (G) for 0 ≤ l ≤ d. As noted in the Introduction, G is the variety generated by the infinite dimensional Grassmann algebra G and F2d coincides with the 2d-generated relatively free algebra in the variety G. The identities of G and related topics have been studied by several authors. See for example the paper [19] by Krakovski and Regev about the ideal of polynomial identities of G. Let X = {x1 , x2 , . . . , xd }, Y = {y1 , y2 , . . . , yd } be two disjoint sets of variables. Of course the relatively free algebra of G of rank 2d in the variables from U := X ∪Y is isomorphic to F2d and we consider the following order inside U : x1 < y1 < x2 < · · · < xd < yd . Moreover we say a polynomial f is homogeneous in the set sof variables S = {u1 , . . . , us } if for each monomial m appearing in f we have i=1 degui m is the same. It is well known (see for example Theorem 5.1.2 of [6]) F2d has a basis consisting of all xa1 1 y1b1 · · · xad2d ydb2d [ui1 , ui2 ] · · · [ui2c−1 , ui2c ], ai , bj ≥ 0, , uil ∈ U, ui1 < ui2 < · · · < ui2c , c ≥ 0. We recall the identity [z1 , z2 , z3 ] ≡ 0 implies the identity [z1 , z2 ][z3 , z4 ] ≡ −[z1 , z3 ][z2 , z4 ]
(26)
in F2d . Consider the following Weitzenb¨ock derivation δ of F2d acting on U such that δ(yi ) = xi , δ(xi ) = 0,
1 ≤ i ≤ d.
δ Our goal is to exhibit a finite set of generators for F2d and we are going to prove it using an induction argument similar to the one used by Drensky and MakarLimanov in [11]. Let α = (α1 , · · · , αd−1 ) ∈ K d−1 and consider the algebra endomomorphism φα of F2d such that
φα (xi ) = xi , φα (yi ) = yi , i = 1, . . . , d − 1, φα (xd ) =
d−1
αi xi , φα (yd ) =
i=1
d−1
α i yi .
i=1
δ δ Notice that φα commutes with δ. Hence if f ∈ F2d , then φα (f ) ∈ F2d−2 too. We have the next result on the purpose and, as above, we consider F2l as generated by the disjoint sets {x1 , . . . , xl } and {y1 , . . . , yl }.
Lemma 3.1. Let d ≥ 2 and f ∈ F2d being homogeneous with respect to the set {xd , yd }. If φα (f ) = 0 for some non-zero α ∈ K d−1 , then f is in the left ideal generated by d−1 d−1
α i xi yd − αi yi xd , [ωα , u], u ∈ U, ωα := i=1
μα := xd ,
i=1 d−1 i=1
αi xi , να := yd ,
d−1 i=1
α i yi
10
LUCIO CENTRONE, S ¸ EHMUS FINDIK
Proof. Before starting the proof remark φ(ωα ) = φ(μα ) = φ(να ) = 0. We shall prove the assertion by induction on degxd ,yd f . Indeed the result is trivial if degxd ,yd f = 0, so we may assume the result true for degxd ,yd f > 0. Let f be as above, then we can write f as ap,u0 [u0 , yd ]ydp−1 + ap−1 ydp−1 xd + ap−1,u1 [u11 , xd ]ydp−1 f = ap ydp + 1
u0
+
u11
ap−1,u2 [u21 , yd ]ydp−2 xd + ap−1 [xd , yd ]ydp−2 + · · · + a0 xpd +
1
a0 [up , xd ]xp−1 , d
up
u21
where the uij ’s are in U whereas the aj ’s, the ar,ui ’s, the as,ui and the at,ui ’s all j j j belong to F2d−2 . Let p ≥ 3 and let us denote g1 := ap ydp−1 + ap,u0 [u0 , yd ]ydp−2 + ap−1,u1 [u11 , xd ]ydp−2 + ap−1 [xd , yd ]ydp−3 1
u0
u11
and g2 := f − g1 yd . Remark g2 can be written as
g2 = g2 xd
for some polynomial g2 ∈ F2d . We consider now d−1
α i xi f i=1
= g1 ωα + g 1
d−1
α i yi
xd −
i=1
ap ydp−1
= g1 ω α −
α i yi
αi [xi , yd ]
i=1
xd −
i=1
d−1
+
g2∗ xd
+
d−1
[xi , yd ]
i=1
αi g2 [xd , xi ]
i=1
+ g2 xd +
d−1
αi g2 [xd , xi ] ,
i=1
d−1 αi yi + g2∗ and g2∗= g2 i=1 αi xi . Because p − 1 > 0 we may d−1 apply the same argument above to f i=1 αi xi . Then using (26) we have
where g2 = g1
d−1
d−1
d−1
f
i=1
d−1 i=1
2 α i xi
= h1 ωα +
d−1
h1 [ωα , xi ] + h2 xd + h3 μα .
i=1
Therefore we get φα (h2 xd ) = 0, i.e., φα (h2 ) = 0 because clearly φα (xd ) = 0. Now we can apply induction on h2 2 d−1 obtaining f is in the left ideal generated by ωα , [ωα , u], u ∈ U , μα i=1 αi xi d−1 and να . Also remark ωα , [ωα , u], μα and να are irreducible in F2d and i=1 αi xi does not depend on xd , yd , then we conclude f is as desired and we are done. The d−1 other cases may be treated similarly also multiplying everything by i=1 αi yi . The next is an easy consequence of Lemma 3.1 and its proof follows verbatim the one of Corollary 2 of [11].
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
11
Corollary 3.2. If φα (f ) = 0 for all non-zero α = (α1 , . . . , αd−1 ) ∈ K d−1 , then f = 0 if d > 2 and f is in the left ideal generated by v12 := x1 y2 − y1 x2 and [v12 , u], where u ∈ U , if d = 2. We define the following objects inside F2d : vij := xi yj − yi xj ,
1 ≤ i, j ≤ d,
wijk := yi [xj , xk ] − xi [yj , xk ], 1 ≤ i, j, k ≤ d, zijkl := yi [xj , vk,l ] − xi [yj , vk,l ], 1 ≤ i, j, k, l ≤ d, Notice that the vij ’s, the wijk ’s and the zijkl ’s are constants of F2d and starting from these objects we shall construct three subsets of elements of F2d . We set V := {vij |1 ≤ i, j ≤ d}, W0 := {wijk |1 ≤ i, j ≤ k ≤ d} and Z0 := {zijkl |1 ≤ i ≤ j ≤ k ≤ l ≤ d}. Suppose s > 0 and we set Ws := {y[x, w] − x[y, w]|w ∈ Ws−1 , x ∈ X, y ∈ Y }, Zs := {y[x, z] − x[y, z]|z ∈ Zs−1 x ∈ X, y ∈ Y }. As above, both the Ws ’s and the Zs ’s are subsets of constants. Moreover it can be easily seen for s ≥ d both Vs and Ws are 0. We shall denote by C the algebra generated by the non-zero X, V, Wl , Zl . Remark 3.3. We have [v12 , x2 ] = x1 v22 + w212 . The next easy-to-prove relation is crucial in the sequel: b(b − 1) b−2 δ(y b ) = bxy b−1 + y [y, x]. 2 Before writing the proof of the main result we still need another technical lemma. Lemma 3.4. For every p ≥ 1 we have xk1 · · · xkp ydp = c + g1 (x1 , y1 , . . . , xd , yd )xd + g2 (x1 , y1 , . . . , xd , yd )[xd , yd ], where c ∈ C and g1 (x1 , y1 , . . . , xd , yd ), g1 (x1 , y1 , . . . , xd , yd ) ∈ F2d . Proof. We shall prove the result by induction on p. If p = 1, then xk yd = vkd +yk xd and we are done because vkd ∈ V . Suppose the result true for p ≥ 1 and let us prove it for p + 1. Notice that xk1 · · · xkp+1 ydp+1 = xk1 (xk2 · · · xkp+1 ydp )yd and we apply induction on the term inside the parenthesis. We have now xk1 · · · xkp+1 ydp+1 = xk1 (c + g1 xd + g2 [xd , yd ])yd = xk1 cyd + xk1 g1 xd yd + xk1 g2 yd [xd , yd ] = cxk1 yd + [xk1 , c]yd + xk1 g1 yd xd + g[xd , yd ]. Now observe that xk1 yd = vkd + yk xd and [xk1 , c]yd = [xk1 , c]yd − [yk1 , c]xd − [yk1 , c]xd = ω − [yk1 , c]xd ,
12
LUCIO CENTRONE, S ¸ EHMUS FINDIK
where ω := [xk1 , c]yd − [yk1 , c]xd belongs to C too and we are done.
