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Remark It seems that for [MATH] Lie superalgebras with root spaces [MATH] can not have NIS since even [MATH] — the analog of [MATH] for [MATH] , see BGL — does not have it. If the form [MATH] is invariant, then nondegeneracy is violated: |
[EQUATION] 3.2. The deforms of [MATH] for [MATH] These deforms constitute two non-isomorphic types, see BLW [MATH] The parametric family [MATH] with Cartan matrix |
[MATH] ; it has NIS by the recipe ( ). Note that [MATH] is simple if [MATH] , and [MATH] [MATH] An exceptional simple Lie algebra [MATH] discovered by A. Kostrikin and Kuznetsov. Recall the description of [MATH] , see BLW , Prop. 3.2] The contact bracket of two polynomials in divided powers [MATH] |
is defined to be [EQUATION] Let [MATH] and [MATH] be the simple roots of [MATH] and [MATH] be the root vector corresponding to root [MATH] . The following is the bracket in [MATH] |
[EQUATION] given in terms a basis of [MATH] expressed via generating functions of [MATH] [EQUATION] 3.2.1. Claim (NIS on deforms of [MATH] ). |
The deform ( ) of [MATH] preserves NIS with the same Gram matrix as the one determined by recipe ( ) applied to [MATH] In Claims 3.3 3.4 we consider all Lie (super)algebras with indecomposable Cartan matrix (not isomorphic to [MATH] ) that can be deformed with even parameter, see BGLd , but whose deforms have no Cartan... |
3.3. Claim (NIS on deforms of [MATH] and [MATH] ). For Lie algebras [MATH] , and [MATH] for [MATH] , where [MATH] if and only if [MATH] , see BGL , and Lie superalgebra [MATH] , all deforms depend on even parameters as proved in BGLd . These deforms preserve NIS with the same Gram matrix as the one determined by recipe... |
[MATH] , where for the same numbering of Chevalley generators [MATH] [MATH] and [MATH] as in BGL for the Cartan matrix obtained from ( ) at [MATH] , we have |
[EQUATION] and [EQUATION] where [MATH] and [MATH] For the deform of [MATH] with cocycle [MATH] , i.e., when the deformation parameter is equal to [MATH] , we only have a degenerate invariant symmetric bilinear form [MATH] for which [MATH] and zero on all other pairs of Chevalley basis vectors. |
3.4. Claim (NIS on deforms of [MATH] ). For Lie algebra [MATH] for [MATH] , where [MATH] if and only if [MATH] for some [MATH] , see BGL , all deforms depend on even parameters |
BGLd These deforms preserve NIS with the same Gram matrix as the one determined by recipe ( ) applied to [MATH] except for the deform with cocycle [MATH] with parameter [MATH] distinct from [MATH] and [MATH] , when the Gram matrix is a different one, of the form |
[MATH] , where for the same numbering of Chevalley generators [MATH] [MATH] and [MATH] as in BGL for the Cartan matrix obtained from the matrix ( 10 ) at [MATH] , we have |
[EQUATION] and [EQUATION] where [MATH] The kernel of [MATH] is the center of [MATH] spanned by the vector [MATH] and the restriction of [MATH] to [MATH] is a NIS. |
For the deform of [MATH] with cocycle [MATH] , i.e., when the deformation parameter is equal to [MATH] , we only have a degenerate invariant symmetric bilinear form [MATH] for which [MATH] and zero on all other pairs of Chevalley basis vectors. |
4. Linear (matrix) and vectorial Lie (super)algebras; their simple relatives In representation theory, it is reasonable to consider Lie (super)algebras [MATH] of vector fields together with a natural topology defined by Weisfeiler filtration determined, in its turn, by a grading vector [MATH] , see LSh BGLLS . In our s... |
We describe each vectorial Lie superalgebra as the result of the (generalized) Cartan prolongation of a pair consisting of a Lie superalgebra and a module over it. |
4.1. Linear (matrix) Lie (super)algebras. The general linear Lie superalgebra of all supermatrices of size [MATH] corresponding to linear operators in the superspace [MATH] over the ground field [MATH] is denoted by |
