Leites' (super)questions

Pasha Zusmanovich


Date: April 30, 2003, last revised January 5, 2005

This is an account of (some of the) questions posed by Dmitry Leites, compiled by me during the last few years. While the essence belongs to him, all wording, (mis)interpretations, as well as possible errors are mine. More thorough and inspirational exposition belonging to Leites himself may be found in his and his collaborators (numerous) papers and preprints ([GL2], [GLS], [LLS], [Lei], [LS1], [LS2], [LS3], [LS4] and references therein).

The questions, being concentrated around one topic - Lie superalgebras - may be roughly divided into three categories: first, the very particular questions about the very particular algebras (though nobody will beat you, except may be Leites himself under very particular circumstances, for a ``proper'' generalizations, like, say, replacing the Laurent polynomials ring by an arbitrary commutative associative ring), second, more ``generic'' or ``theoretical'' questions (like describing superalgebra structures on the whole cohomology space, precise relationship between Spencer and ordinary cohomology, elucidation in the supercase of certain issues well-known in the plain even setting, etc.), and third, to pursue, more or less, entirely new theories.

Cohomology in all dimensions

Cohomology of Hamiltonian (super)algebras

Cohomology of Poisson and Hamiltonian Lie (super)algebras, with emphasis and low-dimensional cohomology and deformations.

Particularly, compute cohomology with trivial coefficients of Hamiltonian Lie superalgebra h(0| n) for n > 4 ( h(0| 4) $ \simeq$ psl (2| 2), so covered by Leites-Fuchs in their paper on cohomology of classical Lie superalgebras ([LF]; later Shapovalov discovered one cocycle missed there - see [LS4, §2.2.1] and [Kor1, §4.1]). Compute cohomology with trivial coefficients of (infinite-dimensional) Hamiltonian Lie algebra H(n) (n = 2 considered by Gelfand-Kalinin-Fuchs in [GKF] with a very partial results). Study the behavior of cohomology of a (finite-dimensional) Hamiltonian Lie algebra in characteristic p as p$ \to$$ \infty$.

Computer calculations

To develop novel approaches to computer calculations of Lie superalgebras cohomology. What kind of sparse matrices arises? What methods are most suitable for dealing with these sparse matrices? (Raised also by Kornyak in [Kor2, §2.3])

Structure of cohomology algebra

([GL2, §4.1-4.2] and [LLS]) Describe the Nijenhuis-Richardson Lie superalgebra structure on H*(L, L) for some ``interesting'' (super)algebras L - not in terms of homogeneous components or generators and relations, like it is done traditionally, but from the structure theory viewpoint - i.e. determine its radical, semisimple part, etc. Say, for maximal nilpotent subalgebras of simple classical and vectorial Lie (super)algebras. The case of maximal nilpotent subalgebras of sl (3) and G2 was treated by Aleksei Lebedev (see [Leb] and [LLS]).

Conjectures ([GL3]):

  1. H*(psl (2| 2), psl (2| 2)) is generated by certain explicitly given cocycles of degrees 0,2 and 3.
  2. H*(osp(4| 2;$ \alpha$), osp(4| 2,$ \alpha$)) is exhausted by explicitly given cocycles in degrees 1 and 4.

A similar question of study of an algebra of simplicial cohomology (in the context of computer calculations) was raised in [DHSW]. It could be that something similar was undertaken for the cohomology of associative algebras and groups.

Structure of homotopy algebra

In his (never finished) ``superbook'' (one of the previous versions, p. 18), Leites writes: ``Whitehead's multiplication of the homotopy groups makes the set $ \oplus_{i}^{}$ $ \pi_{i}^{}$ a Lie superring''. Again, study this superring from the structure theory viewpoint.

