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\begin{document}
\title{Leites' (super)questions}
\author{Pasha Zusmanovich}
\address{}
\email{pasha.zusmanovich@gmail.com}
\date{first written April 30, 2003; last revised August 5, 2018}
\maketitle
This is an account of (some of the) questions posed by Dimitry Leites, compiled
by me mainly during few years at the beginning of 2000s. 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) writings
(\cite[Appendices]{berezin},
\cite{onslie}, \cite{vietnamica}, \cite{lebedev-et-al}, \cite{kdv},
\cite{ls-teormatfiz}, \cite{ls-classif}, \cite{ls-list}, \cite{ls-feigin-fest},
\cite{sos2} and references therein).
Needless to say, there are much more questions there, I account here merely for
what I was able to grasp.
The questions, being concentrated around one topic -- Lie superalgebras --
may be roughly divided
into two 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),
and second, more ``generic'' or ``theoretical'' questions
(like describing superalgebra structures on the whole cohomology space,
elucidation in the supercase of certain issues well-known in the plain even setting,
etc.).
\section{Cohomology in all dimensions}
\subsection{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
(\cite{leites-fuchs}; later Shapovalov discovered one cocycle missed there - see
\cite[\S 2.2.1]{ls-feigin-fest} and \cite[\S 4.1]{kornyak}).
Compute cohomology with trivial coefficients of (infinite-dimensional) Hamiltonian Lie algebra
$H(n)$ ($n=2$ considered by Gelfand--Kalinin--Fuchs in \cite{gelfand-kalinin-fuchs}
with a very partial results).
Study the behavior of cohomology of a (finite-dimensional) Hamiltonian Lie algebra in
characteristic $p$ as $p \to \infty$.
Can we utilize the fact that $H(2)$ is a ``limit'' of $sl(n)$'s in an
appropriate basis? (See \cite[Example 2.12]{arnold-khesin} and \cite{zeitlin}
and references therein).
\subsection{Computer calculations}\label{sec-comput}
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 \cite[\S 2.3]{kornyak-2})
\subsection{Structure of cohomology algebra}
(\cite[\S 4.1-4.2]{onslie} and \cite{lebedev-et-al})
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 $G_2$ was treated by
Lebedev (see \cite{diploma} and \cite{lebedev-et-al}).
Conjectures (\cite{gl-2}):
\begin{enumerate}
\item
$H^*(psl(2|2), psl(2|2))$
is generated by certain explicitly given cocycles of degrees 0,2 and 3.
\item
$H^*(osp(4|2; \alpha), osp(4|2, \alpha))$ is exhausted by explicitly given cocycles
in degrees 1 and 4.
\end{enumerate}
A similar question of study of an algebra of simplicial cohomology (in the context
of computer calculations) was raised in \cite{simplicial}.
It could be that something similar was undertaken for the cohomology of
associative algebras and groups.
\section{Low-dimensional cohomology and related invariants}
\subsection{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:
\subsubsection{Central extensions of vectorial superalgebras}
(\cite[\S 6.2]{ls-feigin-fest})
Prove that there are no central extensions of simple Lie superalgebras of vector
fields with polynomial or formal coefficients, except the following:
\begin{enumerate}
\item
Poisson superalgebra $po(2n|m)$
extending Hamiltonian one $h(2n|m)$;
\item
Deformed Buttin superalgebra $b_\lambda(n)$ extending $le(n)$
(probably values $\lambda = 1$ or $-1$ are exceptional in some sense);
\item
Two amazing extensions of $sle^o(3)$ described by Shchepochkina and Post in \cite{shep-post}.
\end{enumerate}
(Well, may be something else is missing, but the most nontrivial cases are certainly here).
\subsubsection{Central extension of Lie superalgebras of infinite matrices}
There are various superalgebras with different finiteness constraints. See
\cite[\S 3.2.1]{sos2}. Probably some works of Feigin (?), cyclic cohomology, and
``Japanese cocycle'' are relevant.
\subsubsection{Deformations of current algebras} (\cite[Warning 3 on p.~6]{ls-list}).
It is probably proved (Lecomte and Roger, see \cite{roger}) 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 deformations of corresponding current algebras:
namely, Krichever-Novikov algebras, example of P.~Golod (for the latter, see
\cite{golod} or \cite{ls-list}, p.~38), and examples of Fialowski--Schlichenmaier. So:
\begin{enumerate}
\item Accurately compute all ``classical'' (= Gerstenhaber) deformations of
current (super)algebras. Are all they rigid?
\item Build an ``ideologically right'' deformation theory which will include Krichever-Novikov
and Golod algebras. Are there others?
\end{enumerate}
Ideally, this should also include, as a special case, filtered deformations of modular
$W_1(n)\otimes A$ (computed in \cite{me-deformations}). Probably the unification
can be done by considering ``Block (super)algebras'' as defined (in the non-super case)
in \cite{me-tams}.
