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[MATH] is a degree-preserving [MATH] -invariant prolongation of the trace satisfying the following conditions, so that [MATH] , i.e., is a cocycle: |
[EQUATION] 4.11.1. Divergence-free subalgebras. [MATH] In [MATH] , this is [MATH] [MATH] In [MATH] . Since, as is easy to calculate, |
[EQUATION] it follows that the divergence-free subalgebra of the contact Lie superalgebra either coincides with the whole algebra (for [MATH] ), or is isomorphic to the Poisson superalgebra [MATH] |
[MATH] In [MATH] , the situation is more interesting: in the standard grading of [MATH] , the codimension of [MATH] in [MATH] is equal to 2; hence, there are two linearly independent traces and two cohomologically inequivalent divergences corresponding to these traces. Let [MATH] be the restriction of the divergence fr... |
[EQUATION] The divergence-free (relative to [MATH] ) subalgebra of the pericontact superalgebra is [EQUATION] In particular, [EQUATION] |
Set [EQUATION] The other divergence can be selected to be [MATH] , where [MATH] . The subalgebra of [MATH] corresponding to a linear combination of the two divergences is described in Subsection 4.14 |
4.11.2. Traceless subalgebras. In terms of super [MATH] -functor, the superdimension [MATH] is an element [MATH] , where [MATH] . Lie superalgebras [MATH] [MATH] and [MATH] |
have traceless ideals [MATH] [MATH] and [MATH] defined from the exact sequences [EQUATION] For more examples of traceless subalgebras, see ( 68 ). |
4.12. The exceptional simple vectorial Lie superalgebras. The five exceptional simple vectorial Lie superalgebras are given below as Cartan prolongs [MATH] or generalized Cartan prolongs [MATH] |
For depth [MATH] , for [MATH] , we write [MATH] instead of [MATH] . The corresponding terms [MATH] for [MATH] are given in 61 ) and ( 62 ); for notation, see Subsection 4.8 |
The non-positive components of exceptional simple vectorial Lie superalgebras over [MATH] [EQUATION] None of the above [MATH] -graded vectorial Lie superalgebras ( 18 ) and ( 61 ) is of depth [MATH] and only one is of depth 3: |
[EQUATION] 4.13. The modules of tensor fields. Let [MATH] and [MATH] in the standard grading ( [MATH] for all [MATH] ). For any other [MATH] -graded vectorial Lie superalgebra for whose component [MATH] the notion of lowest weight can be defined, the construction is identical. |
Let [MATH] be the [MATH] -module with the lowest weight [MATH] . Let us make [MATH] into a [MATH] -module by setting [MATH] for [MATH] . Let us realize [MATH] by vector fields on the linear supermanifold [MATH] with coordinates [MATH] . The superspace [MATH] is isomorphic, due to the Poincaré–Birkhoff–Witt theorem, to ... |
tensor fields of type [MATH] . When [MATH] we will simply write [MATH] instead of [MATH] . We will usually consider [MATH] -modules coinduced from irreducible |
[MATH] -modules. For example, [MATH] is the superspace of functions; [MATH] (the bar separates the first [MATH] (“even”) coordinates of the weight with respect to the matrix units [MATH] of [MATH] ) is the superspace of |
densities or volume forms We denote the generator of [MATH] , as [MATH] -module, corresponding to the ordered set of coordinates [MATH] by [MATH] The space of [MATH] -densities is [MATH] . In particular, |
[MATH] but [MATH] As [MATH] - and [MATH] -modules, [MATH] 4.14. Deformations of the Buttin superalgebra (after LSq ). As is clear from the definition of the Buttin bracket, there is a regrading (namely, [MATH] given by |
[MATH] for all [MATH] ) under which [MATH] initially of depth 2, takes the form [MATH] with [MATH] and [MATH] . Replace now the [MATH] -module [MATH] of functions (with inverted parity) by the rank 1 (over the algebra of functions) module of |
[MATH] -densities (also with inverted parity), i.e., we set [MATH] , where the [MATH] -action on the generator [MATH] is given by the formula |
[EQUATION] Define [MATH] to be the Cartan prolong [EQUATION] Clearly, this is a deform of [MATH] . The collection of these [MATH] for all [MATH] ’s is called the main deformation , the other deformations, defined in what follows, will be called singular |
The deform [MATH] of [MATH] is a regrading of [MATH] described as follows. For [MATH] , set [EQUATION] For future use, we denote the operator that singles out |