Low-dimensional cohomology and related invariants

Invariants of ``classical'' Lie superalgebras

Describe (in all remaining cases) central extensions, derivations, automorphisms, invariant bilinear forms, deformations and forms (that is, over algebraically nonclosed fields, notably over $ \mathbb {R}$) of all (remaining cases) of ``interesting'' Lie superalgebras: Lie superalgebras of vector fields, ``stringy'' superalgebras and (possibly twisted) current superalgebras (all with polynomial, formal and Laurent coefficients). This is too broad and vague, so let's start with some particular questions:

Central extensions of vectorial superalgebras

([LS4, §6.2]) Prove that there are no central extensions of simple Lie superalgebras of vector fields with polynomial or formal coefficients, except the following:

  1. Poisson superalgebra po(2n| m) extending Hamiltonian one h(2n| m);
  2. Deformed Buttin superalgebra b$\scriptstyle \lambda$(n) extending le(n) (probably values $ \lambda$ = 1 or -1 are exceptional in some sense);
  3. Two amazing extensions of sleo(3) described by Shchepochkina and Post in [SP].
(Well, may be something else is missing, but the most nontrivial cases are certainly here).

Deformations of current algebras

([LS3, Warning 3 on p. 6]). It is probably proved (Lecomte and Roger, see [R]) that current and Kac-Moody (super)algebras are rigid (though probably all the cases, including super ones, are formally not covered). This is, however, ``ideologically wrong'', as there are interesting examples which are by all possible means should count as ``deforms'' of corresponding current algebras: namely, Krichever-Novikov algebras and example of P. Golod (for the latter, see [G] or [LS3], p. 38). So:

  1. Accurately compute all ``classical'' (= Gerstenhaber) deformations of current (super)algebras. Are all they rigid?
  2. Build an ``ideologically right'' deformation theory which will include Krichiver-Novikov and Golod algebras. Are there others?

Ideally, this should also include, as a special case, filtered deformations of modular W1(n) $ \otimes$ A (computed in [Z2]). Probably the unification can be done by considering ``Block (super)algebras'' as defined (in the non-super case) in [Z1].

Probably to some degree (ii) (especially in the context of Krichever-Novikov algebras) is answered by Fialowski.

Deformations and outer derivations of stringy superalgebras

Compute deformations and outer derivations of ``stringy'' superalgebras (i.e. exactly those described in [GLS]; some cocycles formulae there describing central extensions are incorrect). It is known ([HK]) that some of the initial algebras in the series, namely, kL(1| n) and kM(1| n) for n = 0, 1 have zero second degree cohomology in the adjoint module (and, consequently, are rigid), and that svectL(1| n) has at least one deformation (denoted as svect$\scriptstyle \lambda$L(1| n) in [GLS]). No new algebras are expected.

Deformations of Hamiltonian algebras

Accurately describe all deformations of Lie superalgebras of Hamiltonian vector fields h(2n| m). Why the case of h(2| 2) is exceptional? This is discussed, based on the previous (partially unpublished) works of Kochetkov, in [LS1].

Kochetkov in [Koc] described deformations of a Hamiltonian Lie algebra H(2) and Cheng and Kac in [CK] proved that there is no filtered (with respect to the standard grading) deformations of h(2n| m) (though the latter probably should be taken with a caution).

Filtered deformations

  1. Compute filtered deformations of Lie superalgebras of vector fields in all possible ``Weisfeiler''gradings (Cheng and Kac [CK] considered only standard gradings with even deformation parameter).

  2. Provide interesting (whatever it means) examples when different gradings of the same (super)algebra give rise to nonisomorphic filtered deformations. (This is, particularly, related to some gaps in Kac's classification(s); see [LS2], §I.4 and [LS4], §1.12).

Modules of tensor fields

Compute H1 for ``stringy'' superalgebras in modules of ``tensor fields'' and their generalizations (see [GLS, §1.3], [Lei, §1.2-1.3] and [P1]). This sounds to vague, so let's start from the following:

  1. H1(L, F$\scriptstyle \lambda$;$\scriptstyle \mu$) for L = kL(1| n) and kM(1| n).

  2. H1(L, F$\scriptstyle \lambda_{1}$,$\scriptstyle \lambda_{2}$;$\scriptstyle \mu$) for L = kL(1| 2) and kM(1| 3).

  3. for kLo(1| 4) and kMo(1| 5).