\subsubsection{Deformations and outer derivations of stringy superalgebras}
(\cite[Appendix D3, \S 8.5]{berezin})
Compute deformations and outer derivations of ``stringy'' superalgebras
(i.e. exactly those described in \cite{vietnamica} or
\cite[Appendix D3, \S 8]{berezin}); some cocycles formulae in \cite{vietnamica}
describing central extensions are incorrect).
It is known (\cite{hij-koch}) that some of the initial algebras in the series,
namely, $k^L(1|n)$ and $k^M(1|n)$ for $n=0,1$ have zero second degree cohomology
in the adjoint module (and, consequently, are rigid), and that $svect^L(1|n)$
has at least one deformation (denoted as $svect_\lambda^L(1|n)$ in \cite{vietnamica}).
No new algebras are expected.
\subsubsection{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 \cite{ls-teormatfiz}.
Kochetkov in \cite{kochetkov}
described deformations of a Hamiltonian Lie algebra $H(2)$ and Cheng and Kac in \cite{kac}
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).
\subsubsection{Filtered deformations}
\begin{enumerate}
\item
Compute filtered deformations of Lie superalgebras of vector fields in all possible
``Weisfeiler''gradings
(Cheng and Kac \cite{kac} considered only standard gradings with even deformation
parameter).
\item
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 \cite{ls-classif}, \S I.4 and \cite{ls-feigin-fest}, \S 1.12).
\end{enumerate}
\subsubsection{Modules of tensor fields}
Compute $H^1$ for ``stringy'' superalgebras in modules of ``tensor fields''
and their generalizations
(see \cite[\S 1.3]{vietnamica}, \cite[\S 1.2-1.3]{kdv} and \cite{poletaeva}).
This sounds to vague, so let's start
from the following:
\begin{enumerate}
\item $H^1(L, F_{\lambda; \mu})$ for $L=k^L(1|n)$ and $k^M(1|n)$.
\item $H^1(L, F_{\lambda_1,\lambda_2; \mu})$ for $L = k^L(1|2)$ and $k^M(1|3)$.
\item for $k^{Lo}(1|4)$ and $k^{Mo}(1|5)$.
\end{enumerate}
$H^1 (L, F_\lambda)$ for $L=k^L(1|n)$ and $k^\alpha(1|4)$, and
$H^1 (L, T(\lambda, \mu))$ for $L=k^L(1|2)$ and $k^+(1|2)$ were computed by Poletaeva
(unpublished M.Sc. thesis; see \cite{poletaeva} and \cite[\S 3]{kdv}).
Some additional papers possibly instrumental in understanding of the structure of
algebras and modules involved: \cite{pol-superconformal}
and \cite{kac-characters}. Some low-dimensional cohomology related to tensor fields
considered by Wagemann (\cite{wagemann}).
\subsection{Structure functions}
To compute structure functions associated with (twisted) current superalgebras (that is,
``twist'' and superize results of \cite{zus-sf}). 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. \cite[\S 4.1]{onslie}).
\section{Other}
\subsection{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 $3 \times 3$). See \cite{gl-1} and \cite[\S\S 1.8.4 and 1.9]{onslie}.
\subsection{Volichenko algebras}
Further study of Volichenko algebras (which are, roughly speaking, nonhomogeneous subalgebras
of Lie superalgebras). See \cite[Chapter 2]{sos2}.
In particular, to provide an intrinsic definition of Volichenko algebras
(they do not form a variety, see \cite{baranov}, so such description is impossible solely
in the language of identities).
\begin{thebibliography}{DHSW}
\bibitem[AH]{arnold-khesin} V.I. Arnold and B.A. Khesin,
\emph{Topological Methods in Hydrodynamics}, Springer, 1998.
\bibitem[Ba]{baranov}
A.A. Baranov, \emph{Volichenko algebras and nonhomogeneous subalgebras of Lie algebras},
Siber. Math. J. \textbf{36} (1995), 859--868.
\bibitem[Be]{berezin} F.A. Berezin, \emph{Introduction to Superanalysis} (ed. D. Leites),
2nd edition, MCCME, 2011 (in Russian).
\bibitem[CK]{kac}
S.-J. Cheng and V.G. Kac,
\emph{Generalized Spencer cohomology and filtered deformations of $\mathbb{Z}$-graded
Lie superalgebras},
Adv. Theor. Math. Phys. \textbf{2} (1998), 1141--1182; \textsf{arXiv:math/9805039}.
\bibitem[DHSW]{simplicial}
J.-G. Dumas, F. Heckenbach, S. Saunders, and V. Welker,
\emph{Computing simplicial homology based on efficient Smith normal form algorithms},
Algebra, Geometry and Software Systems (ed. M. Joswig and N. Takayama),
Springer, 2003, 177--206.
\bibitem[GKF]{gelfand-kalinin-fuchs}
I.M. Gelfand, D.I. Kalinin and D.B. Fuchs,
\emph{Cohomology of the Lie algebra of Hamiltonian formal vector fields},
Funct. Anal. Appl. \textbf{6} (1972), 193--196.