[MATH] , where [MATH] is an ordered collection of parities of the basis vectors of [MATH] and [MATH] usually, for the standard (simplest) format, [MATH] is abbreviated to [MATH] . Any supermatrix from [MATH] can be uniquely expressed as the sum of its even and odd parts; in the standard format this is the following blo... |
[EQUATION] The supertrace is the map [MATH] [MATH] , where [MATH] . Thus, in the standard format, [MATH] Observe that for Lie superalgebra [MATH] over a supercommutative superalgebra [MATH] we have |
[EQUATION] so on odd supermatrices with entries in [MATH] such that [MATH] , the supertrace coincides with the trace. Since [MATH] , the subsuperspace of supertraceless matrices constitutes the special linear Lie subsuperalgebra |
[MATH] There are, however, at least two super versions of [MATH] , not one; for reasons, see Lsos , Ch1, Ch.7] . The other version — [MATH] — is called the queer |
Lie superalgebra and is defined as the one that preserves the complex structure given by an odd operator [MATH] , i.e., [MATH] is the centralizer [MATH] of [MATH] |
[EQUATION] It is clear that by a change of basis we can reduce [MATH] to the form (shape) [EQUATION] in the standard format and then [MATH] takes the form |
[EQUATION] On [MATH] , the queertrace is defined: [MATH] . Denote by [MATH] the Lie superalgebra of queertraceless matrices; set [MATH] |
4.1.1. Supermatrices of operators. To the linear map of superspaces [MATH] there corresponds the dual map [MATH] between the dual superspaces. In bases consisting of the homogeneous vectors [MATH] of parity [MATH] , and [MATH] of parity [MATH] , the formula |
[MATH] assigns to [MATH] the supermatrix [MATH] . In the dual bases, the supertransposed matrix [MATH] corresponds to [MATH] [EQUATION] |
4.1.2. Supermatrices of bilinear forms. The supermatrices [MATH] such that [EQUATION] constitute the Lie superalgebra [MATH] that preserves the bilinear form [MATH] on [MATH] whose matrix [MATH] is given by the formula |
[EQUATION] In order to identify a bilinear form [MATH] with an operator, an element of [MATH] , the matrix [MATH] of the bilinear form [MATH] is defined in Lsos , Ch.1] by eq. ( 12 ), not by seemingly natural but inappropriate for such an identification formula |
[EQUATION] Moreover, for the odd forms [MATH] , the definition ( 13 ) contradicts the obvious symmetry of the NIS defined by [MATH] on [MATH] . Indeed, the symmetry of a homogeneous form [MATH] means, according to Lsos , Ch.1] , that [MATH] for any [MATH] and [MATH] , i.e., its matrix [MATH] satisfies the condition |
[EQUATION] Similarly, antisymmetry of [MATH] means that [MATH] Thus, we see that the upsetting of bilinear forms [MATH] , which for the spaces |
and the case where [MATH] is expressed on matrices in terms of the transposition, is a new operation, not supertransposition. Observe that the passage from [MATH] to [MATH] turns every symmetric form [MATH] on [MATH] into an antisymmetric one on [MATH] |
Most popular normal forms (shapes) of the even nondegenerate supersymmetric form are the ones whose supermatrices in the standard format are in the following normal forms: |
[EQUATION] The usual notation for [MATH] is [MATH] sometimes one writes more explicitly, [MATH] . Observe that the antisymmetric non-degenerate bilinear form is preserved by the “symplectico-orthogonal” Lie superalgebra, [MATH] or, more prudently, [MATH] , which is isomorphic to |
[MATH] A nondegenerate symmetric odd bilinear form [MATH] can be reduced to a normal shape whose matrix in the standard format is |
[MATH] , see ( 14 ), NOT [MATH] . A normal shape of the anti symmetric odd nondegenerate form in the standard format is [MATH] . The usual notation for |
[MATH] is [MATH] The passage from [MATH] to [MATH] establishes an isomorphism [MATH] . These isomorphic Lie superalgebras are called, as A. Weil suggested, periplectic |
Observe that, though the Lie superalgebras [MATH] and [MATH] , as well as [MATH] and [MATH] , are isomorphic, the difference between them is sometimes crucial, e.g., their Cartan prolongs, see Subsections 4.6 |