H1(L, F$\scriptstyle \lambda$) for L = kL(1| n) and k$\scriptstyle \alpha$(1| 4), and H1(L, T($ \lambda$,$ \mu$)) for L = kL(1| 2) and k+(1| 2) were computed by Poletaeva (unpublished M.Sc. thesis; see [P1] and [Lei, §3]).

Some additional papers possibly instrumental in understanding of the structure of algebras and modules involved: [P2] and [Kac]. Some low-dimensional cohomology related to tensor fields considered by Wagemann ([W]).

Structure functions

To compute structure functions associated with (twisted) current superalgebras (that is, ``twist'' and superize results of [Z3]). Or better yet, consider a ``nonholonomic'' situation, when the underlying Lie (super)algebra is not abelian, but ``negatively-graded'' nilpotent, with corresponding generalized Cartan prolong and generalized Spencer cohomology (see e.g. [GL2, §4.1]).

Other

Presentations of Lie algebras associated with Cartan matrix

Find ``reasonable'' presentation for Kac-Moody-type Lie (super)algebras associated with arbitrary Cartan matrix (the question is open even for ordinary Lie algebras and matrices of size 3 x 3). See [GL1] and [GL2, §§1.8.4 and 1.9].

Applications v narodnom hozyaistve

Apply calculations of Lie superalgebras cohomology to economical/financial problems, via nonholonomic systems. The initial reading is [S1] and [S2].

Volichenko algebras

(Further) study of Volichenko algebras (which are, roughly speaking, nonhomogeneous subalgebras of Lie superalgebras).

Bibliography

CK
S.-J. Cheng and V. G. Kac, Generalized Spencer cohomology and filtered deformations of $ \mathbb {Z}$-graded Lie superalgebras, Adv. Theor. Math. Phys. 2 (1998), 1141-1182; arXiv:math.RT/9805039.

DHSW
J.-G. Dumas, F. Heckenbach, S. Saunders and V. Welker, Computing simplical homology based on efficient Smith normal form algorithms, Algebra, Geometry and Software Systems (ed. M. Joswig and N. Takayama), Springer, 2003, pp. 177-206.

GKF
I. M. Gelfand, D. I. Kalinin and D. B. Fuchs, Cohomology of the Lie algebra of Hamiltonian formal vector fields, Funct. Anal. Appl. 6 (1972), 193-196.

G
P. Golod, A deformation of the affine Lie algebra A1(1) and hamiltonian systems on the orbits of its subalgebras, Group-theoretical methods in physics, Moscow, Nauka, 1986, pp. 368-376 (in Russian).

GL1
P. Grozman and D. Leites, Defining relations for Lie superalgebras with Cartan matrix, Czechoslovak J. Phys. 51 (2001), 1-21; arXiv:hep-th/9702073

GL2
P. Grozman and D. Leites, SuperLie and problems (to be) solved with it, Preprint MPI Bonn, MPI 2003-39.

GL3
P. Grozman and D. Leites, Lie superalgebra structures in H*($ \mathfrak{g},\mathfrak{g})$, Czechoslovak J. Phys 54 (2004).

GLS
P. Grozman, D. Leites and I. Shchepochkina, Lie superalgebras of string theories, Acta Math. Vietnam. 26 (2001), No.1, 27-63; arXiv:hep-th/9702120.

HK
N. W. Hijligenberg and Yu. Yu. Kochetkov, The absolute rigidity of the Neveu-Schwarz and Ramond superalgebras, J. Math. Phys. 37 (1996), No.11, 5858-5868.

Kac
V. Kac, Characters of typical representations of classical Lie superalgebras.

Koc
Yu. Yu. Kochetkov, Deformations of the Hamiltonian Lie algebra H(2), Funct. Anal. Appl. 28 (1994), No.3, 211-213.

