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Invariant differential operators in positive characteristic
Journal of Algebra 499 (2018) 281–297
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Journal of Algebra www.elsevier.com/locate/jalgebra
Invariant differential operators in positive characteristic Sofiane Bouarroudj a,∗ , Dimitry Leites a,b a
New York University Abu Dhabi, Division of Science and Mathematics, P.O. Box 129188, United Arab Emirates b Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
a r t i c l e
i n f o
Article history: Received 13 June 2016 Available online 19 December 2017 Communicated by Vera Serganova MSC: 17B50 17B55 17B56 17B20 17B40 Keywords: Veblen’s problem Invariant differential operator Positive characteristic
a b s t r a c t In 1928, at the IMC, Veblen posed the problem: classify invariant differential operators between spaces of “natural objects” (in modern terms: either tensor fields, or jets) over a real manifold of any dimension. The problem was solved by Rudakov for unary operators (no nonscalar operators except the exterior differential); by Grozman for binary operators. In dimension one, Grozman discovered an indecomposable selfdual operator of order 3 that does not exist in higher dimensions. We solve Veblen’s problem in the 1-dimensional case over any field of positive characteristic. Unary invariant operators are known: these are the exterior differential and analogs of the Berezin integral. We construct new binary operators from these analogs and discovered two more (up to dualizations) types of new indecomposable operators of however high order: analogs of the Grozman operator and a completely new type of operators. Gordan’s transvectants, aka Cohen–Rankin brackets, always invariant with respect to the simple 3-dimensional Lie algebra, are also invariant, in characteristic 2, with respect to the whole Lie algebra of vector fields on the line when the height of the indeterminate is equal to 2. © 2017 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail addresses: sofi[email protected] (S. Bouarroudj), [email protected] (D. Leites). https://doi.org/10.1016/j.jalgebra.2017.11.048 0021-8693/© 2017 Elsevier Inc. All rights reserved.
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1. Introduction 1.1. Discussion of open problems In the representation theory of a given algebra or group, the usual ultimate goal is a clear description of its irreducible modules to begin with, description of indecomposables being the next step. Other goals are sometimes also natural and reachable, cf. those discussed here and in [1]. The history of mathematics shows that problems with a nice (at least in some sense, for example, short) answer often turn out to be “reasonable”. I. Gelfand used to say that, the other way round, easy-to-formulate lists are answers to “reasonable” problems and advised to try to formulate such problems once we got a short answer. Hereafter, K is any field of characteristic p > 0, unless otherwise specified. Speaking of Lie algebras or superalgebras, and their representations over K, which of them is it natural to consider? In [18], Deligne suggests that one should begin with restricted ones: unlike nonrestricted ones, the restricted Lie (super)algebras correspond to groups, in other words: to geometry. Certain problems concerning nonrestricted algebras are also natural (although tough), and have a short answer, e.g., classification of simple Lie algebras for p > 3 (for its long proof, see [26,2]). Rudakov and Shafarevich [24] were the first to describe ALL, not only restricted, irreducible representations of sl(2) for p > 2. Dolotkazin solved the same problem for p = 2, see [10]; more precisely, he described the irreducible representations of the simple 3-dimensional Lie algebra o(1) (3) = [o(3), o(3)]. The difference of Dolotkazin’s problem from that considered in [24] is that, unlike sl(2) for p > 2, the Lie algebra o(1) (3) is not restricted. These two results show that the description of all irreducible representations looks feasible, to an extent, at least for Lie (super)algebras with indecomposable Cartan matrix or their simple “relatives” (for the classification of both types of Lie (super)algebras over K, see [7]). 1.1.1. Veblen’s problem In 1928, at the IMC, O. Veblen formulated a problem, later reformulated in more comprehensible terms by A. Kirillov and further reformulated as a purely algebraic problem by J. Bernstein who interpreted Rudakov’s solution of Veblen’s problem for unary operators; for setting in modern words and review, in particular, for a superization of Veblen’s problem, see [15]. Let vect(m) be the Lie algebra of polynomial vector fields (over a ground field of characteristic 0, say C). For the definition of the vect(m)-module T (V ) of (formal) tensor fields of type V , where V is a gl(m)-module with lowest weight vector, see [15]. Let us briefly recall the results concerning the nonscalar unary and binary invariant differential operators between spaces of tensor fields, although we only need the simplest version of spaces T (V ), namely, the spaces of weighted densities Fa := T (a tr), where tr is the 1-dimensional gl(m)-module given by the trace (supertrace for gl(m|k)) and a ∈ C (or
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another ground field). In what follows we assume that modules T (V ) are coinduced from irreducible modules V ; this is certainly so for modules Fa . (In Veblen’s problem over C, it suffices to consider only irreducible modules V ; for its superization and over K, one has to consider indecomposable modules V .) 1.1.1.1. Unary operators on manifolds and supermanifolds The problem we are solving here for p > 0 resembles its super version for p = 0; we recall the known results. The only vect(m)-invariant differential operators T (V ) −→ T (W ) are the exterior differential operators (so, practically, there is just one such operator) in the de Rham complex, where Ωi = T (E i (id)), E i is the functor of raising to the ith exterior power and id is the tautological m-dimensional gl(m)-module: d
d
d
d
0 −→ C −→ Ω0 −→ Ω1 −→ · · · −→ Ωm−1 −→ Ωm −→ 0.
(1)
In super setting, where sdim(id) = m|k, the analog of the complex (1) is infinite without the “top term” d
d
d
0 −→ C −→ Ω0 (m|k) −→ Ω1 (m|k) −→ Ω2 (m|k) −→ . . . ,
(2)
its dualization brings the complex of integrable (because it is possible to integrate them; Deligne calls them integral) forms which in the purely odd case of superdimension 0|k terminates with an order-k differential operator called the Berezin integral which to a given function f (θ) assigns its coefficient of the highest term, where Σ−i (m|k) := (Ωi (m|k))∗ : d
d
d
· · · −→ Σ−1 (m|k) −→ Σ0 (m|k) −→ 0 d
d
for m = 0,
(3)
· · · −→ Σ−1 (0|k) −→ Σ0 (0|k) −→ K −→ 0 for m = 0. Over C, the de Rham complex (2) is exact, while the complex of integrable forms (3) has 1-dimensional cohomology in the (−m)th term; the cohomology is spanned, see [3, 23,9], by (here the ui are the even indeterminates, the θj are the odd ones for clarity; in Notation 2.1 all indeterminates are called u) θ1 . . . θk ∂u1 . . . ∂um vol(u|θ)
(4)
and the super version of the volume element vol(u) = du1 ∧ · · · ∧ dum is the class vol(u|θ) of the element du1 ∧ · · · ∧ dum ⊗ ∂θ1 ∧ · · · ∧ ∂θk
(5)
in the indecomposable gl(V ) = gl(m|k)-module generated by the element (5) modulo the codimension 1 submodule, see [9,20]. (There are interesting generalizations to pseudod-
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ifferential and pseudointegral forms, but they have no analog for p > 0 we are interested in this note.) Simple Lie (super)algebras of vector fields with polynomial or formal coefficients constitute a class of interesting Lie (super)algebras. Over C and R, they are classified and their continuous (in the x-adic topology) irreducible representations with a vacuum (highest or lowest) weight vector are classified as well in the following cases: • For the four series of simple Lie algebras, and their direct super versions, the answer (if dim V < ∞) is as follows, see the review [15]: the module T (V ) of tensor fields coinduced from any irreducible module V with lowest weight vector over the subalgebra of linear vector fields is irreducible, unless T (V ) Ωk , the module of exterior k-forms. Rudakov proved that the exterior differential is the only (nonscalar) operator between the modules of exterior differential forms invariant with respect to all changes of indeterminates, or, equivalently, for the general vectorial Lie algebra vect. Together with the Poincaré lemma (stating that the sequence (1) is exact over any star-shaped domain) this yields a complete description of continuous irreducible modules with lowest weight. For simple Lie algebras of hamiltonian and contact vector fields (and analogous simple Lie superalgebras in their standard grading) the answer is essentially the same, except that one should consider not the whole spaces Ωk (or their dual) but their subspaces of “primitive” forms (for details, see [15]). In the purely odd case, the Berezin integral has to be added to the list of invariant differential operators, see eq. (3). • For simple vectorial Lie superalgebras in their nonstandard grading (listed in [22]), and for the modules of tensor fields with infinite-dimensional fibers (i.e., dim V = ∞) even over finite-dimensional simple vectorial Lie superalgebras, many never previously known differential operators appear, see [15,19]. • Over K, it is tempting to conjecture that, at least for p “sufficiently large”, the description of irreducible modules over vectorial Lie algebras (at least Z-graded, and “close to simple” ones) would resemble that given by Rudakov over C. For irreducible restricted modules over restricted simple vectorial Lie algebras of the 4 classical series and p ≥ 3, this is almost so, i.e., the spaces of differential k-forms are the only reducible spaces of tensor fields, although the complex of exterior differential forms is not exact, see Theorem 2.2.1 ([17]). 1.1.1.2. Binary operators on manifolds These are classified by P. Grozman, see [13]. Recall that the formal dual of T (V ) on the m-dimensional space is T (V ∗ ⊗ tr). Hence, with every invariant operator D : T (V ) −→ T (W ) its dual D∗ : T (W ∗ ⊗tr) −→ T (V ∗ ⊗tr) is also invariant. In particular, on the line, with every operator D : Fa −→ Fa+deg D of order deg D, there exists its dual D∗ : F1−a−deg D −→ F1−a of the same order. Similarly, with every binary operator D : Fa ⊗ Fb −→ Fa+b+deg D , there exist its two duals D∗1 : F1−a−b−deg D ⊗ Fb −→ F1−a and D∗2 : Fa ⊗ F1−a−b−deg D −→ F1−b , as