[MATH] in [MATH] as follows: [EQUATION] Taking into account the explicit form of the divergence of [MATH] we get [EQUATION] It is subject to a direct verification that [MATH] for [MATH] . This isomorphism shows that [MATH] actually runs over [MATH] , not [MATH] as one might hastily think. The Lie superalgebras [MATH] |
are simple for [MATH] and [MATH] , 1, [MATH] for reasons clear from eq. ( 68 ). It is also clear that the [MATH] are non-isomorphic for distinct [MATH] ’s, bar occasional isomorphisms, see LSh |
The Lie superalgebra [MATH] is not simple: it has a [MATH] -dimensional center. Observe that [MATH] and [MATH] are not simple either. The corresponding exact sequences are |
[EQUATION] Clearly, at the exceptional values of [MATH] , i.e., 0, 1, and [MATH] , the deformations of [MATH] should be investigated extra carefully. As we will see immediately, it pays: at each of the exceptional points we find extra deformations. An exceptional deformation at [MATH] remains inexplicable. Other except... |
4.15. Deforms of [MATH] LSq ). For [MATH] , set [MATH] for brevity. Then 1) [MATH] for [MATH] unless [MATH] [MATH] [MATH] [MATH] for [MATH] . For [MATH] , in addition to the above, [MATH] at [MATH] and |
[MATH] 2) At the exceptional values of [MATH] listed in heading [MATH] we have [MATH] at [MATH] and [MATH] odd, or [MATH] and [MATH] even, or [MATH] and [MATH] or |
[MATH] [MATH] at [MATH] , or [MATH] and [MATH] odd, or [MATH] and [MATH] even The corresponding cocycles [MATH] are given by the following nonzero values in terms of the generating functions [MATH] and [MATH] where [MATH] is the degree of [MATH] with respect to odd indeterminates only; for [MATH] we set |
[MATH] and [MATH] [EQUATION] On [MATH] (the latter being the non-standard W-grading in which [MATH] , other indeterminates being of degree 1) , the cocycle [MATH] is the one induced on [MATH] |
by the usual deformation (quantization) of [MATH] : we first quantize [MATH] and then take the quotient modulo the center 3) The space [MATH] is diagonalizable with respect to the Cartan subalgebra of [MATH] . Let the cocycle [MATH] corresponding to the main deformation be one of the eigenvectors. Let [MATH] be another... |
All the singular deformations of the bracket [MATH] in [MATH] , except the ones for [MATH] or [MATH] and [MATH] , have the following very simple form even for [MATH] |
[EQUATION] Since the elements of [MATH] are encoded by functions (for us: polynomials or formal power series) in [MATH] [MATH] and |
[MATH] subject to one relation with an odd left hand side in which [MATH] enters, it seems plausible that the bracket in [MATH] can be, at least for generic values of parameter |
[MATH] , expressed solely in terms of [MATH] and [MATH] . This is, indeed, the case, and here is an explicit formula (in which [MATH] is the usual antibracket and |
[MATH] ): [EQUATION] and [MATH] is computed with respect to the standard grading [MATH] 4.16. Vectorial Lie (super)algebras for [MATH] |
Recall that we are interested in simple Lie (super)algebras; their central extensions and the algebras of derivations, though no less interesting per se, are next objects on our agenda. Every simple [MATH] -graded Lie (super)algebra [MATH] is transitive , i.e., such that |
[EQUATION] Over any field [MATH] of characteristic [MATH] , in order for the analogs of vectorial Lie (super)algebras be transitive , we must change the definition in the two places: |
(A) Consider not polynomial coefficients but divided powers in [MATH] even and [MATH] odd indeterminates, whose powers are bounded by the shearing vector [MATH] , usually abbreviated to [MATH] , forming the supercommutative superalgebra (here |
[MATH] [EQUATION] where [MATH] . Set [MATH] (B) Introduce distinguished partial derivatives [MATH] each of them serving as several partial derivatives at once, for each of the generators [MATH] [MATH] [MATH] , …(or, in terms of [MATH] ): |
[EQUATION] The (general) Lie (super)algebra of vector fields is [EQUATION] In what follows, speaking about Lie SUPERalgebras, we assume that [MATH] unless otherwise specified. |
4.16.1. Linguistics: names. Priorities. In the old literature, the Lie algebra [MATH] of vector fields with coefficients in the algebra of divided powers [MATH] was called “the general Lie algebra of Cartan type”; in the modern literature, it is called the Jacobson–Witt algebra ; it is usually denoted by [MATH] for any... |