Remark 3.5. We shall use in the sequel the following relation [xk1 , c]yd = ω − [yk1 , c]xd , where c, ω ∈ C. δ is finitely generated as an algebra, Theorem 3.6. The algebra of constants F2d and its generators are:
(27)
X;
(28)
V,
(29)
Wl , l ≤ d − 1;
(30)
Zl , l ≤ d − 1.
Proof. It is easy to see that the elements from (27)-(30) are constants of F2d . δ δ , then its U -homogeneous components lie in F2d Let f = f (X , Y , xd , yd ) ∈ F2d too. Hence we may assume f being homogeneous in U . Arguing analogously we can take f being homogeneous in {xd , yd }. We shall prove the theorem by induction on d and on the total degree with respect to the set {xd , yd }. Suppose d = 1, then f=
n
αk xk y n−k +
n−1
βj xj−1 y n−j−1 [x, y].
j=1
k=0
This means 0 = δ(f ) =
n−1
αk (n − k)xk+1 y n−k−1
k=0
+[x, y]
n−2 k=1
n − k k−1 n−k−1 x (n − k − 1) αk y + βk xk y n−k−2 . 2
The previous relation gives us αk = 0 for k ≤ n − 1, then βj = 0 for 1 ≤ j ≤ n − 2. Thus the only possibility is f = αn xn + βn−1 xn−2 [x, y] and we are done. Assume now d > 1 and the result true for F2d−2 . We recall F2d−2 is generated by the sets of variables X = {x1 , x2 , . . . , xd−1 }, Y = {y1 , y2 , . . . , yd−1 } and we set δ is homogeneous in U , then degX (v) ≥ degY (v) and if U = X ∪ Y . If v ∈ F2d−2 δ degxd ,yd (f ) = 0, then f ∈ F2d−2 , so we shall study only the case degxd ,yd (f ) > 0. Remembering that inside F2d−2 every commutator belongs to the center of F2d−2 , we can write f in the following way: ap,u0 [u0 , yd ]ydp−1 + ap−1 xd ydp−1 + ap−1,u1 [u11 , xd ]ydp−1 f = ap ydp + 1
u0
+
u11
ap−1,u2 [u21 , yd ]xd ydp−2 + ap−1 [xd , yd ]ydp−2 + · · · + a0 xpd +
1
u21
up
a0 [up , xd ]xp−1 , d
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
13
where the uij ’s are in U whereas the aj ’s, the ar,ui ’s, the as,ui and the at,ui ’s all j j j belong to F2d−2 . We derive f and we obtain (31)
+
δ(f ) = δ(ap )ydp + pap xd ydp−1 +
p(p − 1) ap [yd , xd ]ydp−2 2
δ(ap,u0 )[u0 , yd ]ydp−1 + ap,u0 [δ(u0 ), yd ]ydp−1 + ap,u0 [u0 , xd ]ydp−1
u0
+
+(p − 1)ap,u0 [u0 , xd ]xd ydp−2 + δ(ap−1 )xd ydp−1 + (p − 1)ap−1 x2d ydp−2 (p − 1)(p − 2) δ(ap−1,u1 )[u11 , xd ]ydp−1 ap−1 [yd , xd ]xd ydp−3 + + 1 2 u11 +ap−1,u1 [δ(u11 ), xd ]ydp−1 + (p − 1)ap−1,u1 [u11 , xd ]xd ydp−2 1
δ(ap−1,u2 )[u21 , yd ]xd ydp−2 1
+
1
ap−1,u2 [δ(u21 ), yd ]xd ydp−2 1
+ ap−1,u2 [u21 , xd ]xd ydp−2 1
u21
+(p − 2)ap−1,u2 [u21 , yd ]x2d ydp−3 + δ(ap−1 )[yd , xd ]ydp−2 + (p − 2)ap−1 [yd , xd ]xd ydp−3 1 p−1 + · · · + δ(a0 )xpd + + a [δ(u ), x ]x δ(a0 )[up , xd ]xp−1 =0 p d d 0 d up
By (31) we immediately get δ(ap ) = 0, hence: δ . ap ∈ F2d
(32) We also have (33)
p(p−1) ap + δ(ap−1 ) 2
= 0, then by (32) we get δ(ap−1 ) = 0. This means:
δ ap−1 ∈ F2d .
Suppose now u0 = ys , s ≤ d − 1, then ap,u0 = 0. So we can assume u0 = xs , s ≤ d − 1. In this case we get δ(ap,xs ) = 0, i.e.: (34)
δ ap,xs ∈ F2d .
By (32), (33) and (34), using induction, we get ap = b1 (X , V , W , Z ), b2 (X , V , W , Z ), ap−1 = b3,i (X , V , W , Z ). ap,xi = If φα (f ) = 0 for all α = (α1 , . . . , αd−1 ), then by Corollary 3.2 we get d = 2 and f is in the left ideal generated by v12 and [v12 , u], u ∈ U . Hence f = gv12 + u∈U gu [v12 , u] and, after a simple manipulation, we apply inductive arguments to g and gu ’s and we are done. Now we consider the case φα (f ) = 0 for some non-zero α ∈ K n−1 . The next two remarks are crucial. First we have degX (f ) = degX (φα (f )) ≥ degY (φα (f )) = degY (f ). Hence degX (f ) = degX (ap ) ≥ degY (f ) = p + degY (ap )
14
LUCIO CENTRONE, S ¸ EHMUS FINDIK
so ap =
b1 (X , V , W , Z )xk1 · · · xkp .
We can argue analogously and obtain b2 (X , V , W , Z )xl1 · · · xlp , ap−1 = b3,i (X , V , W , Z )xm1 · · · xmp . ap,xi =
Now we apply Lemma 3.4 and Remark 3.5 in order to rewrite ap ydp , ap−1 [xd , yd ]ydp−2 and ap,xi [xi , yd ]ydp−1 and we get f = c + gxd +
d−1
mp,i yd [xi , xd ],
i=1
where c ∈ C. We can rewrite the last summand as g1 xd + degxd hi = 0 and we get finally f = c + g xd +
d−1
d−1 i=1
hi [xi , xd ], where
hi [xi , xd ].
i=1
Hence g and the hi ’s are constants too because xd and [xi , xd ] are. Now we are allowed to apply induction on g and the hi ’s because their degrees with respect to the set {xd , yd } are strictly smaller than the one of f and we conclude the proof. 4. Acknowledgements The authors are very thankful to the referee who found a mistake in the proof of the main theorem in the first version of the paper. References [1] Yu. A. Bahturin, Identical Relations in Lie Algebras (Russian), Nauka, Moscow, 1985. Translation: VNU Science Press, Utrecht, 1987. [2] L. Bedratyuk, A note about the Nowicki conjecture on Weitzenb¨ ock derivations, Serdica Math. J. 35 (2009), 311-316. [3] R. Dangovski, V. Drensky, S ¸ . Fındık, Weitzenb¨ ock derivations of free metabelian Lie algebras, Linear Algebra Appl., 439 (10) (2013), 3279-3296. [4] R. Dangovski, V. Drensky, S ¸ . Fındık, Weitzenb¨ ock derivations of free associative algebras, Journal Algebra Appl., 16 (03) (2017), 1750041/1-26. [5] H. Derksen, G. Kemper, Computational Invariant Theory, Encyclopaedia of Mathematical Sciences, Invariant Theory and Algebraic Transformation Groups 130, Springer-Verlag, Berlin, 2002. [6] V. Drensky, Free Algebras and PI-Algebras, Springer, Singapore, 1999. [7] V. Drensky, Invariants of unipotent transformations acting on noetherian relatively free algebras, Serdica Math. J. 30 (2004), 395–404. [8] V. Drensky, Another proof of the Nowicki conjecture, arXiv:1902.08758 [math.AC]. [9] V. Drensky, S ¸ . Fındık, The Nowicki conjecture for free metabelian Lie algebras, Int. J. Algebra Comp. (to appear). [10] V. Drensky, C.K. Gupta, Constants of Weitzenb¨ ock derivations and invariants of unipotent transformations acting on relatively free algebras, J. Algebra 292 (2) (2005), 393-428. [11] V. Drensky, L. Makar-Limanov, The conjecture of Nowicki on Weitzenb¨ ock derivations of polynomial algebras, J. Algebra Appl. 8 (2009), 41-51. [12] A. Giambruno, M. V. Zaicev, A characterization of varieties of associative algebras of exponent two, Serdica Math. J. 26 (2000), 245-252. [13] A. Giambruno, M. V. Zaicev, Exponential codimension growth of P.I. algebras: an exact estimate, Adv. Math. 142 (1999), 221-243.