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well as the operator r(D) : Fb ⊗ Fa −→ Fa+b+deg D , given by the interchange of the arguments r(D)(X, Y ) : = D(Y, X) for any X ∈ Fb and Y ∈ Fa . Summary: Up to dualizations and interchange of the arguments, there are 8 series of invariant operators of order 1, all invariant differential operators of order 2 and 3 are compositions of order-1 operators, except for the case m = 1, where there is an indecomposable operator Gz of order 3 discovered by Grozman; for explicit expressions, see [13] and Subsec. 2.4. There are no operators of order > 3. Aside remarks: 1) An interesting fact. Among all 8 series of invariant operators of order 1, there are 4 series that determine an associative algebra and 3 series determine a Lie (super)algebra structure on their domain of definition. These Lie (super)algebras are either simple or simple modulo center or contain a simple algebra of codimension 1: the corresponding products are the Poisson bracket, the Schouten bracket (more popular now under the name anti-bracket) and a deformed Schouten bracket, see [13,21]. 2) Kirillov gave an example of the symbol of an invariant with respect to all coordinate changes nonlocal bilinear operator on the circle but did not identify the operator itself; in [16] this operator is described in terms of the integral=residue; it is the only nonlocal invariant bilinear operator on the circle. In characteristic p > 0, same as on superpoints, analogs of various operators nonlocal over R become local because if the space of functions is finite-dimensional, then any operator in this space can be viewed as differential, i.e., composition of derivations and multiplications by some functions. 1.1.1.3. Invariant differential operators of higher arity The only k-ary invariant operators classified so far are antisymmetric ones on C, see [11], and ternary operators between the spaces of weighted densities on Cn , see [4]. 1.2. Main result The classification (Theorem 2.6) of binary vect(1; N )-invariant operators between modules of weighted densities over K; in particular, we discovered three series of indecomposable operators of however high order: 1) related to the analog of Berezin integral, 2) an analog of the Grozman operator, and 3) a completely new series of operators, Bj. If p = 2, Gordan’s transvectants, always o(1) (3)-invariant, are vect(1; 2)-invariant. 1.3. Open problems 1) We conjecture that the Grozman operator has something to do with the contact structure since vect(1) = k(1); see also Conjecture 4) below. Over C, it seems feasible to classify binary k(2n + 1)-invariant differential operators for n > 0 and complete the
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partial result for binary h(2n)-invariant differential operators, solved only for n = 1, see [14]. 2) The only feasible super version of the problem we consider here is, it seems, classification of the k(1; N |k)-invariant operators for k = 1 or 2 when the coinduced modules with lowest weight vectors are of the form Fa . For a (non-trivial if p = 2) description of the Lie superalgebras k(1; N |k), see [6]. 3) There are natural filtered deforms of simple nonrestricted vectorial Lie algebras; besides, for p ≤ 5, there are new types of simple vectorial Lie algebras and no description of their irreducible modules is known; in particular, no description of intertwining operators between modules of tensor fields — the most natural ones over these Lie algebras. 4) Conjecture. The only vect(m; N |k)-invariant unary operators between modules of tensor fields coinduced from irreducible gl(m|k)-modules are the exterior operator d and the depending on N analogs of the Berezin integral . 2. Veblen’s problem over fields of characteristic p > 0 2.1. Notation Introduce parity in the ring of polynomials K[x], where x = (x1 , . . . , xa ) and a = m+k, by setting p(xi ) = ¯0 for i ≤ m and p(xi ) = ¯1 for m < i ≤ a. For any a-tuple of nonnegative integers r = (r1 , . . . , ra ), where ri = 0 or 1 for i > m, set (ri )
ui
:=
xri i ri !
and u(r) :=
(ri )
ui
.
1≤i≤a
The idea is to formally replace fractions with ri ! in denominators by inseparable symbols (r ) ui i which are well-defined over fields of characteristic p; in particular, if k = 0, the factor in the first parentheses below is equal to 1:
u(r) · u(s) = where
r+s r
min(1, 2 − ri − si ) · (−1)m
m+1≤i≤a
:=
1≤i≤m
rj si
·
r+s r
u(r+s) ,
ri +si . ri
For an m +k-tuple of positive integers N = (N1 , ..., Nm , 1, . . . , 1), set N ev = (N1 , ..., Nm ). The following supercommutative superalgebra is the “correct” for p > 0 analog of the algebra of functions: O(m; N ev |k) := K[u; N ] := SpanK (u(r) | ri < pNi for i ≤ m, 0 ≤ ri ≤ 1 for i > m). The Lie superalgebra vect(m; N |k) of its distinguished derivations consists of the vector fields of the form fi (u)∂i , where fi ∈ O(m; N ev |k) and ∂i are distinguished partial (r ) (r −1) derivatives defined by the condition ∂j (ui i ) = δji ui i .
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2.2. Unary operators: De Rham complex for k = 0 (no super) Set τ (N ) = (pN1 − 1, . . . , pNm − 1). Denote: Z i (m; N ) := {ω ∈ Ωi (m; N ) | dω = 0} and B i (m; N ) := {dω | ω ∈ Ωi−1 (m; N )}. 2.2.1. Theorem ([17,25]). The sequence d
d
d
d
d
0 −→ K −→ Ω0 (m; N ) −→ Ω1 (m; N ) −→ Ω2 (m; N ) −→ · · · −→ Ωm (m; N ) −→ 0 is not exact: the space H a (m; N ) := Z a (m; N )/B a (m; N ) is spanned by the elements (τ (N )i1 )
ui1
(τ (N )ik )
. . . uik
dui1 . . . duik where a = i1 + · · · + ik .
. The supercommutative superalgebra H (m; N ) is generated by H 1 (m; N ). 2.2.2. Corollary. There is an invariant differential operator of order |τ (N )| := ( pN i ) − m:
Ωm (m; N ) −→ K; this, depending on N , analog of the Berezin integral is defined as follows:
(τ (N )1 )
f (u) vol(u) = the coefficient of uτ (N ) = u1
(N )m ) . . . u(τ , m
where vol(u) = du1 ∧ · · · ∧ dum . 2.3. Unary operators: De Rham complex for k = 0, i.e., in super setting On the m|k-dimensional space over K, the de Rham complex goes ad infinitum d
d
d
0 −→ K −→ Ω0 (m; N |k) −→ Ω1 (m; N |k) −→ Ω2 (m; N |k) −→ . . .
(6)
The dual to (6) complex of integrable forms is as follows, where Σ0 := Ω0 vol(u|θ): d
d
· · · −→ Σ−1 (m; N |k) −→ Σ0 (m; N |k) −→ K −→ 0.
(7)
Observe that, for p > 0, the integral is defined for m = 0 as well, whereas for p = 0, there is no such an invariant differential operator.
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2.4. The binary differential operators over C for m = 1 (recapitulation, see [13]) For m = 1, we have Fa := {f (u)(du)a | f (u) ∈ Ω0 = F0 } on which the Lie derivative acts as follows: LD (f (u)(du)a ) = (D(f ) + af g )(du)a ) for any D = g(u)∂u ∈ vect(1).
(8)
For any bilinear differential operator
D : Fa ⊗ Fb −→ Fa+b+deg D ,
f (du)a ⊗ g(du)b −→ D(f, g)(du)a+b+deg D ,
it suffices to indicate (a, b) and D(f, g). The operators are listed up to proportionality. 2.4.1. Order 1 operators For generic values (a, b), the invariant operators form a 1-dimensional space. For (a, b) = (0, 0), and only in this case, we have a 2-dimensional space of operators.
(a, b) (0, 0)
D(f, g)
comment
P00 : αf g + βf g
for any α, β ∈ C
(a, b)
{·, ·}P.B. : = af g − bf g
Poisson bracket in two even indeterminates u and du
(−1, −1)
{·, ·} : = f g − f g
contact bracket of generating functions d f −→ Kf := f du (9)
Clearly, the bottom line is a particular case of the penultimate line; we singled it out because in higher dimensions the contact bracket is not a particular case of the Poisson bracket. 2.4.2. Order 2 operators
(a, b) (0, b) (a, 0) (a, −1 − a)
D(f, g) f g − bf g af g − f g af g + (2a + 1)f g + (a + 1)f g
comment {df, g(du)b }P.B. {f (du)a , dg}P.B. ({f (du)a , g(du)−1−a }P.B. )
(10)
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2.4.3. Order 3 operators (a, b) (0, 0) (−2, 0) (0, −2) 2 2 −3, −3
D(f, g) T1 (f, g) = {f , g } T1∗1 (f, g) = f g + 3f g + 2f g T1∗2 (f, g) = f g + 3f g + 2f g f f g + 3 det Gz(f, g) = 2 det f g f
(11) g g
2.5. New binary differential operators: p > 0 and m = 1 For a fixed N = N , we list vect(1; N )-invariant operators up to interchange of arguments. Recall that : Ω1 = F1 −→ K ⊂ F0 is the order pN − 1 operator, see Corollary 2.2.2 and 1 − pN ≡ 1 (mod p). Therefore, for any p > 0 and N , there are vect(1; N )-invariant bilinear operators acting in the spaces Fa ⊗ Fb in the following cases: operator(a, b) ⊗ id(1, a), ( ⊗ id)∗1 (1 − a, a), ( ⊗ id)∗2 (1, 1 − a) ⊗d = ( ⊗d)∗2 (1, 0), ( ⊗d)∗1 (0, 0), r( ⊗d)(0, 1) ∗2 ∗1 ⊗ = ( ⊗ ) = ( ⊗ ) (1, 1)
order k pN − 1 pN 2(pN − 1)
case 1 2 3
(12)
The operator ( ⊗ id)∗1 (1 − a, a) is of the form (L for “long” expression) (k−1)/2
Lp,N (f, g) =
(−1)i f (i) g (k−i) + f (k−i) g (i)
i=0
+
(−1)k/2 f (k/2) g (k/2)
if p > 2 if p = 2.