4.7 4.10 , are totally different, see Sh5 The special periplectic superalgebra is simple; it is defined to be [EQUATION] Of particular interest to us will also be the Lie superalgebras (here [MATH] |
[EQUATION] and the nontrivial central extension [MATH] of [MATH] that we describe after some preparation. Finally, observe that the term super symmetric applied to the bilinear forms in the title of this paper refers to the property [MATH] of the matrix of the bilinear form ( 12 ); according to the definition ( 14 ) th... |
4.2. A. Sergeev’s central extension [MATH] of [MATH] In 1970’s A. Sergeev proved that over [MATH] there is just one nontrivial central extension of [MATH] for [MATH] . It exists only for [MATH] and we denote it by [MATH] . (For a generalization of Sergeev’s result to analogs of [MATH] over fields [MATH] of characterist... |
[MATH] and [MATH] is the central element. The bracket in [MATH] is [EQUATION] where [MATH] is extended via linearity from matrices |
[MATH] on which [MATH] for any even permutation [MATH] The Lie superalgebra [MATH] can also be described with the help of the spinor representation, see LShs . For this, consider the Poisson superalgebra [MATH] , the Lie superalgebra whose superspace is the Grassmann superalgebra [MATH] |
[MATH] and the bracket is the Poisson bracket ( 36 ). Recall that [MATH] . Now, observe that [MATH] can be embedded into [MATH] , see ShM . Indeed, setting [MATH] for all [MATH] we introduce [MATH] -grading on [MATH] which, in turn, induces [MATH] -grading on [MATH] of the form |
[MATH] . Since [MATH] , we can identify [MATH] with [MATH] It is not difficult to see that the elements of degree [MATH] in the standard gradings of [MATH] and [MATH] constitute isomorphic |
[MATH] -modules. It is subject to a direct verification that it is possible to embed [MATH] into [MATH] Sergeev’s extension [MATH] is the result of the restriction of the cocycle that turns [MATH] into |
[MATH] to [MATH] . The quantization deforms [MATH] into [MATH] ; the through maps [MATH] are representations of [MATH] in the [MATH] -dimensional modules [MATH] isomorphic to each other for all [MATH] . The explicit form of |
[MATH] is as follows: [EQUATION] where [MATH] is the unit matrix and [MATH] is defined in the line under eq. ( 16 ). Clearly, [MATH] is an irreducible representation for any [MATH] |
4.3. Projectivization. If [MATH] is a Lie algebra of scalar matrices, and [MATH] is a Lie subsuperalgebra containing [MATH] , then the projective |
Lie superalgebra of type [MATH] is [MATH] . Examples: [MATH] [MATH] [MATH] [MATH] whereas [MATH] if [MATH] 4.4. “Classical” series of vectorial Lie superalgebra over [MATH] |
In the table, FD marks the particular cases of finite dimension. [EQUATION] We continue explaining the notation used in Table ( 18 ) till Subsection 4.15 |
1) General algebras . Let [MATH] , where the [MATH] are even indeterminates and the [MATH] are odd ones. Set [MATH] ; it is called the general vectorial Lie superalgebra |
2) Special algebras . The divergence of the field [MATH] is the function (in our case: a polynomial, or a series) [EQUATION] [MATH] The Lie superalgebra [MATH] is called the special (or |
divergence-free vectorial superalgebra Equivalently, [EQUATION] where [MATH] is the volume form with constant coefficients in coordinates [MATH] and [MATH] |
the Lie derivative along the vector field [MATH] [MATH] The Lie superalgebra [MATH] is called the traceless special vectorial superalgebra |
[MATH] The deform of [MATH] is the Lie superalgebra [EQUATION] where [MATH] . It is called the deformed special (or divergence-free vectorial superalgebra . Clearly, [MATH] for [MATH] . So we briefly denote these deforms by [MATH] Observe that for [MATH] odd the parameter of deformation, [MATH] , is odd. |
3) The algebras preserving Pfaff equations and certain differential 1- and 2-forms [MATH] Set [MATH] ; let [EQUATION] The form [MATH] is called contact , the form |