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
15
[14] A. Giambruno, M. V. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145-155. [15] A. R. Kemer, Ideals of Identities of Associative Algebras, Translations of Math. Monographs 84, AMS, Providence, RI, 1991. [16] A.R. Kemer, Finite basis property of identities of associative algebras, Algebra Logic 26 (5) (1987), 362-397. [17] J. Khoury, Locally Nilpotent Derivations and Their Rings of Constants, Ph.D. Thesis, Univ. Ottawa, 2004. [18] J. Khoury, A Groebner basis approach to solve a conjecture of Nowicki, J. Symbolic Comput. 43 (2008), 908-922. [19] D. Krakovski, A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973), 429-438. [20] K. Kurano, Positive characteristic finite generation of symbolic Rees algebras and Roberts’ counterexamples to the fourteenth problem of Hilbert, Tokyo J. Math. 16 (2) (1993), 473-496. [21] S. Kuroda, A simple proof of Nowicki’s conjecture on the kernel of an elementary derivation, Tokyo J. Math. 32 (2009), 247-251. [22] J. Lewin, A matrix representation for associative algebras I, Trans. Amer. Math. Soc. 188 (1974), 29-308. [23] A. Nowicki, Polynomial Derivations and Their Rings of Constants, Uniwersytet Mikolaja Kopernika, Torun, 1994. www-users.mat.umk.pl/˜anow/ps-dvi/pol-der.pdf. [24] P. Roberts, An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert’s fourteenth problem, J. Algebra 132 (2) (1990), 461-473. [25] A.L. Shmel’kin, Wreath products of Lie algebras and their application in the theory of groups (Russian), Trudy Moskov. Mat. Obshch. 29 (1973), 247-260. Translation: Trans. Moscow Math. Soc. 29 (1973), 239-252. [26] B. Sturmfels, Algorithms in Invariant Theory. 2nd ed. Texts and Monographs in Symbolic Computation, Springer-Verlag, Wien, 2008. ¨ [27] R. Weitzenb¨ ock, Uber die Invarianten von linearen Gruppen, Acta Math. 58 (1932), 231-293. [28] M. V. Zaitsev, Integrality of exponents of growth of identities of finite-dimensional Lie algebras, Izv. Math. 66 (2002), 463-487. IMECC, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda 651,13083859, Campinas (SP), Brazil Email address: [email protected] Department of Mathematics, C ¸ ukurova University, 01330 Balcalı, Adana, Turkey Email address: [email protected]
Lucio Centrone, Sehmus ¸ Fındık
PII:
S0021-8693(20)30054-5
DOI:
https://doi.org/10.1016/j.jalgebra.2020.01.017
Reference:
YJABR 17532
To appear in:
Journal of Algebra
Received date:
23 August 2019
Please cite this article as: L. Centrone, S. ¸ Fındık, The Nowicki conjecture for relatively free algebras, J. Algebra (2020), doi: https://doi.org/10.1016/j.jalgebra.2020.01.017.
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier.
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS LUCIO CENTRONE AND S ¸ EHMUS FINDIK
Abstract. A linear locally nilpotent derivation of the polynomial algebra ock K[Xm ] in m variables over a field K of characteristic 0 is called a Weitzenb¨ derivation. It is well known from the classical theorem of Weitzenb¨ ock that ock derivation δ is finitely genthe algebra of constants K[Xm ]δ of a Weitzenb¨ erated. Assume that δ acts on the polynomial algebra K[X2d ] in 2d variables as follows: δ(x2i ) = x2i−1 , δ(x2i−1 ) = 0, i = 1, . . . , d. The Nowicki conjecture states that the algebra K[X2d ]δ is generated by x1 , x3 . . . . , x2d−1 , and x2i−1 x2j − x2i x2j−1 , 1 ≤ i < j ≤ d. The conjecture was proved by several authors based on different techniques. We apply the same idea to two relatively free algebras of rank 2d. We give the infinite set of generators of the algebra of constants in the the free metabelian associative algebras F2d (A), and finite set of generators in the free algebra F2d (G) in the variety determined by the identities of the infinite dimensional Grassmann algebra.
1. Introduction Let K be a field of characteristic zero, Xm = {x1 , . . . , xm } be a set of variables, and KXm be the vector space with basis Xm . Now consider a non-zero nilpotent linear operator δ of KXm . Certainly δ can be extended to a derivation of the polynomial algebra K[Xm ], the free algebra of rank m in the class of all commutative unitary algebras. Such derivations are called Weitzenb¨ ock due to the classical theorem of Weitzenb¨ock [27], dating back to 1932, which states that the algebra K[Xm ]δ = kerδ of constants of the derivation δ in the algebra K[Xm ] is finitely generated. Obviously, any Weitzenb¨ock derivation δ is locally nilpotent and the linear operator exp(δ) which acts on the vector space KXm is unipotent. Hence the algebra K[Xm ]δ of constants of δ is equal to the algebra of invariants K[Xm ]exp(δ) = {x ∈ K[Xm ] | exp(δ)(x) = x}. The algebra of invariants of exp(δ) is equal to the algebra of invariants K[Xm ]G , where the group G consists of all elements exp(cδ), c ∈ K. As an abstract group the group G is isomorphic to the unitriangular group consisting of all 2 × 2 upper triangular matrices with 1’s on the diagonal. The group G is embedded into the general linear group GLm (K) via the representation corresponding to the Jordan normal form of the linear operator δ acting on KXm . This means that the methods inherited from the classical invariant theory are linked to the study of the algebra 2010 Mathematics Subject Classification. 16R10; 16S15;16W25; 13N15; 13A50. Key words and phrases. Free metabelian associative algebras; algebras of constants; Weitzenb¨ ock derivations, the Nowicki conjecture. The first author is partially supported by FAPESP grant 2018/02108-7 and by CNPq. The research of the second named author was partially supported by C ¸ ukurova University (BAP/10746) for Bilateral Scientific Cooperation between Turkey and Brazil. 1
2
LUCIO CENTRONE, S ¸ EHMUS FINDIK
of constants K[Xm ]δ . One can see computational aspects of algebras of constants and invariant theory in the books by Nowicki [23], Derksen and Kemper [5], and Sturmfels [26]. The Jordan normal form of the linear operator δ consists of Jordan cells with zero diagonals. The Weitzenb¨ ock derivations are in one-to-one correspondence with the partitions of m, up to a linear change of variables. Thus there are essentially finite number of Weitzenb¨ock derivations for a fixed dimension m. In particular, let m = 2d, d ≥ 1, and assume that the Jordan form of δ contains the Jordan cells of size 2 × 2 only, i.e., ⎛ ⎞ 0 1 ··· 0 0 ⎜0 0 · · · 0 0⎟ ⎜ ⎟ ⎜ ⎟ J(δ) = ⎜ ... ... . . . ... ... ⎟ . ⎜ ⎟ ⎝0 0 · · · 0 1⎠ 0 0 ··· 0 0 We assume that δ(x2j ) = x2j−1 , δ(x2j−1 ) = 0, j = 1, . . . , d. Nowicki conjectured in his book [23] (Section 6.9, page 76) that in this case the algebra K[X2d ]δ is generated by x1 , x3 . . . . , x2d−1 , and x2i−1 x2j − x2i x2j−1 , 1 ≤ i < j ≤ d. The conjecture was proved by several authors with different techniques. The first published proof appeared in 2004 by Khoury [17] in his PhD thesis, followed by his paper [18], where he makes use of a computational approach involving Gr¨ obner basis techniques. Derksen and Panyushev applied ideas of classical invariant theory in order to prove the Nowicki conjecture but their proof remained unpublished. Later in 2009, Drensky and Makar-Limanov [11] confirmed the conjecture by an elementary proof from undergraduate algebra, without involving invariant theory. Another simple short proof was given by Kuroda [21] in the same year. Bedratyuk [2] proved the Nowicki conjecture by reducing it to a well known problem of classical invariant theory. There are three more proofs of the conjecture of Nowicki. Two of them are mentioned in the paper by Kuroda [21], and the latest proof is given by Drensky [8]. The following problem arises naturally. Let δ be a Weitzenb¨ock derivation of the free algebra Fm (V) of rank m in a given variety of algebras V (or relatively free algebra of rank m of V), then give an explicit set of generators of the algebra δ (V). In [10] Drensky and Gupta studied Weitzenb¨ock derivations of constants Fm δ acting on Fm (V) proving that if the algebra U T2 (K) of 2 × 2 upper triangular δ (V) matrices over a field K of characteristic zero belongs to a variety V, then Fm δ is not finitely generated whereas if U T2 (K) does not belong to V, then Fm (V) is finitely generated by a result of Drensky in [7]. Recently, Dangovski et al. [3, 4] gave some new results in this direction. They showed that the algebra of constants (Fd (V))δ in the commutator ideal Fd (V) of Fd (V) is a finitely generated module of K[Xd ]δ and K[Ud , Vd ]δ , respectively. Here δ acts on Ud and Vd in the same way as on Xd . When studying varieties of associative unitary algebras over a field of characteristic zero, the polynomial identities of U T2 (K) play a crucial role. It is well known the identities of U T2 (K) follow from the metabelian identity [x1 , x2 ][x3 , x4 ] ≡ 0, so every variety V contains U T2 (K) or satisfies the Engel identity [x2 , x1 , . . . , x1 ] ≡ 0. Another precious tool in the study of varieties is given by the identities of the infinitely generated Grassmann algebra G. In a famous work by Kemer (see [16] or his monograph [15]) the author proves that any variety V of associative algebras
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
3
satisfies the Specht property, i.e, any proper subvariety of V is finitely generated. In particular the identities of V are the same as the identities of the so called Grassmann envelope of a finite dimensional superalgebra. In the description of varieties of (not necessarily associative) algebras and their rate of growth the exponent of an algebra is worthy used. In the next lines we shall recall the definition of the exponent. Let V be a variety, then by a well known multilinearization process, in order to study the polynomial identities of V, it is enough studying its multilinear identities. Hence, if we denote by Pn the vector space of multilinear polynomials of degree n, we denote by cn (V) the dimension of Pn modulo the multilinear identities of V of degree n. In [13], [14] Giambruno and Zaicev proved that if V is a variety of associative algebras, then there exists the limit exp(V) = lim n cn (V) n→∞
and it is a non-negative integer called the PI-exponent of V denoted by exp(V). In the Lie case the existence of the exponent has been proved in the finite dimensional case by Zaicev in [28]. Indeed varieties of exponent one are called polynomial growth varieties, hence varieties having a non-polynomial growth may have exponent at least two. The algebras U T2 (K) and G are also fundamental tools in the study of the growth of varieties of exponent two. In particular in [12] the authors constructed five “minimal” varieties {Ai }i=1,...,5 and they proved that a variety V has exponent two if and only if none of the Ai belongs to V and either G or U T2 (K) belongs to V. At the light of the above discussion, the goal of the paper is giving experimental results supporting the computational aspects of Nowicki conjecture for any 2d-generated relatively free algebra of any variety of associative algebras. In the second section we give the full infinite set of generators of the free metabelian associative algebra of rank 2d, d ≥ 1, whose commutator ideal has a K[X2d ]-bimodule structure. In this case its algebra of constants is a K[U2d , V2d ]δ -module, where the elements uj and vj stand for the associative (left side) and Lie (right side) multiplications, respectively. This allows to use the results from Nowicki conjecture as a counterpart of the work. For this purpose, we worked in the abelian wreath product where the algebra of constants is embedded into, and δ acts in the same way, and finally the results were pulled back to the algebra of constants. In Section 3, we give the explicit form of the set of generators for the algebra of constants in the variety G of associative algebras generated by the infinite dimensional Grassmann algebra. Of course the free metabelian associative algebra coincides with the variety V generated by U T2 (K). As noticed above, both G and V have exponent two although G does not belong to the variety generated by U T2 (K). At the light of the results [7] and [10] the algebra of constants in G is finitely generated not as in the metabelian case. 2. The free metabelian associative algebras We start off our investigation with the relatively free algebra of the associative metabelian algebra. Let K be a field of characteristic zero, P2d be the free unitary = P2d [P2d , P2d ]P2d be its commutator associative algebra of rank 2d over K, P2d ideal generated by all elements of the form [x, y] = xy − yx,
x, y ∈ P2d .
4
LUCIO CENTRONE, S ¸ EHMUS FINDIK
2 Let us consider the quotient algebra F2d = P2d /(P2d ) . The algebra F2d is the free algebra of rank 2d in the variety of all associative algebras satisfying the polynomial identity [x, y][z, t] ≡ 0. Let the free associative metabelian algebra F2d be generated by X2d = {x1 , . . . , x2d }. We assume that all Lie commutators are left normed; i.e.,
[x, y, z] = [[x, y], z], The commutator ideal xξ11
F2d
x, y, z ∈ F2d .
of F2d has the following basis (see e.g. [1, 6])
· · · xξ2d2d [xi , xj , xj1 , . . . , xjm ],
ξi ≥ 0,
i > j ≤ j1 ≤ · · · ≤ jm ≤ 2d.
As a consequence of the metabelian identity in F2d we have the following identity:
xiπ(1) · · · xiπ(m) [xi , xj , xjσ(1) , . . . , xjσ(n) ] ≡ xi1 · · · xim [xi , xj , xj1 , . . . , xjn ], for any permutation π ∈ Sm and σ ∈ Sn . Thus the commutator ideal F2d can be “seen” as a module of polynomial algebra from both sides via the associative (left side) and Lie (right side) multiplication. We recall now some of the results and constructions given in [4]. Let U2d = {u1 , . . . , u2d } and V2d = {v1 , . . . , v2d } be two sets of commuting variables and let as follows. If f ∈ F2d , then K[U2d , V2d ] be the polynomial algebra acting on F2d
f ui = xi f,
f vi = [f, xi ],
i = 1, . . . , 2d.
This action defines a K[Ud , Vd ]-module structure on the vector space F2d . We are going to construct a wreath product which is the same as the one used in [4]. It is a particular case of the construction of Lewin [22] given in [10] and is similar to the construction of Shmel’kin [25] in the case of free metabelian Lie algebras as appeared in [3]. = {v1 , . . . , v2d } be sets of commuting variables Let Y2d = {y1 , . . . , y2d } and V2d ]and let A2d = {a1 , . . . , a2d } in 2d variables. Now let M2d be the free K[U2d , V2d module generated by A2d equipped with with trivial multiplication M2d · M2d = 0. We endow M2d with a structure of a free K[Y2d ]-bimodule structure via the action
yj ai = ai uj ,
ai yj = ai vj ,
i, j = 1, . . . , 2d.
The wreath product W2d = K[Y2d ] M2d is an algebra satisfying the metabelian identity. As well as in [22] F2d can be embedded into W2d . In fact we have the following result. Proposition 2.1. The mapping ε : xj → yj + aj , j = 1, . . . , 2d, extends to an embedding ε of F2d into W2d . We identify vi = vi − ui , i = 1, . . . , 2d, and get (1)
ε(xi1 · · · xim [xi , xj , xj1 , . . . , xjn ]) = (ai vj − aj vi )vj1 · · · vjn ui1 · · · uim
Thus we may assume that M2d is a free K[U2d , V2d ]-module. Clearly the commuta is embedded into M2d , too. An element ai f (U2d , V2d ) ∈ M2d is an tor ideal F2d if and only if vi f (U2d , V2d ) = 0, as a consequence image of some element from F2d of (1). Let δ be the Weitzenb¨ock derivation of F2d acting on the variables U2d , V2d , as well as explained in the Introduction on X2d . By [4] we know that the vector space δ of the constants of δ in the K[U2d , V2d ]-module M2d is a K[U2d , V2d ]δ -module. M2d The following results are particular cases of [7] (Proposition 3) and [4]. δ δ Theorem 2.2. The vector spaces (F2d ) and M2d are finitely generated K[U2d , V2d ]δ modules.