Because [MATH] , we may assume that [MATH] speaking about the contact Lie algebras [MATH] . In what follows, for [MATH] , just ignore the shearing vector [MATH] . For the proof on non-existence of NIS on [MATH] , see Zus |
Dzhumadildaev generalized a result due to Block (1958) and listed simple [MATH] -graded vectorial Lie algebras of the four series of “Cartan type” in characteristic [MATH] that have NIS, see Dz1 . Proofs appeared later in Dz2 , §2] . (Dzhumadildaev also did the same, albeit in a somewhat implicit way and also, as in Dz... |
To extend Dzumadildaev’s result to filtered deforms of the algebras he considered, we reproduce an explicit description of these NISes from SF , and of deforms, see BGLd and DK |
4.17. Filtered deforms of [MATH] and [MATH] The Lie (super)algebras of series [MATH] and [MATH] are not simple, but several of their simple relatives are subquotients of their filtered deforms. |
4.17.1. Types of Lie algebras [MATH] described by Tyurin and Wilson Tyu Let [MATH] be the algebra of formal power series in [MATH] for [MATH] . For [MATH] , suppose that |
[EQUATION] Wilson proved that there are only the following three types of non-equivalent classes of volume forms, and hence filtered deforms of divergence-free algebras preserving them: |
[EQUATION] For brevity, set [MATH] and [MATH] Remarks 1) Kac observed that if [MATH] , then [MATH] with all coordinates of [MATH] finite although [MATH] , see KfiD |
2) For [MATH] and 2, the deforms ( 72 ) of [MATH] are also possible; nobody knows if there are no other, non-isomorphic, ones, whereas for [MATH] there definitely is at least one more deform, its existence is the most spectacular result of BGLLS |
3) S. Tyurin Tyu described the Lie algebras of divergence-free type and got an extra type of volume forms, as compared with Wilson’s list ( 72 ); Tyurin missed an equivalence. |
3) S. Kirillov Kir verified Skryabin’s remark in Sk2 , namely for which [MATH] is the [MATH] th derived algebra from Wilson’s list ( 72 ) simple and what is its dimension: |
[EQUATION] 4.17.2. Types of Hamiltonian Lie algebras classified by S. Skryabin Sk1 Sk2 Let [MATH] be the [MATH] -graded Lie algebra preserving the symplectic form |
[MATH] . Its nonisomorphic filtered deforms are only the following [MATH] , where [MATH] , preserving the respective forms. For [MATH] , let [MATH] be the Lie algebra that preserves the form (Skryabin calls it of [MATH] nd type |
[EQUATION] S. Skryabin proved ( Sk2 ) that for [MATH] the only other, inequivalent to ( 74 ) and to each other indecomposable symplectic forms (Skryabin calls them of [MATH] st type ) are those that can be reduced to the following normal shapes |
[EQUATION] and [MATH] can only be equal to one of the following: [EQUATION] where [MATH] is a Jordan [MATH] block with eigenvalue [MATH] , and [MATH] is a [MATH] block matrix with blocks of size [MATH] , so [MATH] for some [MATH] |
[EQUATION] 4.17.3. The two conditions on [MATH] and [MATH] 1) The case with [MATH] occurs only when [EQUATION] and, furthermore, [MATH] for all [MATH] and all [MATH] , i.e., the [MATH] indeterminates in each of the [MATH] successive groups have equal heights. |
The case with [MATH] occurs only when condition ( 76 ) is violated. 2) Let [MATH] be the group permutations of the coordinates of vectors in [MATH] . The 2nd condition requires the identity element to be the only element in [MATH] that fixes the two vectors [MATH] and [MATH] simultaneously. It suffices to consider repr... |
Implicit brackets For a basis in [MATH] we take Hamiltonian vector fields [EQUATION] where the generating functions run over the union of a basis in the maximal ideal of [MATH] and the collection of non-existing in [MATH] for the vector [MATH] with finite coordinates Hamiltonians [MATH] for [MATH] . These Hamiltonians,... |
4.18. How conditions for simplicity change under the passage from [MATH] to [MATH] The “natural” objects are vectorial Lie algebras obtained as a result of a (generalized) Cartan prolongation. Such objects are often not simple, the simple “derived” (figuratively speaking) of these objects are their first or second deri... |
[MATH] S. Kirillov Kir checked for which [MATH] the [MATH] th derived algebra of the Hamiltonian Lie algebra from Skryabin’s list is simple and what is its dimension: |