(13)
2.5.1. Lemma. For (a, b) = (1, 1), there exists a vect(1; N )-invariant bilinear operator Bjp,m of order k = pm − 1 for any m ≤ N given by the expression Bjp,m (f, g) = det
f
g
f (k)
g (k)
.
(14)
Its duals for (a, b) = (1, 0) and (0, 1), respectively, are as follows: (k−1)/2 ∗2 Bjp,m (f, g) = f g (k) +
(−1)i f (i) g (k−i) + f (k−i) g (i)
i=1
+
(−1)k/2 f (k/2) g (k/2)
if p > 2 if p = 2
(15)
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and (k−1)/2 ∗1 Bjp,m (f, g)
=f
(k)
g+
(−1)i f (i) g (k−i) + f (k−i) g (i)
i=1
+
(−1)k/2 f (k/2) g (k/2)
if p > 2 if p = 2.
(16)
Proof. Every order k bilinear differential operator D : Fa ⊗ Fb −→ Fa+b+deg D is of the form ⎛ ⎞
(f (dx)a , g(dx)b ) −→ ⎝ αi,j f (i) g (j) ⎠ (dx)c , where αi,j ∈ K. (17) i+j=k
For N big enough (such that pN > deg D + 1) and a given pair (a, b), the vect(1; N )-invariance of D implies the following system of equations Ci,r for i = 1, . . . , k and 2 ≤ r ≤ k + 1: Ci,r
i i : a + αi,k−i r−1 r k−i+r−1 k−i+r−1 + b + αi−r+1,k−i+r−1 = 0. r−1 r
(18)
In the case of Lemma, (a, b) = (1, 1); besides αi,j = 0 whenever i + j = k. The above system becomes
i+1 k−i+r αi,k−i + αi−r+1,k−i+r−1 = 0 for i = 1, . . . , k and 2 ≤ r ≤ k + 1. r r (19)
In the line above, if r = k + 1, then i must be equal to k, so we get the condition α0,k + αk,0 = 0. If r < k + 1, then the above equations are either identically zero (since αi,j = 0 whenever i = 0, k) or of the form
k+1 k+1 αk,0 = 0 and α0,k = 0. r r
But k+1 = 0 (mod p), since k + 1 = pm . r The claim on dual operators is subject to a direct verification. 2 2.5.2. The transvectants aka Cohen–Rankin brackets Recall that for any p = 2, there is an embedding sl(2) → vect(1; N ). The transvectants are sl(2)-invariant bilinear differential operators discovered by Gordan. As a matter of
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k fact, there is a misprint in the explicit expressions of the transvectants Ja,b in [8]; the corrected expression is (up to a numerical factor)
k Ja,b (f, g)
=
i
(−1)
0≤i≤k
2b + k − 1 (i) (k−i) . f g i
2a + k − 1 k−i
(20)
This expression is for generic a and b, and p = 2. For the exceptional values of a and b over C, see [5], Prop. 2. In this paper, we are interested in the explicit expressions of the transvectants for p = 2. For this, we first normalize the expression (20) by means of a factor (a power of 2) if needed, then reduce the result modulo 2. Recall that if p = 2, then the simple 3-dimensional Lie algebra — the analog of sl(2) for p = 2 — is o(1) (3). Theorem 2.6 claims that if p = 2 and p = 3, then Gordan’s transvectants are invariant not only over o(1) (3) if p = 2 or sl(2) if p = 3, subalgebras of vect(1; N ), but over the 2 whole vect(1; N ) for N small. More exactly, Ja,b is sl(2) vect(1; 1)-invariant for p = 3 3 and Ja,b is vect(1; 2)-invariant for p = 2. 2.5.3. Lemma. For (a, b) = (1, 1), there exists a vect(1; N )-invariant bilinear operator of order k = pm − 2 for any m ≤ N if p = 2 whereas for p = 2 it should be any m ≤ N + 1 (because of the term f (k/2) g (k/2) ) given by the expression
(k−1)/2
Gzp,m (f, g) =
(−1)i det
i=0
f
f (i)
g (i)
(k−i)
(k−i)
g
+
f (k/2) g (k/2)
if p = 2 otherwise.
(21)
This operator coincides with the Grozman operator if p = 5 and m = 1. Observe that Gz is selfdual over any field it is defined, but Gz ◦ (d ⊗ d) is not selfdual. Proof. Let, for simplicity, p = 2 (for p = 2 the proof is more or less the same). Since (a, b) = (1, 1), the invariance with respect to vect(1; N ) is equivalent to the system (19). For i = k or i = 0, we have αk,0 + α0,k = 0 which is certainly satisfied. Now, for i = 0, k, the system becomes
i+1 k−i+r i (−1) + (−1)i−r+1 = 0 for i = 1, . . . , k and 2 ≤ r ≤ k + 1. r r
To show that the system is satisfied, we proceed by induction. If i = 1, then r must be equal to 2. The result follows since
m k+1 p −1 = = 1 (mod p). 2 2
Now suppose that the equality is true at i for every r such that r ≤ i + 1. It follows that
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k − (i + 1) + r r
k−i+r k−i+r−1 − r r−1 induction hypothesis r i+1 r−1 i + 1 ≡ (−1) − (−1) (mod p) r r−1 i+1 i+1 r + (mod p) = (−1) r r−1 i+2 “Pascal’s triangle” = (−1)r (mod p). 2 r “Pascal’s triangle”
=
2.6. Theorem. Up to the interchange of arguments, the indecomposable vect(1; N )-invariant bilinear operators D : Fa ⊗ Fb −→ Fc are only those of the form (12)–(16) and (20)–(21). For the complete list of vect(1; N )-invariant bilinear operators of order ≤ 7, see tables (22)–(29). Proof. Although we have found all invariant bilinear operators, the complete list is not that interesting, we think, unlike the indecomposable operators. We illustrate the answer with a part of the complete list (of order ≤ 7). • Order 1 operators are the same as for p = 0, but for p = 2 we have (a, b) (1, b) and (a, 1)
D(f, g) ⊗ id : f g and f g
comment for N = 1
for any α, β ∈ K and N = 1
(a, b)
P00 : αf g + βf g
(0, 0)
P00 : αf g + βf g
for any α, β ∈ K
(a, b)
{·, ·}P.B. : = af g + bf g
Poisson bracket
(1, 1)
{·, ·} : = f g + f g
contact bracket of generating functions (22)
Clearly, the bottom line is a particular case of the penultimate line; we singled it out here and in (9) because in higher dimensions the contact bracket is not a particular case of the Poisson bracket. • Order 2 operators are the same as for p = 0, but the condition a + b + 1 = 0 is to be replaced with a + b + 1 = 0 (mod p) and for p = 2 we have (a, b) (a, b) (0, b) (a, 1 + a) (a, b)
D(f, g) f g f g + bf g af g + f g + (a + 1)f g ag f + f g + bf g
whereas for p = 3 we have
comment for N = 1; if (a, b) = (1, 1), then D = ⊗ {df, g(du)b }P.B. {f (du)a , dg(du)1+a }P.B. generalizes the above (23)
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(a, b) (a, b) for a, b = 0, 1 (1, b) (0, b) (a, −1 − a) (1, 1) (a, 0)
D(f, g) J 2 (f, g) a,b ⊗ id(f, g) = f g f g − bf g {f, g} = af g + (1 − a)f g + (a + 1)f g Bj3,1 (f, g) {f, g } = −af g + f g
293
comment for N = 1 for N = 1 for N = 1
(24)
• Order 3 operators are the same as for p = 0, but the Grozman operator Gz does not survive for p = 2 and 3. For p = 2, we additionally have (this might be unclear; in fact, for (a, b) = (1, 1), we have ⊗ id and id ⊗ and Bj2,2 = ⊗ id − id ⊗ ): (a, b) (1, b) for b = 0 (1, 0) (a, b) for a, b = 0, 1
D(f, g) ⊗ id(f, g) = f g ∗2 Bj2,2 , ⊗ id 3 Ja,b (f, g)
comment for N = 2 for N = 2 for N = 2
(25)
• Order 4 operators: no operators if p > 5; for p ≤ 5, we have (a, b)
D(f, g)
(0, 0) (1, 0) (0, 1)
p
comments
2 2 2
N =2 N =2 N =2
3
for N = 1
(1, 1)
Gz2,2 (df, dg) = f g + f g + f g ∗1 Gz2,2 (df, dg) = f g (4) + f g Gz ∗2 (df, dg) = f (4) g + f g 2,2 ⊗ = f g
(0, 0)
Bj3,1 ◦ (df, dg)
3
(1, 1) (1, 0) (0, 1)
Bj5,1 (f, g) ∗2 Bj5,1 (f, g) = f g (4) − f g − f g + f g Bj ∗1 (f, g) = f (4) g − f g + f g − f g 5,1 ⊗ id : f (4) g ∗1 id ⊗ , Bj5,1 L5,1 (f, g) = f (4) g + f g (4) − f g − f g + f g
5 5 5
(1, b), b = 0 (0, 1) (a, 1 − a), a = 0, 1
5 5 5
for N = 1 for N = 1 (26)
• Order 5 operators: no invariant operators if p > 7; for p ≤ 7, we have (a, b)
D(f, g)
p (4)
+f
(4)
(0, 0)
Bj2,2 (df, dg) = f g
(0, 0) (0, 1) (1, 0)
Gz5,1 (df, dg) = f (4) g − g (4) f + f g − f g ∗2 Gz5,1 (df, dg) = f (5) g − f g (4) ∗1 Gz5,1 (df, dg) = f g (5) − f (4) g
g
5 5 5
(1, 1)
Gz7,1 (f, g)
7
comments
2 (27)
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• Order 6 operators: no invariant operators if p > 7; for p ≤ 7, we have (a, b) (1, 1) (1, 1) (0, 0) (0, 1) (1, 1) (1, 0) (1, b) for b = 0 (1, 0) (0, 1) (1, 1) (a, 1 − a), a = 0, 1
D(f, g) ⊗ Gz2,3 (f, g) Bj5,1 (df, dg) = f (5) g − f g (5) ∗1 Bj7,1 (f, g) Bj7,1 (f, g) Bj ∗2 (f, g) 7,1 ⊗ id ∗2 ⊗ id, Bj7,1 ∗1 Bj7,1 , id ⊗ Bj7,1 (f, g) L7,1 (f, g)
p 2 2 5 7 7 7 7 7 7 7 7
comment for N = 2 for N > 2
(28) for for for for
N N N N
=1 =1 =1 =1
• Order 7 operators: no invariant operators if p > 7; for p ≤ 7, we have (a, b) (1, 0) (1, 1) (1, b) for b = 0 (1, 0) (a, 1 − a), a = 0, 1 (1, 1) (0, 0) (1, 0) (0, 1)