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
5
In the sequel we shall write down an explicit set of generators for the algebra of constants K[U2d , V2d ]δ as a consequence of the Nowicki conjecture [23] (proved in [17, 18, 11, 2, 21]), and one of the results by Drensky and Makar-Limanov [11]. Let the Weitzenb¨ock derivation δ act on X2d , A2d , U2d , V2d by the rule δ(x2i ) = x2i−1 , δ(x2i−1 ) = 0,
δ(a2i ) = a2i−1 , δ(a2i−1 ) = 0,
δ(u2i ) = u2i−1 , δ(u2i−1 ) = 0, δ(v2i ) = v2i−1 , δ(v2i−1 ) = 0, for i = 1, . . . , d. Then the algebra of constants K[U2d , V2d ]δ is generated by u1 , u3 , . . . , u2d−1 , v1 , v3 , . . . , v2d−1 and the determinants
u αpq = u2p−1 u2q − u2p u2q−1 =
2p−1 u2q−1
v βpq = v2p−1 v2q − v2p v2q−1 =
2p−1 v2q−1
u γpq = u2p−1 v2q − u2p v2q−1 =
2p−1 v2q−1 with the following defining relations (2)
u2p
, 1 ≤ p < q ≤ d, u2q
v2p
, 1 ≤ p < q ≤ d, v2q
u2p
, p, q = 1, . . . , d, v2q
u2i−1 αjk − u2j−1 αik + u2k−1 αij = 0,
1 ≤ i < j < k ≤ d,
(3)
u2i−1 γjk − u2j−1 γik + v2k−1 αij = 0, 1 ≤ i < j ≤ d, 1 ≤ k ≤ d,
(4)
u2i−1 βjk − v2j−1 γik + v2k−1 γij = 0, 1 ≤ i ≤ d, 1 ≤ j < k ≤ d,
(5)
v2i−1 βjk − v2j−1 βik + v2k−1 βij = 0,
1 ≤ i < j < k ≤ d,
(6)
αij αkl − αik αjl + αil αjk = 0, 1 ≤ i < j < k < l ≤ d,
(7)
αij γkl − αik γjl + γil αjk = 0, 1 ≤ i < j < k ≤ d, 1 ≤ l ≤ d,
(8)
αij βkl − γik γjl + γil γjk = 0, 1 ≤ i < j ≤ d, 1 ≤ k < l ≤ d,
(9)
γij βkl − γik βjl + γil βjk = 0, 1 ≤ i ≤ d, 1 ≤ j < k < l ≤ d,
(10)
βij βkl − βik βjl + βil βjk = 0,
1 ≤ i < j < k < l ≤ d.
The vector space K[U2d , V2d ]δ has a canonical linear basis consisting of the elements of the form (11) v2i1 −1 · · · v2im −1 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1 such that among the generators βpq , γp q , and αp q there is no intersection, and no one covers v2ik −1 or u2jl −1 . Note that each βpq , γp q , and αp q is identified with the open interval (p+d, q + d), (p , q + d), and (p , q ), respectively, on the real line. The generators intersect each other if the corresponding open intervals have a nonempty intersection and are not contained in each other. On the other hand the generators v2i−1 and u2j−1 are identified with the points i + d and j, respectively. We say also that a generator among βpq , γp q , or αp q covers v2i−1 or u2j−1 if the corresponding open interval covers the corresponding point.
6
LUCIO CENTRONE, S ¸ EHMUS FINDIK
The pairs of indices are ordered in the following way: p1 ≤ · · · ≤ pr and if ; pξ = pξ+1 , then qξ ≤ qξ+1 ; p1 ≤ · · · ≤ ps and if pμ = pμ+1 , then qμ ≤ qμ+1 p1 ≤ · · · ≤ pt and if pσ = pσ+1 , then qσ ≤ qσ+1 ; and i1 ≤ · · · ≤ im , j1 ≤ · · · ≤ jn . it is sufficient to In order to detect the constants in the commutator ideal F2d δ δ δ work in the K[U2d , V2d ] -submodule C2d of M2d , which is generated by a1 , a3 , . . . , a2d−1 and the determinants wpq = a2p−1 v2q − a2q v2p−1
a =
2p−1 v2p−1
a2q
, v2q
p, q = 1, . . . , d.
and spanned as a vector space on the elements of the form (12) a2i0 −1 v2i1 −1 · · · v2im −1 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1 (13) wp0 q0 v2i1 −1 · · · v2im −1 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1 δ for each i0 , p0 , q0 = 1 . . . , d. We also have the following relations in the algebra C2d as a consequence of [11].
(14) (15)
a2i−1 βjk − wik v2j−1 + wij v2j−1 = 0, wij βkl − wik βjl + wil βjk = 0,
1 ≤ i ≤ d, 1 ≤ j < k ≤ d,
1 ≤ i ≤ d, 1 ≤ j < k < l ≤ d.
δ
δ We denote by L the K[U2d , V2d ] -submodule of C2d generated by the following elements
(16)
wii ,
1 ≤ i ≤ d, 1 ≤ i < j ≤ d,
(17)
wij + wji ,
(18)
a2i−1 v2j−1 − a2j−1 v2i−1 ,
1 ≤ i < j ≤ d,
(19)
a2i−1 βpq − wpq v2i−1 , 1 ≤ i ≤ d, 1 ≤ p < q ≤ d,
(20)
wij βpq − wpq βij , 1 ≤ i < j ≤ d, 1 ≤ p < q ≤ d,
(21)
a2i−1 βjk − a2j−1 βik + a2k−1 βij ,
1 ≤ i < j < k ≤ d,
(22)
wkl γij − wjl γik + wjk γil , 1 ≤ i ≤ d, 1 ≤ j < k < l ≤ d.
(23)
wjk u2i−1 − a2j−1 γik + a2k−1 γij , 1 ≤ i ≤ d, 1 ≤ j < k ≤ d.
One can observe that the generating elements (16)-(23) of L are the images of some of the free associative algebra F2d . elements in the commutator ideal F2d δ /L is spanned on the following elements. Lemma 2.3. The quotient space C2d (24) a2i0 −1 v2i1 −1 · · · v2im −1 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1
(25)
wp0 q0 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1
such that when we replace a2i0 −1 and wp0 q0 by v2i0 −1 and βp0 q0 , respectively, we obtain the basis elements of the algebra K[U2d , V2d ]δ .
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
7
Proof. We start by showing that the spanning elements of the form (13) reduce to the form (25). By (16) and (17) we assume p0 < q0 . Then (22) ensures that βp0 q0 does not intersect with each γpξ qξ . Now by the relation (15) we assume βp0 q0 does not intersect with each βpμ qμ . By definition, βp0 q0 does not intersect with each αpσ qσ and it does not cover each u2ic −1 . By (20) we can fix the order among βp0 q0 , βp1 q1 , . . . , βpr qr . Now if m ≥ 1 in (13), by (19) we may replace wp0 q0 v2im −1 by a2im −1 βp0 q0 . Hence we may assume m = 0, and the expression (13) reduces to (25) with the desired conditions. Let us consider the spanning element of the form (12). We have to check whether we obtain a basis element of K[U2d , V2d ]δ after replacing a2i0 −1 by v2i0 −1 . If v2i0 −1 is covered by some γpξ qξ , then by (23) we replace a2i0 −1 γpξ qξ by wiqξ u2pξ −1 + a2qξ −1 γpξ i . The elements v2qξ −1 is not covered by γpξ i . Additionally we have wiqξ u2pξ −1 v2i1 −1 · · · v2im −1 βp1 q1 · · · βpr qr γp1 q1 · · · γps qs αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1 which is not of the form (25). In this expression, if m = 0, then applying the argument above we reduce it to (25). If m ≥ 1, we apply (19) and get a2im −1 v2i1 −1 · · · v2im−1 −1 βiqξ βp1 q1 · · · βpr qr γp1 q1 · · · γps qs × ×αp1 q1 · · · αpt qt u2j1 −1 · · · u2jn −1 u2pξ −1 which has a number of v2ic −1 ’s and γp q ’s strictly less than in the previous expression. Hence this process terminates when there is no more γpξ qξ covering v2i0 −1 . Now by (21), we may assume v2i0 −1 is not covered by each βpξ qξ . Then by (18) we are able to fix the order among v2i0 −1 , v2i1 −1 , . . . , v2im −1 . Finally the order among the generators of constants of K[U2d , V2d ]δ is fixed by the defining relations (2)-(10). Now we are in the position to prove the main result of this section. δ Theorem 2.4. The K[U2d , V2d ]δ -submodule L of C2d consists of all commutator δ δ elements in C2d ; i.e., the images in (F2d ) ⊂ F2d .