Kor1
V. V. Kornyak, Cohomology of Lie superalgebras of Hamiltonian vector fields: Computer analysis, Computer Algebra in Scientific Computing (ed. V. G. Ganzha, E. W. Mayr and E. V. Vorontsov), Springer, 1999, pp. 241-249; arXiv:math.NA/9906046.

Kor2
V. V. Kornyak, Modular algorithm for computing cohomology: Lie superalgebra of special vector fields on (2| 2)-dimensional odd-symplectic superspace, Computer Algebra in Scientific Computing (ed. V. G. Ganzha, E. W. Mayr and E. V. Vorozhtsov), TUM, Munich, 2003, pp. 227-240; arXiv:math.RT/0305155.

Leb
A. Lebedev, Invariants of nonholonomic systems and cohomology superalgebras of nilpotent Lie algebras, B.Sc. Thesis, Nizhnii Novgorod Univ., 2003 (in Russian).

LLS
A. Lebedev, D. Leites and I. Shereshevskii, Lie superalgebra structures in cohomology spaces of Lie algebras with coefficients in the adjoint representation, arXiv:math.KT/0404139.

Lei
D. Leites, Supersymmetry of the Sturm-Liouville and Korteveg-de Vries operators, Operator Methods in Ordinary and Partial Differential Equations, S. Kovalevski Symposium, Univ. of Stockholm, June 2000 (S. Albeverio, N. Elander, W. N. Everitt and P. Kurasov, eds.), Birkhäuser, Operator Methods: Advances and Applications 132 (2002), 267-285.

LF
D. Leites and D. B. Fuchs, Cohomology of Lie superalgebras, C. R. Acad. Bulg. Sci. 37 (1984), 1595-1596.

LS1
D. Leites and I. Shchepochkina, How to quantize antibracket, Theor. Math. Phys. 126 (2001), No.3, 281-306.

LS2
D. Leites and I. Shchepochkina, The classification of the simple Lie superalgebras of vector fields, Preprint MPI Bonn, MPI 2003-28.

LS3
D. Leites and I. Shchepochkina, List of classical Lie superalgebras, Preprint, 2003, 48pp.

LS4
D. Leites and I. Shchepochkina, Deformations and central extensions of the simple Lie superalgebras of polynomial vector fields, planned for the Feigin Festschrift, 2003.

P1
E. Poletaeva, Cohomology of Lie superalgebras and string theories: problems and results, Unfinished draft, July 1996.

P2
E. Poletaeva, Superconformal algebras and Lie superalgebras of the Hodge theory, J. Nonlin. Math. Phys. 10 (2003), no.2, 141-147; hep-th/0209168.

R
C. Roger, Cohomology of current Lie algebras, Deformation theory of algebras and structures and applications, Kluwer, 1988, 357-374.

S1
V. Sergeev, Limits of Rationality: A Thermodynamic Approach to The Problem of Economic Equilibrium, Fazis, Moscow, 1999 (in Russian).

S2
V. Sergeev, The thermodynamic approach to the market equilibrium, Sante Fe Institute Working Paper #03-04-027.

SP
I. Shchepochkina and G. Post, Explicit bracket in an exceptional simple Lie superalgebra $ \mathfrak{cvect}(0\vert 3)_*$, Internat. J. Algebra Comput. 8 (1998), No.4, 479-495; arXiv:physics/9703022.

W
F. Wagemann, arXiv:math-ph/0003035.

Z1
P. Zusmanovich, Central extensions of current algebras, Trans. Amer. Math. Soc. 334 (1992), 143-152.

Z2
P. Zusmanovich, Deformations of W1(n) $ \otimes$ A and modular semisimple Lie algebras with a solvable maximal subalgebra, J. Algebra 268 (2003), 603-635.

Z3
P. Zusmanovich, Low-dimensional cohomology of current Lie algebras and structure functions associated with loop manifolds, arXiv:math.RA/0302334.

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Leites' (super)questions

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