D(f, g) ∗2 Bj2,3 (f, g) Bj (f, g) 2,3 ⊗ id ∗2 ⊗ id, Bj2,3 L2,3 (f, g) Gz3,2 (f, g) Gz7,1 (df, dg) ∗1 Gz7,1 (df, dg) = f g (7) − f (6) g ∗2 Gz7,1 (df, dg) = f (7) g − f g (6)
p 2 2 2 2 2 3 7 7 7
comment
N =3 N =3 N =3
(29)
If deg D = k ≤ 7, the result can be checked directly. Let as assume that k > 7. We give the detailed proof for the case where k ≤ pN − 2: the proof for the cases where k = pN − 1, k = pN and k > pN is absolutely similar. (1) If a = b = 0. Eqs. Ck,k and Ck−1,k , see eq. (18), imply that αk,0 = 0 and α0,k = 0. Moreover, eq. Ck−1,k−1 implies that α1,k−1 + αk−1,1 = 0. Similarly, eqs. Ck−1,k−2 and Ck−2,k−2 imply that α2,k−2 − (k − 1)α1,k−1 = 0 and (k − 1)α1,k−1 + αk−2,2 = 0. It follows that eq. C2,2 implies that k2 α1,k−1 = 0. If α1,k−1 = 0, then by induction we can show that all other coefficients vanish. Indeed, by solving eq. Ci,i for i = 3, . . . , k − 3, we see that αi,k−i + k−1 α1,k−1 = 0. Therefore, D = 0. Now combining eqs. C2,2 , Ci,i i and Ck−1,i for i = 3, . . . , k − 3, we get ki α1,k−1 = 0 for every i = 2, . . . , k − 3. It follows k that i = 0 (mod p). Using Luca’s theorem we have m k kj (mod p), = ij i j=0
where k =
m
i=0
kj pj and i =
m
i=0
ij pj .
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Therefore, ki = 0 (mod p), for every i = 2, . . . , k − 3, if and only if k = pm or k = pm + 1 for some m ∈ N. If k = pm for some m ∈ N, it follows that all coefficients are expressed in terms of α1,k−1 . Now using Lemma 2.5.3 we see that D is proportional to Gz ◦ (d ⊗ d). If k = pm + 1 for some m ∈ N, we use the same arguments as just above to show that D is proportional to Bj ◦ (d ⊗ d). (2) If a = 0 and b = 0. In this case, eqs. Ck,k+1 and Ck,k imply that α0,k = 0 and αk,0 = −bα1,k−1 . Moreover, eqs. Ck,k−1 and Ck−1,k−1 imply that bα2,k−2 − bkα1,k−1 = 0 and (1 + b(k − 1))α1,k−1 + αk−1,1 = 0. Now, eq. Ck−1,k−2 implies that (1 − b)α1,k−1 = 0. If α1,k−1 = 0, then by induction we can show that all other coefficients vanish. Indeed, computing Cr,r for r = 2, . . . , k − 2, we get k−1 k−1 αr,k−r + b + α1,k−1 = 0. r−1 r Therefore, D = 0. It follows that b = 1, and therefore eq. C2,2 implies that k+1 k+1 = 0 2 α1,k−1 = 0. Using the same arguments as before we can show that i m (mod p) for every i = 2, . . . , k − 2. Luca’s theorem implies that either k = p or k = pm + 1 for some m ∈ N. In both cases, we have either Gz ∗1 ◦ (d ⊗ d) or Bj ∗1 ◦ (d ⊗ d), cf. Lemma 2.5.1 and Lemma 2.5.3. (3) If a, b = 0. In this case, eq. Ck,k+1 implies that αk,0 = −bτ and α0,k = aτ for some τ ∈ K. Besides, eqs. Ck,k and Ck−1,k imply that α1,k−1 = (1 + ak)τ and αk−1,1 = −(1 + bk)τ . Now, eq. Ck−1,k−1 implies that (a − b)τ = 0. If τ = 0, then D = 0 and the proof is completed by induction as we did above. Therefore, a = b. On the other hand, eqs. Ck,k−1 , Ck−2,k−1 and Ck−1,k−2 imply b = 1. Using the same arguments as before we can show that k+2 = 0 (mod p) for every i = 2, . . . , k. Luca’s theorem implies that i m either k = p − 2 or k = pm − 1 for some m ∈ N. These two cases produce either Gz or Bj, respectively, cf. Lemma 2.5.1 and Lemma 2.5.3. 2 Remark. We got the answer with the aid of Grozman’s Mathematica-based code SuperLie, see [12]: we fixed D with deg D ≤ 20 and p ∈ {2, 3, 5, . . . , 19}. For these fixed deg D and p, we looked for which N and (a, b) there is a vect(1; N )-invariant operator. The results inspired us to prove the theorem for any k. The above proof does not rely on any computer. Acknowledgments We are thankful to Grozman for his wonderful code SuperLie, see [12]. We heartily thank the referee for inspiring questions. S.B. was partly supported by the grant AD 065 NYUAD.
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References [1] G. Benkart, J. Feldvoss, Some problems in the representation theory of simple modular Lie algebras, in: M. Avitabile, J. Feldvoss, Th. Weigel (Eds.), Lie Algebras and Related Topics, Milano, Italy, 2013, in: Contemp. Math., vol. 652, Amer. Math. Soc., Providence, RI, 2015, pp. 207–228, arXiv:1503.06762. [2] G. Benkart, Th. Gregory, A. Premet, The Recognition Theorem for Graded Lie Algebras in Prime Characteristic, American Mathematical Society, 2009, 145 pp. [3] J. Bernstein, D. Leites, Invariant differential operators and irreducible representations of Lie superalgebras of vector fields, Sel. Math. Sov. 1 (2) (1981) 143–160. [4] S. Bouarroudj, Ternary invariant differential operators acting on spaces of weighted densities, Teoret. Mat. Fiz. 158 (2009) 165–180. [5] S. Bouarroudj, On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys. 14 (1) (2007) 112–127. [6] S. Bouarroudj, P. Grozman, A. Lebedev, D. Leites, I. Shchepochkina, Simple vectorial Lie algebras in characteristic 2 and their superizations, arXiv:1510.07255. [7] S. Bouarroudj, P. Grozman, D. Leites, Classification of simple finite dimensional modular Lie superalgebras with Cartan matrix, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009) 060, 63 pp., arXiv:0710.5149 [math.RT]. [8] S. Bouarroudj, V. Ovsienko, Three cocycles on Diff(S 1 ) generalizing the Schwarzian derivative, Int. Math. Res. Not. IMRN 1 (1998) 25–39. [9] P. Deligne, P. Etingof, D. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D. Morrison, E. Witten (Eds.), Quantum Fields and Strings: A Course for Mathematicians, vol. 1, Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997, American Mathematical Society/Institute for Advanced Study (IAS), Providence, RI/Princeton, NJ, 1999, Vol. 1: xxii+723 pp. [10] A. Dolotkazin, Irreducible representations of the simple three-dimensional Lie algebra characteristic p = 2, Math. Notes 24 (2) (1978) 588–590. [11] B. Feigin, D. Fuchs, Invariant skew-symmetric differential operators on the line and Verma modules over the Virasoro algebra, Funktsional. Anal. i Prilozhen. 16 (2) (1982) 47–63. [12] P. Grozman, SuperLie, http://www.equaonline.com/math/SuperLie. [13] P. Grozman, Classification of bilinear invariant operators on tensor fields, Funct. Anal. Appl. 14 (2) (1980) 127–128, for proofs, see arXiv:math/0509562. [14] P. Grozman, On bilinear invariant differential operators acting on tensor fields on the symplectic manifold, J. Nonlinear Math. Phys. 8 (1) (2001) 31–37, arXiv:math/0101266. [15] P. Grozman, D. Leites, I. Shchepochkina, Invariant differential operators on supermanifolds and The Standard Model, in: M. Olshanetsky, A. Vainshtein (Eds.), M. Marinov Memorial Volume, World Sci., 2002, pp. 508–555, arXiv:math.RT/0202193. [16] K. Iohara, O. Mathieu, A global version of Grozman’s theorem, Math. Z. 274 (3) (2013) 955–992, arXiv:1204.0695. [17] Ya.S. Krylyuk, Indices of standard maximal subalgebras of Cartan-type p-algebras of finite characteristic, Math. USSR, Sb. 67 (1) (1990) 1–21. [18] A. Lebedev, D. Leites, On realizations of the Steenrod algebras (with an appendix by Deligne P.), J. Prime Res. Math. 2 (1) (2006) 1–13; MPIMiS preprint 131/2006, see http://www.mis.mpg.de. [19] D. Leites, On unconventional integration on supermanifolds and cross ratio on classical superspaces, in: E. Ivanov, S. Krivonos, J. Lukierski, A. Pashnev (Eds.), Proceedings of the International Workshop “Supersymmetries and Quantum Symmetries”, Karpacz, Poland, September 21–25, 2001, JINR, Dubna, 2002, pp. 251–262, arXiv:math.RT/0202194. [20] D. Leites (Ed.), Seminar on Supersymmetry (v. 1. Algebra and Calculus: Main Chapters), (J. Bernstein, D. Leites, V. Molotkov, V. Shander), MCCME, Moscow, 2011, 410 pp. (in Russian; a version in English is in preparation but available for perusal). [21] D. Leites, I. Shchepochkina, How to quantize the antibracket, Theoret. Math. Phys. 126 (3) (2001) 339–369, arXiv:math-ph/0510048. [22] D. Leites, I. Shchepochkina, The classification of simple Lie superalgebras of vector fields, Preprint MPIM-2003-28, http://www.mpim-bonn.mpg.de/preblob/2178. [23] Yu.I. Manin, Gauge Field Theory and Complex Geometry, second edition, Springer-Verlag, Berlin, 1997, xii+346 pp.