Proof. Let Bpq = βp1 q1 · · · βpr qr , Gp q = γp1 q1 · · · γps qs , Ap q = αp1 q1 · · · αpt qt , Vi = v2i1 −1 · · · v2im −1 , Uj = u2j1 −1 · · · u2jn −1 , and let σ pqp q p q j wp0 q0 Bpq Gp q Ap q Uj π ipqp q p q j a2i0 −1 Vi Bpq Gp q Ap q Uj + be a commutator element. By the metabelian identity we get π ipqp q p q j v2i0 −1 Vi Bpq Gp q Ap q Uj + σ pqp q p q j βp0 q0 Bpq Gp q Ap q Uj = 0. On the other hand the two sums are linearly independent basis elements of the algebra of constants K[U2d , V2d ]δ . Then by Lemma 2.3 we have π ipqp q p q j = pqp q p q j = 0, and σ π ipqp q p q j a2i0 −1 Vi Bpq Gp q Ap q Uj + σ pqp q p q j wp0 q0 Bpq Gp q Ap q Uj = 0,
because the constants π ipqp q p basis elements.
q j
and σ pqp q p
q j
are uniquely determined by the
8
LUCIO CENTRONE, S ¸ EHMUS FINDIK
δ As a consequence of the previous result we get the full list of generators of (F2d ) as a K[U2d , V2d ]-module. δ Corollary 2.5. The K[U2d , V2d ]δ -module (F2d ) is generated by the following polynomials:
g1 (i) = [x2i−1 , x2i ], g2 (i, j) = [x2i−1 , x2j−1 ],
1 ≤ i ≤ d, 1 ≤ i < j ≤ d,
g3 (i, j) = [x2i−1 , x2j ] + [x2j−1 , x2i ],
1 ≤ i < j ≤ d,
g4 (i, p, q) = [x2i−1 , x2p−1 , x2q ] − [x2i−1 , x2p , x2q−1 ], 1 ≤ i ≤ d, 1 ≤ p < q ≤ d, g5 (i, j, k) = [x2i−1 , x2j−1 , x2k ] − [x2i−1 , x2k−1 , x2j ] + [x2j−1 , x2k−1 , x2i ], 1 ≤ i < j < k ≤ d, g6 (i, j, p, q) = [x2i−1 , x2p−1 , x2j , x2q ] + [x2i , x2p , x2j−1 , x2q−1 ] −[x2i−1 , x2p , x2j , x2q−1 ] − [x2i , x2p−1 , x2j−1 , x2q ], 1 ≤ i < j ≤ d, 1 ≤ p < q ≤ d, g7 (i, j, k, l) = x2i [x2j−1 , x2k−1 , x2l ] + x2i−1 [x2j , x2k , x2l−1 ] −x2i [x2j−1 , x2k , x2l−1 ] − x2i−1 [x2j , x2k−1 , x2l ] 1 ≤ i ≤ d, 1 ≤ j < k < l ≤ d, g8 (i, j, k) = x2i [x2j−1 , x2k−1 ] − x2i−1 [x2j , x2k−1 ], 1 ≤ i ≤ d, 1 ≤ j < k ≤ d. We have to add for the generating set of the whole algebra (F2d )δ the constants x2i−1 and x2i−1 x2j −x2i x2j−1 , 1 ≤ i < j ≤ d, which are needed for the generation of the factor algebra of (F2d )δ modulo the commutator ideal of F2d . These generators are the ones lifted from the algebra K[X2d ]δ of constants of the polynomial algebra to the algebra (F2d )δ by the fact stated in Corollary 4.3 of the paper [10] by Drensky and Gupta. The following result gives an infinite generating set of the subalgebra of constants (F2d )δ of the free metabelian associative algebra F2d as an algebra. Corollary 2.6. The algebra (F2d )δ of the constants is generated by x1 , x3 , . . . , x2d−1 , x2i−1 x2j − x2i x2j−1 , g 1 f1 , . . . , g 8 f8 , where 1 ≤ i < j ≤ d, f1 , . . . , f8 ∈ K[U2d , V2d ]δ , and gi s are as in Corollary 2.5.
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
9
3. The variety G generated by the Grassmann algebra The variety G consists of all associative unitary algebras satisfying the polynomial identity [z1 , z2 , z3 ] ≡ 0. We shall set F2l := F2l (G) for 0 ≤ l ≤ d. As noted in the Introduction, G is the variety generated by the infinite dimensional Grassmann algebra G and F2d coincides with the 2d-generated relatively free algebra in the variety G. The identities of G and related topics have been studied by several authors. See for example the paper [19] by Krakovski and Regev about the ideal of polynomial identities of G. Let X = {x1 , x2 , . . . , xd }, Y = {y1 , y2 , . . . , yd } be two disjoint sets of variables. Of course the relatively free algebra of G of rank 2d in the variables from U := X ∪Y is isomorphic to F2d and we consider the following order inside U : x1 < y1 < x2 < · · · < xd < yd . Moreover we say a polynomial f is homogeneous in the set sof variables S = {u1 , . . . , us } if for each monomial m appearing in f we have i=1 degui m is the same. It is well known (see for example Theorem 5.1.2 of [6]) F2d has a basis consisting of all xa1 1 y1b1 · · · xad2d ydb2d [ui1 , ui2 ] · · · [ui2c−1 , ui2c ], ai , bj ≥ 0, , uil ∈ U, ui1 < ui2 < · · · < ui2c , c ≥ 0. We recall the identity [z1 , z2 , z3 ] ≡ 0 implies the identity [z1 , z2 ][z3 , z4 ] ≡ −[z1 , z3 ][z2 , z4 ]
(26)
in F2d . Consider the following Weitzenb¨ock derivation δ of F2d acting on U such that δ(yi ) = xi , δ(xi ) = 0,
1 ≤ i ≤ d.
δ Our goal is to exhibit a finite set of generators for F2d and we are going to prove it using an induction argument similar to the one used by Drensky and MakarLimanov in [11]. Let α = (α1 , · · · , αd−1 ) ∈ K d−1 and consider the algebra endomomorphism φα of F2d such that
φα (xi ) = xi , φα (yi ) = yi , i = 1, . . . , d − 1, φα (xd ) =
d−1
αi xi , φα (yd ) =
i=1
d−1
α i yi .
i=1
δ δ Notice that φα commutes with δ. Hence if f ∈ F2d , then φα (f ) ∈ F2d−2 too. We have the next result on the purpose and, as above, we consider F2l as generated by the disjoint sets {x1 , . . . , xl } and {y1 , . . . , yl }.