[EQUATION] [MATH] Over [MATH] , the supervarieties of parameters of deformations of Poisson and Hamiltonian Lie superalgebras can differ, see LSq . For [MATH] , the forms [MATH] , see eq. ( 75 ), do not exist; but instead there is a 1-parametric family of non-isomorphic deforms different from the above — desuperisation... |
[EQUATION] 5. On existence of NIS on vectorial Lie (super)algebras We see two types of reasons why there is a NIS on a given [MATH] ; let us consider them in detail. |
(1) There is only one geometric structure leading to NIS: volume . Invariant with respect to any changes of indeterminates (coordinates) unary operators between sections of tensor fields on a (super)manifold [MATH] with fibers irreducible under the action of the Lie (super)algebra of linear changes of indeterminates ar... |
(2) An algebraic reason for existence of a NIS on the [MATH] -graded Lie (super)algebras [MATH] is a necessary condition, see Lemma 5.1 |
5.1. Lemma (Corollary 4.9 in Ch.3 in SF ). Let [MATH] be a finite-dimensional simple [MATH] -graded Lie algebra, [MATH] a NIS on [MATH] . Then |
[EQUATION] From Lemma 5.1 one deduces — with certain effort — the following statement. 5.2. Corollary (Theorems 6.3 and 6.4 in Ch. 4 in SF ). |
1) The Lie algebra [MATH] has a NIS if and only if either [MATH] and [MATH] in which case NIS is [EQUATION] or [MATH] in which case NIS is |
[EQUATION] where [MATH] and [MATH] 2) If [MATH] , then [MATH] has a NIS if and only if [MATH] ; explicitly [EQUATION] where (for possible future use we give the formula for the general super case) |
[EQUATION] and extending the form ( 82 ) to other pairs of elements by invariance. 5.3. Series [MATH] Classification allows us to forget about deforms of [MATH] for [MATH] and study other series. For example, the true deforms of Lie algebras [MATH] are described in DK ; they are either certain filtered deforms, or isom... |
5.4. Series [MATH] Due to the following “occasional isomorphisms” (taking place for reasons given in parentheses below) we do not consider [MATH] for the following values of parameters: |
[EQUATION] 5.4.1. Conjecture. [Proved for [MATH] and [MATH] There is no NIS on simple Lie algebras [MATH] and [MATH] 5.5. Series [MATH] |
For [MATH] , consider the Lie algebra [MATH] and a standard symplectic form [MATH] , set [EQUATION] 5.5.1. Lemma. For all symplectic forms [MATH] given by the expressions ( 74 ) and ( 75 ), a NIS on [MATH] is given by the following formula |
[EQUATION] Proof. Since [MATH] is a derivation, the invariance of ( 83 ) can be written as [EQUATION] It is easy to check that [EQUATION] |
Observe that if [MATH] , then [MATH] for all symplectic forms given by formulas ( 74 ) and ( 75 ). We see that [MATH] , where [MATH] is the term proportional to [MATH] If [MATH] for some non-zero [MATH] , then |
[EQUATION] Since [MATH] and [MATH] is arbitrary, we see that [MATH] . Therefore, the form [MATH] is non-degenerate on [MATH] 5.6. Series [MATH] and its subalgebras. |
On the supermanifold [MATH] with a contact structure given by [MATH] , consider the space of [MATH] -densities, [EQUATION] where [MATH] is the space of “functions” and [MATH] . We have [MATH] , where [MATH] is the space of volume forms. For [MATH] , NIS obviously exists on the space of [MATH] -densities, i.e., on [MATH... |
Observe that this NIS can be odd, but there are two natural ways to define the parity of it, cf. MaG ; in our situation, where the integral is considered to be even for any [MATH] if [MATH] , the adequate definition is [MATH] |
[EQUATION] Since [EQUATION] it follows that [EQUATION] 5.7. No NIS on [MATH] and [MATH] Eq. ( 84 ) shows there is a NIS on [MATH] , so we have to verify if its restriction to [MATH] , see eq. ( 61 ), is non-degenerate. Since [MATH] , the only element it can be paired with a non-zero result is the highest possible power... |
The Lie superalgebra [MATH] has no NIS by argument ( 84 ) and thanks to Lemma 5.1 5.8. [MATH] : NIS on the Melikyan algebra. This fact is known, see, e.g., . The condition in ( 84 ) is satisfied for [MATH] [MATH] . For descriptions of the Melikyan algebra [MATH] as a subalgebra in [MATH] , see GL . Therefore, to prove ... |
5.9. [MATH] : NIS on some of Skryabin algebras. No NIS on Ermolaev and Frank algebras. The exceptional simple [MATH] -graded vectorial Lie algebras known today to be indigenous to [MATH] are described in GL , where the number [MATH] of parameters, the shearing vector depends on, is correctly computed for the first time... |