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[24] A.N. Rudakov, I.R. Shafarevich, Irreducible representations of a simple three-dimensional Lie algebra over a field of finite characteristic, Mat. Zametki 2 (1967) 760–767. [25] S.M. Skryabin, Classification of Hamiltonian forms over divided power algebras, Math. USSR, Sb. 69 (1) (1991) 121–141. [26] H. Strade, Simple Lie Algebras over Fields of Positive Characteristic. I–III. Structure Theory, de Gruyter Expositions in Mathematics, vol. 38, Walter de Gruyter & Co., Berlin, 2004, viii+540 pp.; 2009, vi+385 pp.; 2012, x+239 pp.
Contents lists available at ScienceDirect
Journal of Algebra www.elsevier.com/locate/jalgebra
Invariant differential operators in positive characteristic Sofiane Bouarroudj a,∗ , Dimitry Leites a,b a
New York University Abu Dhabi, Division of Science and Mathematics, P.O. Box 129188, United Arab Emirates b Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
a r t i c l e
i n f o
Article history: Received 13 June 2016 Available online 19 December 2017 Communicated by Vera Serganova MSC: 17B50 17B55 17B56 17B20 17B40 Keywords: Veblen’s problem Invariant differential operator Positive characteristic
a b s t r a c t In 1928, at the IMC, Veblen posed the problem: classify invariant differential operators between spaces of “natural objects” (in modern terms: either tensor fields, or jets) over a real manifold of any dimension. The problem was solved by Rudakov for unary operators (no nonscalar operators except the exterior differential); by Grozman for binary operators. In dimension one, Grozman discovered an indecomposable selfdual operator of order 3 that does not exist in higher dimensions. We solve Veblen’s problem in the 1-dimensional case over any field of positive characteristic. Unary invariant operators are known: these are the exterior differential and analogs of the Berezin integral. We construct new binary operators from these analogs and discovered two more (up to dualizations) types of new indecomposable operators of however high order: analogs of the Grozman operator and a completely new type of operators. Gordan’s transvectants, aka Cohen–Rankin brackets, always invariant with respect to the simple 3-dimensional Lie algebra, are also invariant, in characteristic 2, with respect to the whole Lie algebra of vector fields on the line when the height of the indeterminate is equal to 2. © 2017 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail addresses: sofi[email protected] (S. Bouarroudj), [email protected] (D. Leites). https://doi.org/10.1016/j.jalgebra.2017.11.048 0021-8693/© 2017 Elsevier Inc. All rights reserved.
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1. Introduction 1.1. Discussion of open problems In the representation theory of a given algebra or group, the usual ultimate goal is a clear description of its irreducible modules to begin with, description of indecomposables being the next step. Other goals are sometimes also natural and reachable, cf. those discussed here and in [1]. The history of mathematics shows that problems with a nice (at least in some sense, for example, short) answer often turn out to be “reasonable”. I. Gelfand used to say that, the other way round, easy-to-formulate lists are answers to “reasonable” problems and advised to try to formulate such problems once we got a short answer. Hereafter, K is any field of characteristic p > 0, unless otherwise specified. Speaking of Lie algebras or superalgebras, and their representations over K, which of them is it natural to consider? In [18], Deligne suggests that one should begin with restricted ones: unlike nonrestricted ones, the restricted Lie (super)algebras correspond to groups, in other words: to geometry. Certain problems concerning nonrestricted algebras are also natural (although tough), and have a short answer, e.g., classification of simple Lie algebras for p > 3 (for its long proof, see [26,2]). Rudakov and Shafarevich [24] were the first to describe ALL, not only restricted, irreducible representations of sl(2) for p > 2. Dolotkazin solved the same problem for p = 2, see [10]; more precisely, he described the irreducible representations of the simple 3-dimensional Lie algebra o(1) (3) = [o(3), o(3)]. The difference of Dolotkazin’s problem from that considered in [24] is that, unlike sl(2) for p > 2, the Lie algebra o(1) (3) is not restricted. These two results show that the description of all irreducible representations looks feasible, to an extent, at least for Lie (super)algebras with indecomposable Cartan matrix or their simple “relatives” (for the classification of both types of Lie (super)algebras over K, see [7]). 1.1.1. Veblen’s problem In 1928, at the IMC, O. Veblen formulated a problem, later reformulated in more comprehensible terms by A. Kirillov and further reformulated as a purely algebraic problem by J. Bernstein who interpreted Rudakov’s solution of Veblen’s problem for unary operators; for setting in modern words and review, in particular, for a superization of Veblen’s problem, see [15]. Let vect(m) be the Lie algebra of polynomial vector fields (over a ground field of characteristic 0, say C). For the definition of the vect(m)-module T (V ) of (formal) tensor fields of type V , where V is a gl(m)-module with lowest weight vector, see [15]. Let us briefly recall the results concerning the nonscalar unary and binary invariant differential operators between spaces of tensor fields, although we only need the simplest version of spaces T (V ), namely, the spaces of weighted densities Fa := T (a tr), where tr is the 1-dimensional gl(m)-module given by the trace (supertrace for gl(m|k)) and a ∈ C (or
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another ground field). In what follows we assume that modules T (V ) are coinduced from irreducible modules V ; this is certainly so for modules Fa . (In Veblen’s problem over C, it suffices to consider only irreducible modules V ; for its superization and over K, one has to consider indecomposable modules V .) 1.1.1.1. Unary operators on manifolds and supermanifolds The problem we are solving here for p > 0 resembles its super version for p = 0; we recall the known results. The only vect(m)-invariant differential operators T (V ) −→ T (W ) are the exterior differential operators (so, practically, there is just one such operator) in the de Rham complex, where Ωi = T (E i (id)), E i is the functor of raising to the ith exterior power and id is the tautological m-dimensional gl(m)-module: d
d
d
d
0 −→ C −→ Ω0 −→ Ω1 −→ · · · −→ Ωm−1 −→ Ωm −→ 0.