Lemma 3.1. Let d ≥ 2 and f ∈ F2d being homogeneous with respect to the set {xd , yd }. If φα (f ) = 0 for some non-zero α ∈ K d−1 , then f is in the left ideal generated by d−1 d−1
α i xi yd − αi yi xd , [ωα , u], u ∈ U, ωα := i=1
μα := xd ,
i=1 d−1 i=1
αi xi , να := yd ,
d−1 i=1
α i yi
10
LUCIO CENTRONE, S ¸ EHMUS FINDIK
Proof. Before starting the proof remark φ(ωα ) = φ(μα ) = φ(να ) = 0. We shall prove the assertion by induction on degxd ,yd f . Indeed the result is trivial if degxd ,yd f = 0, so we may assume the result true for degxd ,yd f > 0. Let f be as above, then we can write f as ap,u0 [u0 , yd ]ydp−1 + ap−1 ydp−1 xd + ap−1,u1 [u11 , xd ]ydp−1 f = ap ydp + 1
u0
+
u11
ap−1,u2 [u21 , yd ]ydp−2 xd + ap−1 [xd , yd ]ydp−2 + · · · + a0 xpd +
1
a0 [up , xd ]xp−1 , d
up
u21
where the uij ’s are in U whereas the aj ’s, the ar,ui ’s, the as,ui and the at,ui ’s all j j j belong to F2d−2 . Let p ≥ 3 and let us denote g1 := ap ydp−1 + ap,u0 [u0 , yd ]ydp−2 + ap−1,u1 [u11 , xd ]ydp−2 + ap−1 [xd , yd ]ydp−3 1
u0
u11
and g2 := f − g1 yd . Remark g2 can be written as
g2 = g2 xd
for some polynomial g2 ∈ F2d . We consider now d−1
α i xi f i=1
= g1 ωα + g 1
d−1
α i yi
xd −
i=1
ap ydp−1
= g1 ω α −
α i yi
αi [xi , yd ]
i=1
xd −
i=1
d−1
+
g2∗ xd
+
d−1
[xi , yd ]
i=1
αi g2 [xd , xi ]
i=1
+ g2 xd +
d−1
αi g2 [xd , xi ] ,
i=1
d−1 αi yi + g2∗ and g2∗= g2 i=1 αi xi . Because p − 1 > 0 we may d−1 apply the same argument above to f i=1 αi xi . Then using (26) we have
where g2 = g1
d−1
d−1
d−1
f
i=1
d−1 i=1
2 α i xi
= h1 ωα +
d−1
h1 [ωα , xi ] + h2 xd + h3 μα .
i=1
Therefore we get φα (h2 xd ) = 0, i.e., φα (h2 ) = 0 because clearly φα (xd ) = 0. Now we can apply induction on h2 2 d−1 obtaining f is in the left ideal generated by ωα , [ωα , u], u ∈ U , μα i=1 αi xi d−1 and να . Also remark ωα , [ωα , u], μα and να are irreducible in F2d and i=1 αi xi does not depend on xd , yd , then we conclude f is as desired and we are done. The d−1 other cases may be treated similarly also multiplying everything by i=1 αi yi . The next is an easy consequence of Lemma 3.1 and its proof follows verbatim the one of Corollary 2 of [11].
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
11
Corollary 3.2. If φα (f ) = 0 for all non-zero α = (α1 , . . . , αd−1 ) ∈ K d−1 , then f = 0 if d > 2 and f is in the left ideal generated by v12 := x1 y2 − y1 x2 and [v12 , u], where u ∈ U , if d = 2. We define the following objects inside F2d : vij := xi yj − yi xj ,
1 ≤ i, j ≤ d,
wijk := yi [xj , xk ] − xi [yj , xk ], 1 ≤ i, j, k ≤ d, zijkl := yi [xj , vk,l ] − xi [yj , vk,l ], 1 ≤ i, j, k, l ≤ d, Notice that the vij ’s, the wijk ’s and the zijkl ’s are constants of F2d and starting from these objects we shall construct three subsets of elements of F2d . We set V := {vij |1 ≤ i, j ≤ d}, W0 := {wijk |1 ≤ i, j ≤ k ≤ d} and Z0 := {zijkl |1 ≤ i ≤ j ≤ k ≤ l ≤ d}. Suppose s > 0 and we set Ws := {y[x, w] − x[y, w]|w ∈ Ws−1 , x ∈ X, y ∈ Y }, Zs := {y[x, z] − x[y, z]|z ∈ Zs−1 x ∈ X, y ∈ Y }. As above, both the Ws ’s and the Zs ’s are subsets of constants. Moreover it can be easily seen for s ≥ d both Vs and Ws are 0. We shall denote by C the algebra generated by the non-zero X, V, Wl , Zl . Remark 3.3. We have [v12 , x2 ] = x1 v22 + w212 . The next easy-to-prove relation is crucial in the sequel: b(b − 1) b−2 δ(y b ) = bxy b−1 + y [y, x]. 2 Before writing the proof of the main result we still need another technical lemma. Lemma 3.4. For every p ≥ 1 we have xk1 · · · xkp ydp = c + g1 (x1 , y1 , . . . , xd , yd )xd + g2 (x1 , y1 , . . . , xd , yd )[xd , yd ], where c ∈ C and g1 (x1 , y1 , . . . , xd , yd ), g1 (x1 , y1 , . . . , xd , yd ) ∈ F2d . Proof. We shall prove the result by induction on p. If p = 1, then xk yd = vkd +yk xd and we are done because vkd ∈ V . Suppose the result true for p ≥ 1 and let us prove it for p + 1. Notice that xk1 · · · xkp+1 ydp+1 = xk1 (xk2 · · · xkp+1 ydp )yd and we apply induction on the term inside the parenthesis. We have now xk1 · · · xkp+1 ydp+1 = xk1 (c + g1 xd + g2 [xd , yd ])yd = xk1 cyd + xk1 g1 xd yd + xk1 g2 yd [xd , yd ] = cxk1 yd + [xk1 , c]yd + xk1 g1 yd xd + g[xd , yd ]. Now observe that xk1 yd = vkd + yk xd and [xk1 , c]yd = [xk1 , c]yd − [yk1 , c]xd − [yk1 , c]xd = ω − [yk1 , c]xd ,
12
LUCIO CENTRONE, S ¸ EHMUS FINDIK
where ω := [xk1 , c]yd − [yk1 , c]xd belongs to C too and we are done.
Remark 3.5. We shall use in the sequel the following relation [xk1 , c]yd = ω − [yk1 , c]xd , where c, ω ∈ C. δ is finitely generated as an algebra, Theorem 3.6. The algebra of constants F2d and its generators are:
(27)
X;
(28)
V,
(29)
Wl , l ≤ d − 1;
(30)
Zl , l ≤ d − 1.
Proof. It is easy to see that the elements from (27)-(30) are constants of F2d . δ δ , then its U -homogeneous components lie in F2d Let f = f (X , Y , xd , yd ) ∈ F2d too. Hence we may assume f being homogeneous in U . Arguing analogously we can take f being homogeneous in {xd , yd }. We shall prove the theorem by induction on d and on the total degree with respect to the set {xd , yd }. Suppose d = 1, then f=
n
αk xk y n−k +
n−1
βj xj−1 y n−j−1 [x, y].
j=1
k=0
This means 0 = δ(f ) =
n−1
αk (n − k)xk+1 y n−k−1
k=0
+[x, y]
n−2 k=1
n − k k−1 n−k−1 x (n − k − 1) αk y + βk xk y n−k−2 . 2
The previous relation gives us αk = 0 for k ≤ n − 1, then βj = 0 for 1 ≤ j ≤ n − 2. Thus the only possibility is f = αn xn + βn−1 xn−2 [x, y] and we are done. Assume now d > 1 and the result true for F2d−2 . We recall F2d−2 is generated by the sets of variables X = {x1 , x2 , . . . , xd−1 }, Y = {y1 , y2 , . . . , yd−1 } and we set δ is homogeneous in U , then degX (v) ≥ degY (v) and if U = X ∪ Y . If v ∈ F2d−2 δ degxd ,yd (f ) = 0, then f ∈ F2d−2 , so we shall study only the case degxd ,yd (f ) > 0. Remembering that inside F2d−2 every commutator belongs to the center of F2d−2 , we can write f in the following way: ap,u0 [u0 , yd ]ydp−1 + ap−1 xd ydp−1 + ap−1,u1 [u11 , xd ]ydp−1 f = ap ydp + 1
u0
+
u11
ap−1,u2 [u21 , yd ]xd ydp−2 + ap−1 [xd , yd ]ydp−2 + · · · + a0 xpd +
1
u21
up
a0 [up , xd ]xp−1 , d
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
13
where the uij ’s are in U whereas the aj ’s, the ar,ui ’s, the as,ui and the at,ui ’s all j j j belong to F2d−2 . We derive f and we obtain (31)
+
δ(f ) = δ(ap )ydp + pap xd ydp−1 +
p(p − 1) ap [yd , xd ]ydp−2 2
δ(ap,u0 )[u0 , yd ]ydp−1 + ap,u0 [δ(u0 ), yd ]ydp−1 + ap,u0 [u0 , xd ]ydp−1
u0
+
+(p − 1)ap,u0 [u0 , xd ]xd ydp−2 + δ(ap−1 )xd ydp−1 + (p − 1)ap−1 x2d ydp−2 (p − 1)(p − 2) δ(ap−1,u1 )[u11 , xd ]ydp−1 ap−1 [yd , xd ]xd ydp−3 + + 1 2 u11 +ap−1,u1 [δ(u11 ), xd ]ydp−1 + (p − 1)ap−1,u1 [u11 , xd ]xd ydp−2 1
δ(ap−1,u2 )[u21 , yd ]xd ydp−2 1
+
1
ap−1,u2 [δ(u21 ), yd ]xd ydp−2 1
+ ap−1,u2 [u21 , xd ]xd ydp−2 1
u21
+(p − 2)ap−1,u2 [u21 , yd ]x2d ydp−3 + δ(ap−1 )[yd , xd ]ydp−2 + (p − 2)ap−1 [yd , xd ]xd ydp−3 1 p−1 + · · · + δ(a0 )xpd + + a [δ(u ), x ]x δ(a0 )[up , xd ]xp−1 =0 p d d 0 d up
By (31) we immediately get δ(ap ) = 0, hence: δ . ap ∈ F2d
(32) We also have (33)
p(p−1) ap + δ(ap−1 ) 2
= 0, then by (32) we get δ(ap−1 ) = 0. This means:
δ ap−1 ∈ F2d .