[MATH] is called symplectic . Sometimes it is more convenient to set [MATH] where [EQUATION] and in place of [MATH] or [MATH] take |
[MATH] and [MATH] , respectively, where [EQUATION] The contact Lie superalgebra is the one that preserves the Pfaff equation [EQUATION] |
or, equivalently, preserves the distribution singled out by the form [MATH] , i.e., the superalgebra [EQUATION] The Lie superalgebra |
[EQUATION] is called the Poisson superalgebra. The above “symmetric” expression of [MATH] is popular among algebraists; due to its symmetry it is convenient in computations. In mechanics and differential geometry (in pre-super era), and in characteristic [MATH] , the following expression of the form [MATH] is used: |
[EQUATION] [MATH] Similarly, set [MATH] let [MATH] be odd. Set (expressions in parentheses are for characteristic [MATH] [EQUATION] |
and call these forms, as A. Weil advised, the pericontact and periplectic , respectively. The pericontact Lie superalgebra is the one that preserves the Pfaff equation |
[EQUATION] or, equivalently, preserves the distribution singled out by the form [MATH] [EQUATION] The Lie superalgebra [EQUATION] |
is called the Buttin superalgebra in honor of C. Buttin who was the first to show that the Schouten bracket of the functions that generate [MATH] , see ( 38 ), satisfies the super Jacobi identity. |
The following are respective divergence-free (or special ) Lie subsuperalgebras [EQUATION] 4.5. Generating functions. [MATH] Odd form [MATH] . For any [MATH] , set: |
[EQUATION] where [MATH] (here the [MATH] are all the coordinates except [MATH] ) is the Euler operator and [MATH] is the hamiltonian field with Hamiltonian [MATH] that preserves [MATH] |
[EQUATION] The choice of the form [MATH] instead of [MATH] only affects the shape of [MATH] that we give for [MATH] [EQUATION] [MATH] |
Even form [MATH] . For any [MATH] , set: [EQUATION] where [MATH] (here the [MATH] are all the coordinates except [MATH] ), and [EQUATION] |
Since [EQUATION] it follows that [MATH] and [MATH] Observe that [EQUATION] [MATH] To the (super)commutators [MATH] or [MATH] there correspond contact brackets of the generating functions: |
[EQUATION] The explicit formulas for the contact brackets are as follows. Let us first define the brackets on functions that do not depend on [MATH] |
(resp. [MATH] ). The Poisson bracket [MATH] (in the realization with the form [MATH] ) is given by the formula [EQUATION] and in the realization with the form [MATH] for [MATH] it is given by the formula |
[EQUATION] The Buttin bracket , or Schouten bracket , a.k.a. antibracket [MATH] is given by the formula [EQUATION] In terms of the Poisson and Buttin brackets, respectively, the contact brackets are |
[EQUATION] and [EQUATION] The Lie superalgebra of Hamiltonian fields (or Hamiltonian superalgebra ) and its special subalgebra (defined only if [MATH] are |
[EQUATION] The “odd” analogues of the Lie superalgebra of Hamiltonian fields are the Lie superalgebra of vector fields [MATH] introduced in L1 and its special subalgebra: |
[EQUATION] It is not difficult to prove the following isomorphisms (as superspaces): [EQUATION] We see that the commutants (first derived algebras) in the purely odd case can be described as |
[EQUATION] 4.6. The Cartan prolongs. We will repeatedly use the Cartan prolongation. So let us recall the definition and generalize it somewhat. Let [MATH] be a Lie algebra, |
[MATH] [MATH] -module, [MATH] the operator of the [MATH] th symmetric power. Set [MATH] and [MATH] Recall that, for any (finite-dimensional) vector space [MATH] , we have |
[EQUATION] where [MATH] is the space of [MATH] -linear maps and we have [MATH] -many [MATH] ’s on both sides. Now, we recursively define, for any [MATH] |
[EQUATION] The space [MATH] is said to be the [MATH] th Cartan prolong (the result of the Cartan prolongation ) of the pair [MATH] |