[EQUATION] Lemma 5.1 implies that among the algebras ( 85 ), only the following simple Skryabin Lie algebras may have NIS, and they do have it (we proved this for [MATH] only): |
[EQUATION] Observe that all of the Skryabin algebras are results of generalized, perhaps partial, Cartan prolongation of the non-positive part of [MATH] in one of the [MATH] -gradings of [MATH] ; both of its Cartan matrices were discovered by Skryabin (for a method proving that these are the only possible Cartan matric... |
5.10. NIS on the two simple stringy Lie (super)algebras over [MATH] For the classification of simple stringy Lie superalgebras over [MATH] , i.e., vectorial ones on a supercircle, see GLS . We see that [MATH] , where superscript [MATH] is for Laurent, and hence the Neveu–Schwarz type stringy Lie superalgebra [MATH] has... |
For the Ramond-type stringy Lie superalgebra [MATH] preserving the distribution given by the form equivalent to [MATH] (so the weight of [MATH] is equal to 0) on the supercircle associated with the Whitney sum of the [MATH] -dimensional cylinder and the Moebius bundle, for [MATH] , we have |
[EQUATION] Therefore, [MATH] has NIS if [MATH] , i.e., [MATH] ; this NIS is an odd one. (For [MATH] , the condition ( 84 ) turns into [MATH] . Hence, no NIS.) |
In terms of tensor fields, we see that [EQUATION] where [MATH] is a 1-dimensional module given by the supertrace, hence [MATH] always. |
The cases [MATH] , and its deform [MATH] preserving [MATH] , are similar to the cases of [MATH] , except for [MATH] with its decomposition [MATH] where [MATH] and [MATH] for the tautological [MATH] -module [MATH] and the [MATH] -valued space of loops [MATH] in this case, a NIS could have existed but does not, see Theor... |
6. Odd NIS. Queer Lie superalgebras, queerified Lie algebras, and exceptions 6.1. Odd NIS on [MATH] The queertrace [MATH] on [MATH] , see ( 11 ), defines a natural NIS on [MATH] |
[EQUATION] In the same way as the (super)trace on the associative superalgebra of supermatrices [MATH] gives rise to a NIS on [MATH] , the queertrace gives rise to an odd NIS on the Lie superalgebra [MATH] which is simple if [MATH] , and also if [MATH] and [MATH] |
6.2. Queerified Lie algebras. The Lie superalgebra [MATH] is a “queerification” of the associative algebra [MATH] . If [MATH] , it is possible to “queerify” any Lie algebra, see BLLSq |
Let [MATH] and [MATH] restricted Lie algebra with a NIS [MATH] . Then [MATH] has an odd NIS [MATH] defined as follows, where [MATH] is the change of parity operator: |
[EQUATION] 6.3. Poisson Lie superalgebras. Let us show that both Poisson Lie superalgebras and their particular deforms resulting from quantization have NIS if [MATH] , see ( 90 ). We do not know if this is so if [MATH] |
6.3.1. Poisson Lie superalgebras over [MATH] On [MATH] realized on the space [MATH] , where [MATH] and [MATH] for [MATH] , or on the space [MATH] for [MATH] , NIS of parity [MATH] is defined by the formula |
[EQUATION] and where [MATH] is the Berezin integral [MATH] the coefficient of the [MATH] th degree monomial of [MATH] in the Taylor series expansion. |
Tyutin classified the deforms of [MATH] , see Ty . There is just one class [MATH] of deformations, called quantization [MATH] sends the integral into the super trace (resp. queer trace): |
[EQUATION] 6.3.2. NIS on Poisson Lie superalgebras over [MATH] for [MATH] NIS is defined by the direct analog of formula ( 88 ). |
Aside . For the case where the shearing vector is of the form [MATH] with [MATH] , and odd indeterminates [MATH] and [MATH] , the description of quantizations is of the same form as over [MATH] , i.e., |
[EQUATION] There are, however, other deforms of [MATH] : e.g., for [MATH] , there are Melikyan algebras, see KD ; for [MATH] , there is a 1-parametric family — desuperization of [MATH] , see BGLLS |
6.4. NIS on [MATH] and [MATH] Recall that by [MATH] we denote the Lie superalgebra on the space of functions (divided powers if [MATH] ) in [MATH] even and [MATH] odd indeterminates with Schouten bracket a.k.a. antibracket. Let [MATH] and |
[EQUATION] The direct analog of ( 88 ) defines NIS on the Lie superalgebra [MATH] , and its restriction to its subquotient [MATH] is a NIS. Observe that [MATH] |
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