(1)
In super setting, where sdim(id) = m|k, the analog of the complex (1) is infinite without the “top term” d
d
d
0 −→ C −→ Ω0 (m|k) −→ Ω1 (m|k) −→ Ω2 (m|k) −→ . . . ,
(2)
its dualization brings the complex of integrable (because it is possible to integrate them; Deligne calls them integral) forms which in the purely odd case of superdimension 0|k terminates with an order-k differential operator called the Berezin integral which to a given function f (θ) assigns its coefficient of the highest term, where Σ−i (m|k) := (Ωi (m|k))∗ : d
d
d
· · · −→ Σ−1 (m|k) −→ Σ0 (m|k) −→ 0 d
d
for m = 0,
(3)
· · · −→ Σ−1 (0|k) −→ Σ0 (0|k) −→ K −→ 0 for m = 0. Over C, the de Rham complex (2) is exact, while the complex of integrable forms (3) has 1-dimensional cohomology in the (−m)th term; the cohomology is spanned, see [3, 23,9], by (here the ui are the even indeterminates, the θj are the odd ones for clarity; in Notation 2.1 all indeterminates are called u) θ1 . . . θk ∂u1 . . . ∂um vol(u|θ)
(4)
and the super version of the volume element vol(u) = du1 ∧ · · · ∧ dum is the class vol(u|θ) of the element du1 ∧ · · · ∧ dum ⊗ ∂θ1 ∧ · · · ∧ ∂θk
(5)
in the indecomposable gl(V ) = gl(m|k)-module generated by the element (5) modulo the codimension 1 submodule, see [9,20]. (There are interesting generalizations to pseudod-
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ifferential and pseudointegral forms, but they have no analog for p > 0 we are interested in this note.) Simple Lie (super)algebras of vector fields with polynomial or formal coefficients constitute a class of interesting Lie (super)algebras. Over C and R, they are classified and their continuous (in the x-adic topology) irreducible representations with a vacuum (highest or lowest) weight vector are classified as well in the following cases: • For the four series of simple Lie algebras, and their direct super versions, the answer (if dim V < ∞) is as follows, see the review [15]: the module T (V ) of tensor fields coinduced from any irreducible module V with lowest weight vector over the subalgebra of linear vector fields is irreducible, unless T (V ) Ωk , the module of exterior k-forms. Rudakov proved that the exterior differential is the only (nonscalar) operator between the modules of exterior differential forms invariant with respect to all changes of indeterminates, or, equivalently, for the general vectorial Lie algebra vect. Together with the Poincaré lemma (stating that the sequence (1) is exact over any star-shaped domain) this yields a complete description of continuous irreducible modules with lowest weight. For simple Lie algebras of hamiltonian and contact vector fields (and analogous simple Lie superalgebras in their standard grading) the answer is essentially the same, except that one should consider not the whole spaces Ωk (or their dual) but their subspaces of “primitive” forms (for details, see [15]). In the purely odd case, the Berezin integral has to be added to the list of invariant differential operators, see eq. (3). • For simple vectorial Lie superalgebras in their nonstandard grading (listed in [22]), and for the modules of tensor fields with infinite-dimensional fibers (i.e., dim V = ∞) even over finite-dimensional simple vectorial Lie superalgebras, many never previously known differential operators appear, see [15,19]. • Over K, it is tempting to conjecture that, at least for p “sufficiently large”, the description of irreducible modules over vectorial Lie algebras (at least Z-graded, and “close to simple” ones) would resemble that given by Rudakov over C. For irreducible restricted modules over restricted simple vectorial Lie algebras of the 4 classical series and p ≥ 3, this is almost so, i.e., the spaces of differential k-forms are the only reducible spaces of tensor fields, although the complex of exterior differential forms is not exact, see Theorem 2.2.1 ([17]). 1.1.1.2. Binary operators on manifolds These are classified by P. Grozman, see [13]. Recall that the formal dual of T (V ) on the m-dimensional space is T (V ∗ ⊗ tr). Hence, with every invariant operator D : T (V ) −→ T (W ) its dual D∗ : T (W ∗ ⊗tr) −→ T (V ∗ ⊗tr) is also invariant. In particular, on the line, with every operator D : Fa −→ Fa+deg D of order deg D, there exists its dual D∗ : F1−a−deg D −→ F1−a of the same order. Similarly, with every binary operator D : Fa ⊗ Fb −→ Fa+b+deg D , there exist its two duals D∗1 : F1−a−b−deg D ⊗ Fb −→ F1−a and D∗2 : Fa ⊗ F1−a−b−deg D −→ F1−b , as
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well as the operator r(D) : Fb ⊗ Fa −→ Fa+b+deg D , given by the interchange of the arguments r(D)(X, Y ) : = D(Y, X) for any X ∈ Fb and Y ∈ Fa . Summary: Up to dualizations and interchange of the arguments, there are 8 series of invariant operators of order 1, all invariant differential operators of order 2 and 3 are compositions of order-1 operators, except for the case m = 1, where there is an indecomposable operator Gz of order 3 discovered by Grozman; for explicit expressions, see [13] and Subsec. 2.4. There are no operators of order > 3. Aside remarks: 1) An interesting fact. Among all 8 series of invariant operators of order 1, there are 4 series that determine an associative algebra and 3 series determine a Lie (super)algebra structure on their domain of definition. These Lie (super)algebras are either simple or simple modulo center or contain a simple algebra of codimension 1: the corresponding products are the Poisson bracket, the Schouten bracket (more popular now under the name anti-bracket) and a deformed Schouten bracket, see [13,21]. 2) Kirillov gave an example of the symbol of an invariant with respect to all coordinate changes nonlocal bilinear operator on the circle but did not identify the operator itself; in [16] this operator is described in terms of the integral=residue; it is the only nonlocal invariant bilinear operator on the circle. In characteristic p > 0, same as on superpoints, analogs of various operators nonlocal over R become local because if the space of functions is finite-dimensional, then any operator in this space can be viewed as differential, i.e., composition of derivations and multiplications by some functions. 1.1.1.3. Invariant differential operators of higher arity The only k-ary invariant operators classified so far are antisymmetric ones on C, see [11], and ternary operators between the spaces of weighted densities on Cn , see [4]. 1.2. Main result The classification (Theorem 2.6) of binary vect(1; N )-invariant operators between modules of weighted densities over K; in particular, we discovered three series of indecomposable operators of however high order: 1) related to the analog of Berezin integral, 2) an analog of the Grozman operator, and 3) a completely new series of operators, Bj. If p = 2, Gordan’s transvectants, always o(1) (3)-invariant, are vect(1; 2)-invariant. 1.3. Open problems 1) We conjecture that the Grozman operator has something to do with the contact structure since vect(1) = k(1); see also Conjecture 4) below. Over C, it seems feasible to classify binary k(2n + 1)-invariant differential operators for n > 0 and complete the
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partial result for binary h(2n)-invariant differential operators, solved only for n = 1, see [14]. 2) The only feasible super version of the problem we consider here is, it seems, classification of the k(1; N |k)-invariant operators for k = 1 or 2 when the coinduced modules with lowest weight vectors are of the form Fa . For a (non-trivial if p = 2) description of the Lie superalgebras k(1; N |k), see [6]. 3) There are natural filtered deforms of simple nonrestricted vectorial Lie algebras; besides, for p ≤ 5, there are new types of simple vectorial Lie algebras and no description of their irreducible modules is known; in particular, no description of intertwining operators between modules of tensor fields — the most natural ones over these Lie algebras. 4) Conjecture. The only vect(m; N |k)-invariant unary operators between modules of tensor fields coinduced from irreducible gl(m|k)-modules are the exterior operator d and the depending on N analogs of the Berezin integral . 2. Veblen’s problem over fields of characteristic p > 0 2.1. Notation Introduce parity in the ring of polynomials K[x], where x = (x1 , . . . , xa ) and a = m+k, by setting p(xi ) = ¯0 for i ≤ m and p(xi ) = ¯1 for m < i ≤ a. For any a-tuple of nonnegative integers r = (r1 , . . . , ra ), where ri = 0 or 1 for i > m, set (ri )
ui
:=
xri i ri !
and u(r) :=
(ri )
ui
.
1≤i≤a
The idea is to formally replace fractions with ri ! in denominators by inseparable symbols (r ) ui i which are well-defined over fields of characteristic p; in particular, if k = 0, the factor in the first parentheses below is equal to 1:
u(r) · u(s) = where
r+s r
min(1, 2 − ri − si ) · (−1)m
m+1≤i≤a
:=
1≤i≤m
rj si
·
r+s r
u(r+s) ,
ri +si . ri
For an m +k-tuple of positive integers N = (N1 , ..., Nm , 1, . . . , 1), set N ev = (N1 , ..., Nm ). The following supercommutative superalgebra is the “correct” for p > 0 analog of the algebra of functions: O(m; N ev |k) := K[u; N ] := SpanK (u(r) | ri < pNi for i ≤ m, 0 ≤ ri ≤ 1 for i > m). The Lie superalgebra vect(m; N |k) of its distinguished derivations consists of the vector fields of the form fi (u)∂i , where fi ∈ O(m; N ev |k) and ∂i are distinguished partial (r ) (r −1) derivatives defined by the condition ∂j (ui i ) = δji ui i .
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2.2. Unary operators: De Rham complex for k = 0 (no super) Set τ (N ) = (pN1 − 1, . . . , pNm − 1). Denote: Z i (m; N ) := {ω ∈ Ωi (m; N ) | dω = 0} and B i (m; N ) := {dω | ω ∈ Ωi−1 (m; N )}. 2.2.1. Theorem ([17,25]). The sequence d
d
d
d
d
0 −→ K −→ Ω0 (m; N ) −→ Ω1 (m; N ) −→ Ω2 (m; N ) −→ · · · −→ Ωm (m; N ) −→ 0 is not exact: the space H a (m; N ) := Z a (m; N )/B a (m; N ) is spanned by the elements (τ (N )i1 )
ui1
(τ (N )ik )
. . . uik
dui1 . . . duik where a = i1 + · · · + ik .
. The supercommutative superalgebra H (m; N ) is generated by H 1 (m; N ). 2.2.2. Corollary. There is an invariant differential operator of order |τ (N )| := ( pN i ) − m:
Ωm (m; N ) −→ K; this, depending on N , analog of the Berezin integral is defined as follows:
(τ (N )1 )
f (u) vol(u) = the coefficient of uτ (N ) = u1
(N )m ) . . . u(τ , m
where vol(u) = du1 ∧ · · · ∧ dum . 2.3. Unary operators: De Rham complex for k = 0, i.e., in super setting On the m|k-dimensional space over K, the de Rham complex goes ad infinitum d
d
d
0 −→ K −→ Ω0 (m; N |k) −→ Ω1 (m; N |k) −→ Ω2 (m; N |k) −→ . . .
(6)
The dual to (6) complex of integrable forms is as follows, where Σ0 := Ω0 vol(u|θ): d
d
· · · −→ Σ−1 (m; N |k) −→ Σ0 (m; N |k) −→ K −→ 0.