Equivalently, let [EQUATION] be the natural maps. Then [MATH] The Cartan prolong of the pair [MATH] is [MATH] If the [MATH] -module [MATH] is faithful there exists an injective linear map [MATH] such that |
[EQUATION] It is subject to a direct verification that the Lie algebra structure on [MATH] induces a Lie algebra structure on [MATH] . In what follows we do not indicate [MATH] ; the space [MATH] has a natural Lie algebra structure even if the [MATH] -module [MATH] is not faithful. |
4.7. A generalization of the Cartan prolong. Let [MATH] be a [MATH] -graded Lie algebra and [MATH] a Lie subalgebra in the algebra of the [MATH] -grading-preserving derivations. Let |
[EQUATION] be natural analogs of maps ( 45 ). For [MATH] , define the [MATH] th prolong of the pair [MATH] to be: [EQUATION] where the subscript [MATH] in the right hand side singles out the component of degree [MATH] |
Set [MATH] then, as is easy to verify, [MATH] is a Lie algebra. Superization of (generalized) Cartan prolongation procedure is immediate. |
4.8. More notation. The tautological representation of a matrix Lie superalgebra [MATH] or its subalgebra [MATH] in [MATH] over the ground field [MATH] , and sometimes the module [MATH] itself are denoted by [MATH] or, for clarity, |
[MATH] . The context prevents confusion of these notations with that of the identity (scalar) operator [MATH] on the space [MATH] as in the next paragraph: |
For [MATH] , the trivial representation of [MATH] is denoted by [MATH] (if [MATH] is simple) whereas [MATH] denotes a representation of [MATH] |
trivial on the semisimple part of [MATH] and such that [MATH] is the value of the central element [MATH] from [MATH] , where [MATH] is chosen so that |
[MATH] Hereafter, [MATH] , the trivial central extension of [MATH] 4.9. Vectorial Lie superalgebras as Cartan prolongs. Superizations of the constructions described in sec. 4.6 are straightforward: via Sign Rule. For its application to [MATH] , see Subsetion 6.7.2 . We thus get infinite-dimensional Lie superalgebras (s... |
[EQUATION] The contact Lie superalgebras and exceptional ones are Cartan prolongs [MATH] with nilpotent (and noncommutative) negative part [MATH] ; let us describe them. |
[MATH] Define the Lie superalgebra [MATH] , where [MATH] and [MATH] is endowed with a nondegenerate antisupersymmetric bilinear form [MATH] , with the following relations: |
[EQUATION] Clearly, we have [EQUATION] [MATH] The “odd” analog of [MATH] is associated with the following “odd” analog of [MATH] . Denote by [MATH] the |
antibracket Lie superalgebra ( [MATH] is Anti-Bracket read backwards), where [MATH] is an [MATH] -dimensional superspace endowed with a nondegenerate antisupersymmetric odd bilinear form [MATH] ; the bracket in [MATH] is given by the following relations: |
[EQUATION] Clearly, [EQUATION] 4.10. A partial Cartan prolong: prolongation of a positive part. Let [MATH] be [MATH] -submodule such that [MATH] . If such |
[MATH] exists (usually, [MATH] ), define the 2nd prolong of [MATH] to be [EQUATION] The terms [MATH] , where [MATH] , are similarly defined. Set |
[MATH] for [MATH] and [MATH] Examples [MATH] is a subalgebra of [MATH] . The former is obtained as the Cartan prolong of the same nonpositive part as [MATH] and a submodule of [MATH] . The simple exceptional superalgebra [MATH] introduced in |
Sh5 Sh14 , see table ( 61 ), is another example. 4.11. Traces and divergencies on vectorial Lie superalgebras. On any Lie algebra [MATH] over a field |
[MATH] , a trace is any linear map [MATH] such that [EQUATION] Now, let [MATH] be a [MATH] -graded vectorial Lie algebra with [MATH] |
[MATH] , and let [MATH] be a trace on [MATH] . Recall that any [MATH] -grading of a given vectorial Lie algebra is given by degrees of the indeterminates, so the space of functions is also |
[MATH] -graded. Let [MATH] be the superalgebra of “functions” (divided powers in indeterminates [MATH] on the [MATH] -dimensional superspace, if [MATH] ). The divergence |
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