Suppose now u0 = ys , s ≤ d − 1, then ap,u0 = 0. So we can assume u0 = xs , s ≤ d − 1. In this case we get δ(ap,xs ) = 0, i.e.: (34)
δ ap,xs ∈ F2d .
By (32), (33) and (34), using induction, we get ap = b1 (X , V , W , Z ), b2 (X , V , W , Z ), ap−1 = b3,i (X , V , W , Z ). ap,xi = If φα (f ) = 0 for all α = (α1 , . . . , αd−1 ), then by Corollary 3.2 we get d = 2 and f is in the left ideal generated by v12 and [v12 , u], u ∈ U . Hence f = gv12 + u∈U gu [v12 , u] and, after a simple manipulation, we apply inductive arguments to g and gu ’s and we are done. Now we consider the case φα (f ) = 0 for some non-zero α ∈ K n−1 . The next two remarks are crucial. First we have degX (f ) = degX (φα (f )) ≥ degY (φα (f )) = degY (f ). Hence degX (f ) = degX (ap ) ≥ degY (f ) = p + degY (ap )
14
LUCIO CENTRONE, S ¸ EHMUS FINDIK
so ap =
b1 (X , V , W , Z )xk1 · · · xkp .
We can argue analogously and obtain b2 (X , V , W , Z )xl1 · · · xlp , ap−1 = b3,i (X , V , W , Z )xm1 · · · xmp . ap,xi =
Now we apply Lemma 3.4 and Remark 3.5 in order to rewrite ap ydp , ap−1 [xd , yd ]ydp−2 and ap,xi [xi , yd ]ydp−1 and we get f = c + gxd +
d−1
mp,i yd [xi , xd ],
i=1
where c ∈ C. We can rewrite the last summand as g1 xd + degxd hi = 0 and we get finally f = c + g xd +
d−1
d−1 i=1
hi [xi , xd ], where
hi [xi , xd ].
i=1
Hence g and the hi ’s are constants too because xd and [xi , xd ] are. Now we are allowed to apply induction on g and the hi ’s because their degrees with respect to the set {xd , yd } are strictly smaller than the one of f and we conclude the proof. 4. Acknowledgements The authors are very thankful to the referee who found a mistake in the proof of the main theorem in the first version of the paper. References [1] Yu. A. Bahturin, Identical Relations in Lie Algebras (Russian), Nauka, Moscow, 1985. Translation: VNU Science Press, Utrecht, 1987. [2] L. Bedratyuk, A note about the Nowicki conjecture on Weitzenb¨ ock derivations, Serdica Math. J. 35 (2009), 311-316. [3] R. Dangovski, V. Drensky, S ¸ . Fındık, Weitzenb¨ ock derivations of free metabelian Lie algebras, Linear Algebra Appl., 439 (10) (2013), 3279-3296. [4] R. Dangovski, V. Drensky, S ¸ . Fındık, Weitzenb¨ ock derivations of free associative algebras, Journal Algebra Appl., 16 (03) (2017), 1750041/1-26. [5] H. Derksen, G. Kemper, Computational Invariant Theory, Encyclopaedia of Mathematical Sciences, Invariant Theory and Algebraic Transformation Groups 130, Springer-Verlag, Berlin, 2002. [6] V. Drensky, Free Algebras and PI-Algebras, Springer, Singapore, 1999. [7] V. Drensky, Invariants of unipotent transformations acting on noetherian relatively free algebras, Serdica Math. J. 30 (2004), 395–404. [8] V. Drensky, Another proof of the Nowicki conjecture, arXiv:1902.08758 [math.AC]. [9] V. Drensky, S ¸ . Fındık, The Nowicki conjecture for free metabelian Lie algebras, Int. J. Algebra Comp. (to appear). [10] V. Drensky, C.K. Gupta, Constants of Weitzenb¨ ock derivations and invariants of unipotent transformations acting on relatively free algebras, J. Algebra 292 (2) (2005), 393-428. [11] V. Drensky, L. Makar-Limanov, The conjecture of Nowicki on Weitzenb¨ ock derivations of polynomial algebras, J. Algebra Appl. 8 (2009), 41-51. [12] A. Giambruno, M. V. Zaicev, A characterization of varieties of associative algebras of exponent two, Serdica Math. J. 26 (2000), 245-252. [13] A. Giambruno, M. V. Zaicev, Exponential codimension growth of P.I. algebras: an exact estimate, Adv. Math. 142 (1999), 221-243.
THE NOWICKI CONJECTURE FOR RELATIVELY FREE ALGEBRAS
15
[14] A. Giambruno, M. V. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145-155. [15] A. R. Kemer, Ideals of Identities of Associative Algebras, Translations of Math. Monographs 84, AMS, Providence, RI, 1991. [16] A.R. Kemer, Finite basis property of identities of associative algebras, Algebra Logic 26 (5) (1987), 362-397. [17] J. Khoury, Locally Nilpotent Derivations and Their Rings of Constants, Ph.D. Thesis, Univ. Ottawa, 2004. [18] J. Khoury, A Groebner basis approach to solve a conjecture of Nowicki, J. Symbolic Comput. 43 (2008), 908-922. [19] D. Krakovski, A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973), 429-438. [20] K. Kurano, Positive characteristic finite generation of symbolic Rees algebras and Roberts’ counterexamples to the fourteenth problem of Hilbert, Tokyo J. Math. 16 (2) (1993), 473-496. [21] S. Kuroda, A simple proof of Nowicki’s conjecture on the kernel of an elementary derivation, Tokyo J. Math. 32 (2009), 247-251. [22] J. Lewin, A matrix representation for associative algebras I, Trans. Amer. Math. Soc. 188 (1974), 29-308. [23] A. Nowicki, Polynomial Derivations and Their Rings of Constants, Uniwersytet Mikolaja Kopernika, Torun, 1994. www-users.mat.umk.pl/˜anow/ps-dvi/pol-der.pdf. [24] P. Roberts, An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert’s fourteenth problem, J. Algebra 132 (2) (1990), 461-473. [25] A.L. Shmel’kin, Wreath products of Lie algebras and their application in the theory of groups (Russian), Trudy Moskov. Mat. Obshch. 29 (1973), 247-260. Translation: Trans. Moscow Math. Soc. 29 (1973), 239-252. [26] B. Sturmfels, Algorithms in Invariant Theory. 2nd ed. Texts and Monographs in Symbolic Computation, Springer-Verlag, Wien, 2008. ¨ [27] R. Weitzenb¨ ock, Uber die Invarianten von linearen Gruppen, Acta Math. 58 (1932), 231-293. [28] M. V. Zaitsev, Integrality of exponents of growth of identities of finite-dimensional Lie algebras, Izv. Math. 66 (2002), 463-487. IMECC, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda 651,13083859, Campinas (SP), Brazil Email address: [email protected] Department of Mathematics, C ¸ ukurova University, 01330 Balcalı, Adana, Turkey Email address: [email protected]