(7)
Observe that, for p > 0, the integral is defined for m = 0 as well, whereas for p = 0, there is no such an invariant differential operator.
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2.4. The binary differential operators over C for m = 1 (recapitulation, see [13]) For m = 1, we have Fa := {f (u)(du)a | f (u) ∈ Ω0 = F0 } on which the Lie derivative acts as follows: LD (f (u)(du)a ) = (D(f ) + af g )(du)a ) for any D = g(u)∂u ∈ vect(1).
(8)
For any bilinear differential operator
D : Fa ⊗ Fb −→ Fa+b+deg D ,
f (du)a ⊗ g(du)b −→ D(f, g)(du)a+b+deg D ,
it suffices to indicate (a, b) and D(f, g). The operators are listed up to proportionality. 2.4.1. Order 1 operators For generic values (a, b), the invariant operators form a 1-dimensional space. For (a, b) = (0, 0), and only in this case, we have a 2-dimensional space of operators.
(a, b) (0, 0)
D(f, g)
comment
P00 : αf g + βf g
for any α, β ∈ C
(a, b)
{·, ·}P.B. : = af g − bf g
Poisson bracket in two even indeterminates u and du
(−1, −1)
{·, ·} : = f g − f g
contact bracket of generating functions d f −→ Kf := f du (9)
Clearly, the bottom line is a particular case of the penultimate line; we singled it out because in higher dimensions the contact bracket is not a particular case of the Poisson bracket. 2.4.2. Order 2 operators
(a, b) (0, b) (a, 0) (a, −1 − a)
D(f, g) f g − bf g af g − f g af g + (2a + 1)f g + (a + 1)f g
comment {df, g(du)b }P.B. {f (du)a , dg}P.B. ({f (du)a , g(du)−1−a }P.B. )
(10)
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2.4.3. Order 3 operators (a, b) (0, 0) (−2, 0) (0, −2) 2 2 −3, −3
D(f, g) T1 (f, g) = {f , g } T1∗1 (f, g) = f g + 3f g + 2f g T1∗2 (f, g) = f g + 3f g + 2f g f f g + 3 det Gz(f, g) = 2 det f g f
(11) g g
2.5. New binary differential operators: p > 0 and m = 1 For a fixed N = N , we list vect(1; N )-invariant operators up to interchange of arguments. Recall that : Ω1 = F1 −→ K ⊂ F0 is the order pN − 1 operator, see Corollary 2.2.2 and 1 − pN ≡ 1 (mod p). Therefore, for any p > 0 and N , there are vect(1; N )-invariant bilinear operators acting in the spaces Fa ⊗ Fb in the following cases: operator(a, b) ⊗ id(1, a), ( ⊗ id)∗1 (1 − a, a), ( ⊗ id)∗2 (1, 1 − a) ⊗d = ( ⊗d)∗2 (1, 0), ( ⊗d)∗1 (0, 0), r( ⊗d)(0, 1) ∗2 ∗1 ⊗ = ( ⊗ ) = ( ⊗ ) (1, 1)
order k pN − 1 pN 2(pN − 1)
case 1 2 3
(12)
The operator ( ⊗ id)∗1 (1 − a, a) is of the form (L for “long” expression) (k−1)/2
Lp,N (f, g) =
(−1)i f (i) g (k−i) + f (k−i) g (i)
i=0
+
(−1)k/2 f (k/2) g (k/2)
if p > 2 if p = 2.
(13)
2.5.1. Lemma. For (a, b) = (1, 1), there exists a vect(1; N )-invariant bilinear operator Bjp,m of order k = pm − 1 for any m ≤ N given by the expression Bjp,m (f, g) = det
f
g
f (k)
g (k)
.
(14)
Its duals for (a, b) = (1, 0) and (0, 1), respectively, are as follows: (k−1)/2 ∗2 Bjp,m (f, g) = f g (k) +
(−1)i f (i) g (k−i) + f (k−i) g (i)
i=1
+
(−1)k/2 f (k/2) g (k/2)
if p > 2 if p = 2
(15)
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and (k−1)/2 ∗1 Bjp,m (f, g)
=f
(k)
g+
(−1)i f (i) g (k−i) + f (k−i) g (i)
i=1
+
(−1)k/2 f (k/2) g (k/2)
if p > 2 if p = 2.
(16)
Proof. Every order k bilinear differential operator D : Fa ⊗ Fb −→ Fa+b+deg D is of the form ⎛ ⎞
(f (dx)a , g(dx)b ) −→ ⎝ αi,j f (i) g (j) ⎠ (dx)c , where αi,j ∈ K. (17) i+j=k
For N big enough (such that pN > deg D + 1) and a given pair (a, b), the vect(1; N )-invariance of D implies the following system of equations Ci,r for i = 1, . . . , k and 2 ≤ r ≤ k + 1: Ci,r
i i : a + αi,k−i r−1 r k−i+r−1 k−i+r−1 + b + αi−r+1,k−i+r−1 = 0. r−1 r
(18)
In the case of Lemma, (a, b) = (1, 1); besides αi,j = 0 whenever i + j = k. The above system becomes
i+1 k−i+r αi,k−i + αi−r+1,k−i+r−1 = 0 for i = 1, . . . , k and 2 ≤ r ≤ k + 1. r r (19)
In the line above, if r = k + 1, then i must be equal to k, so we get the condition α0,k + αk,0 = 0. If r < k + 1, then the above equations are either identically zero (since αi,j = 0 whenever i = 0, k) or of the form
k+1 k+1 αk,0 = 0 and α0,k = 0. r r
But k+1 = 0 (mod p), since k + 1 = pm . r The claim on dual operators is subject to a direct verification. 2 2.5.2. The transvectants aka Cohen–Rankin brackets Recall that for any p = 2, there is an embedding sl(2) → vect(1; N ). The transvectants are sl(2)-invariant bilinear differential operators discovered by Gordan. As a matter of
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k fact, there is a misprint in the explicit expressions of the transvectants Ja,b in [8]; the corrected expression is (up to a numerical factor)
k Ja,b (f, g)
=
i
(−1)
0≤i≤k
2b + k − 1 (i) (k−i) . f g i
2a + k − 1 k−i
(20)
This expression is for generic a and b, and p = 2. For the exceptional values of a and b over C, see [5], Prop. 2. In this paper, we are interested in the explicit expressions of the transvectants for p = 2. For this, we first normalize the expression (20) by means of a factor (a power of 2) if needed, then reduce the result modulo 2. Recall that if p = 2, then the simple 3-dimensional Lie algebra — the analog of sl(2) for p = 2 — is o(1) (3). Theorem 2.6 claims that if p = 2 and p = 3, then Gordan’s transvectants are invariant not only over o(1) (3) if p = 2 or sl(2) if p = 3, subalgebras of vect(1; N ), but over the 2 whole vect(1; N ) for N small. More exactly, Ja,b is sl(2) vect(1; 1)-invariant for p = 3 3 and Ja,b is vect(1; 2)-invariant for p = 2. 2.5.3. Lemma. For (a, b) = (1, 1), there exists a vect(1; N )-invariant bilinear operator of order k = pm − 2 for any m ≤ N if p = 2 whereas for p = 2 it should be any m ≤ N + 1 (because of the term f (k/2) g (k/2) ) given by the expression
(k−1)/2
Gzp,m (f, g) =
(−1)i det
i=0
f
f (i)
g (i)
(k−i)
(k−i)
g
+
f (k/2) g (k/2)
if p = 2 otherwise.
(21)
This operator coincides with the Grozman operator if p = 5 and m = 1. Observe that Gz is selfdual over any field it is defined, but Gz ◦ (d ⊗ d) is not selfdual. Proof. Let, for simplicity, p = 2 (for p = 2 the proof is more or less the same). Since (a, b) = (1, 1), the invariance with respect to vect(1; N ) is equivalent to the system (19). For i = k or i = 0, we have αk,0 + α0,k = 0 which is certainly satisfied. Now, for i = 0, k, the system becomes
i+1 k−i+r i (−1) + (−1)i−r+1 = 0 for i = 1, . . . , k and 2 ≤ r ≤ k + 1. r r
To show that the system is satisfied, we proceed by induction. If i = 1, then r must be equal to 2. The result follows since
m k+1 p −1 = = 1 (mod p). 2 2
Now suppose that the equality is true at i for every r such that r ≤ i + 1. It follows that
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k − (i + 1) + r r
k−i+r k−i+r−1 − r r−1 induction hypothesis r i+1 r−1 i + 1 ≡ (−1) − (−1) (mod p) r r−1 i+1 i+1 r + (mod p) = (−1) r r−1 i+2 “Pascal’s triangle” = (−1)r (mod p). 2 r “Pascal’s triangle”
=
2.6. Theorem. Up to the interchange of arguments, the indecomposable vect(1; N )-invariant bilinear operators D : Fa ⊗ Fb −→ Fc are only those of the form (12)–(16) and (20)–(21). For the complete list of vect(1; N )-invariant bilinear operators of order ≤ 7, see tables (22)–(29). Proof. Although we have found all invariant bilinear operators, the complete list is not that interesting, we think, unlike the indecomposable operators. We illustrate the answer with a part of the complete list (of order ≤ 7). • Order 1 operators are the same as for p = 0, but for p = 2 we have (a, b) (1, b) and (a, 1)
D(f, g) ⊗ id : f g and f g
comment for N = 1
for any α, β ∈ K and N = 1
(a, b)
P00 : αf g + βf g
(0, 0)
P00 : αf g + βf g
for any α, β ∈ K
(a, b)
{·, ·}P.B. : = af g + bf g
Poisson bracket
(1, 1)
{·, ·} : = f g + f g
contact bracket of generating functions (22)
Clearly, the bottom line is a particular case of the penultimate line; we singled it out here and in (9) because in higher dimensions the contact bracket is not a particular case of the Poisson bracket. • Order 2 operators are the same as for p = 0, but the condition a + b + 1 = 0 is to be replaced with a + b + 1 = 0 (mod p) and for p = 2 we have (a, b) (a, b) (0, b) (a, 1 + a) (a, b)
D(f, g) f g f g + bf g af g + f g + (a + 1)f g ag f + f g + bf g
whereas for p = 3 we have
comment for N = 1; if (a, b) = (1, 1), then D = ⊗ {df, g(du)b }P.B. {f (du)a , dg(du)1+a }P.B. generalizes the above (23)
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(a, b) (a, b) for a, b = 0, 1 (1, b) (0, b) (a, −1 − a) (1, 1) (a, 0)
D(f, g) J 2 (f, g) a,b ⊗ id(f, g) = f g f g − bf g {f, g} = af g + (1 − a)f g + (a + 1)f g Bj3,1 (f, g) {f, g } = −af g + f g
293
comment for N = 1 for N = 1 for N = 1
(24)
• Order 3 operators are the same as for p = 0, but the Grozman operator Gz does not survive for p = 2 and 3. For p = 2, we additionally have (this might be unclear; in fact, for (a, b) = (1, 1), we have ⊗ id and id ⊗ and Bj2,2 = ⊗ id − id ⊗ ): (a, b) (1, b) for b = 0 (1, 0) (a, b) for a, b = 0, 1
D(f, g) ⊗ id(f, g) = f g ∗2 Bj2,2 , ⊗ id 3 Ja,b (f, g)
comment for N = 2 for N = 2 for N = 2
(25)
• Order 4 operators: no operators if p > 5; for p ≤ 5, we have (a, b)
D(f, g)
(0, 0) (1, 0) (0, 1)
p
comments
2 2 2
N =2 N =2 N =2
3
for N = 1
(1, 1)
Gz2,2 (df, dg) = f g + f g + f g ∗1 Gz2,2 (df, dg) = f g (4) + f g Gz ∗2 (df, dg) = f (4) g + f g 2,2 ⊗ = f g
(0, 0)
Bj3,1 ◦ (df, dg)
3
(1, 1) (1, 0) (0, 1)
Bj5,1 (f, g) ∗2 Bj5,1 (f, g) = f g (4) − f g − f g + f g Bj ∗1 (f, g) = f (4) g − f g + f g − f g 5,1 ⊗ id : f (4) g ∗1 id ⊗ , Bj5,1 L5,1 (f, g) = f (4) g + f g (4) − f g − f g + f g
5 5 5
(1, b), b = 0 (0, 1) (a, 1 − a), a = 0, 1
5 5 5
for N = 1 for N = 1 (26)
• Order 5 operators: no invariant operators if p > 7; for p ≤ 7, we have (a, b)
D(f, g)
p (4)
+f
(4)
(0, 0)
Bj2,2 (df, dg) = f g
(0, 0) (0, 1) (1, 0)
Gz5,1 (df, dg) = f (4) g − g (4) f + f g − f g ∗2 Gz5,1 (df, dg) = f (5) g − f g (4) ∗1 Gz5,1 (df, dg) = f g (5) − f (4) g
g
5 5 5
(1, 1)
Gz7,1 (f, g)
7
comments
2 (27)
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• Order 6 operators: no invariant operators if p > 7; for p ≤ 7, we have (a, b) (1, 1) (1, 1) (0, 0) (0, 1) (1, 1) (1, 0) (1, b) for b = 0 (1, 0) (0, 1) (1, 1) (a, 1 − a), a = 0, 1
D(f, g) ⊗ Gz2,3 (f, g) Bj5,1 (df, dg) = f (5) g − f g (5) ∗1 Bj7,1 (f, g) Bj7,1 (f, g) Bj ∗2 (f, g) 7,1 ⊗ id ∗2 ⊗ id, Bj7,1 ∗1 Bj7,1 , id ⊗ Bj7,1 (f, g) L7,1 (f, g)
p 2 2 5 7 7 7 7 7 7 7 7
comment for N = 2 for N > 2
(28) for for for for
N N N N
=1 =1 =1 =1
• Order 7 operators: no invariant operators if p > 7; for p ≤ 7, we have (a, b) (1, 0) (1, 1) (1, b) for b = 0 (1, 0) (a, 1 − a), a = 0, 1 (1, 1) (0, 0) (1, 0) (0, 1)
D(f, g) ∗2 Bj2,3 (f, g) Bj (f, g) 2,3 ⊗ id ∗2 ⊗ id, Bj2,3 L2,3 (f, g) Gz3,2 (f, g) Gz7,1 (df, dg) ∗1 Gz7,1 (df, dg) = f g (7) − f (6) g ∗2 Gz7,1 (df, dg) = f (7) g − f g (6)
p 2 2 2 2 2 3 7 7 7
comment
N =3 N =3 N =3
(29)
If deg D = k ≤ 7, the result can be checked directly. Let as assume that k > 7. We give the detailed proof for the case where k ≤ pN − 2: the proof for the cases where k = pN − 1, k = pN and k > pN is absolutely similar. (1) If a = b = 0. Eqs. Ck,k and Ck−1,k , see eq. (18), imply that αk,0 = 0 and α0,k = 0. Moreover, eq. Ck−1,k−1 implies that α1,k−1 + αk−1,1 = 0. Similarly, eqs. Ck−1,k−2 and Ck−2,k−2 imply that α2,k−2 − (k − 1)α1,k−1 = 0 and (k − 1)α1,k−1 + αk−2,2 = 0. It follows that eq. C2,2 implies that k2 α1,k−1 = 0. If α1,k−1 = 0, then by induction we can show that all other coefficients vanish. Indeed, by solving eq. Ci,i for i = 3, . . . , k − 3, we see that αi,k−i + k−1 α1,k−1 = 0. Therefore, D = 0. Now combining eqs. C2,2 , Ci,i i and Ck−1,i for i = 3, . . . , k − 3, we get ki α1,k−1 = 0 for every i = 2, . . . , k − 3. It follows k that i = 0 (mod p). Using Luca’s theorem we have m k kj (mod p), = ij i j=0
where k =
m
i=0
kj pj and i =
m
i=0
ij pj .
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Therefore, ki = 0 (mod p), for every i = 2, . . . , k − 3, if and only if k = pm or k = pm + 1 for some m ∈ N. If k = pm for some m ∈ N, it follows that all coefficients are expressed in terms of α1,k−1 . Now using Lemma 2.5.3 we see that D is proportional to Gz ◦ (d ⊗ d). If k = pm + 1 for some m ∈ N, we use the same arguments as just above to show that D is proportional to Bj ◦ (d ⊗ d). (2) If a = 0 and b = 0. In this case, eqs. Ck,k+1 and Ck,k imply that α0,k = 0 and αk,0 = −bα1,k−1 . Moreover, eqs. Ck,k−1 and Ck−1,k−1 imply that bα2,k−2 − bkα1,k−1 = 0 and (1 + b(k − 1))α1,k−1 + αk−1,1 = 0. Now, eq. Ck−1,k−2 implies that (1 − b)α1,k−1 = 0. If α1,k−1 = 0, then by induction we can show that all other coefficients vanish. Indeed, computing Cr,r for r = 2, . . . , k − 2, we get k−1 k−1 αr,k−r + b + α1,k−1 = 0. r−1 r Therefore, D = 0. It follows that b = 1, and therefore eq. C2,2 implies that k+1 k+1 = 0 2 α1,k−1 = 0. Using the same arguments as before we can show that i m (mod p) for every i = 2, . . . , k − 2. Luca’s theorem implies that either k = p or k = pm + 1 for some m ∈ N. In both cases, we have either Gz ∗1 ◦ (d ⊗ d) or Bj ∗1 ◦ (d ⊗ d), cf. Lemma 2.5.1 and Lemma 2.5.3. (3) If a, b = 0. In this case, eq. Ck,k+1 implies that αk,0 = −bτ and α0,k = aτ for some τ ∈ K. Besides, eqs. Ck,k and Ck−1,k imply that α1,k−1 = (1 + ak)τ and αk−1,1 = −(1 + bk)τ . Now, eq. Ck−1,k−1 implies that (a − b)τ = 0. If τ = 0, then D = 0 and the proof is completed by induction as we did above. Therefore, a = b. On the other hand, eqs. Ck,k−1 , Ck−2,k−1 and Ck−1,k−2 imply b = 1. Using the same arguments as before we can show that k+2 = 0 (mod p) for every i = 2, . . . , k. Luca’s theorem implies that i m either k = p − 2 or k = pm − 1 for some m ∈ N. These two cases produce either Gz or Bj, respectively, cf. Lemma 2.5.1 and Lemma 2.5.3. 2 Remark. We got the answer with the aid of Grozman’s Mathematica-based code SuperLie, see [12]: we fixed D with deg D ≤ 20 and p ∈ {2, 3, 5, . . . , 19}. For these fixed deg D and p, we looked for which N and (a, b) there is a vect(1; N )-invariant operator. The results inspired us to prove the theorem for any k. The above proof does not rely on any computer. Acknowledgments We are thankful to Grozman for his wonderful code SuperLie, see [12]. We heartily thank the referee for inspiring questions. S.B. was partly supported by the grant AD 065 NYUAD.
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