\bibitem[G]{golod}
P. Golod, \emph{A deformation of the affine Lie algebra $A_1^{(1)}$
and hamiltonian systems on the orbits of its subalgebras},
Group-theoretical methods in physics, Moscow, Nauka, 1986, 368--376 (in Russian).
\bibitem[GL1]{gl-1}
P. Grozman and D. Leites, \emph{Defining relations for Lie superalgebras with
Cartan matrix}, Czechoslovak J. Phys. \textbf{51} (2001), 1--21;
\textsf{arXiv:hep-th/9702073}.
\bibitem[GL2]{onslie}
\bysame{} and \bysame{},
\emph{SuperLie and problems (to be) solved with it}, Preprint MPI Bonn, MPI 2003-39.
\bibitem[GL3]{gl-2}
\bysame{} and \bysame{},
\emph{Lie superalgebra structures in $H^*(\mathfrak g,\mathfrak g)$},
Czechoslovak J. Phys. \textbf{54} (2004), 1313--1319; \textsf{arXiv:math/0509469}.
\bibitem[GLS]{vietnamica}
\bysame{}, \bysame{}, and I. Shchepochkina,
\emph{Lie superalgebras of string theories},
Acta Math. Vietnam. \textbf{26} (2001), No.1, 27--63; \textsf{arXiv:hep-th/9702120}.
\bibitem[HK]{hij-koch}
N.W. Hijligenberg and Yu.Yu. Kochetkov,
\emph{The absolute rigidity of the Neveu-Schwarz and Ramond superalgebras},
J. Math. Phys. \textbf{37} (1996), 5858--5868.
\bibitem[Kac]{kac-characters}
V. Kac, \emph{Characters of typical representations of classical Lie superalgebras}.
\bibitem[Koc]{kochetkov}
Yu.Yu. Kochetkov, \emph{Deformations of the Hamiltonian Lie algebra $H(2)$},
Funct. Anal. Appl. \textbf{28} (1994), No.3, 211--213.
\bibitem[Kor1]{kornyak}
V.V. Kornyak,
\emph{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. Vorozhtsov), Springer, 1999, 241--249;
\textsf{arXiv:math/9906046}.
\bibitem[Kor2]{kornyak-2}
\bysame{}, \emph{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, 227--240;
\textsf{arXiv:math/0305155}.
\bibitem[Leb]{diploma}
A. Lebedev,
\emph{Invariants of nonholonomic systems and cohomology superalgebras of nilpotent Lie algebras},
B.Sc. Thesis, Nizhnii Novgorod Univ., 2003 (in Russian).
\bibitem[LLS]{lebedev-et-al}
\bysame{}, D. Leites, and I. Shereshevskii,
\emph{Lie superalgebra structures in $C^*(\mathfrak n,\mathfrak n)$ and
$H^*(\mathfrak n,\mathfrak n)$},
Lie Groups and Invariant Theory (ed. E. Vinberg),
AMS Transl. \textbf{213} (2005), 157--172; \textsf{arXiv:math/0404139}.
\bibitem[Lei]{kdv}
D. Leites, \emph{Supersymmetry of the Sturm-Liouville and Korteveg-de Vries operators},
Operator Methods in Ordinary and Partial Differential Equations
(ed. S. Albeverio, N. Elander, W.N. Everitt, and P. Kurasov), Birkh\"auser,
Operator Methods: Advances and Applications \textbf{132} (2002), 267--285.
\bibitem[LF]{leites-fuchs}
\bysame{} and D.B. Fuchs, \emph{Cohomology of Lie superalgebras},
C. R. Acad. Bulg. Sci. \textbf{37} (1984), 1595--1596.
\bibitem[LS1]{ls-teormatfiz}
\bysame{} and I. Shchepochkina, \emph{How to quantize antibracket},
Theor. Math. Phys. \textbf{126} (2001), 281--306.
\bibitem[LS2]{ls-classif}
\bysame{} and \bysame{},
\emph{The classification of the simple Lie superalgebras of vector fields},
Preprint MPI Bonn, MPI 2003-28.
\bibitem[LS3]{ls-list}
\bysame{} and \bysame{}, \emph{List of classical Lie superalgebras},
Preprint, 2003, 48pp.
\bibitem[LS4]{ls-feigin-fest}
\bysame{} and \bysame{}, \emph{Deformations and central extensions of the simple Lie
superalgebras of polynomial vector fields}, Preprint, 2003.
\bibitem[P1]{poletaeva}
E. Poletaeva,
\emph{Cohomology of Lie superalgebras and string theories: problems and results},
Draft, 1996, 3pp.
\bibitem[P2]{pol-superconformal}
\bysame{}, \emph{Superconformal algebras and Lie superalgebras of the Hodge theory},
J. Nonlin. Math. Phys. \textbf{10} (2003), 141--147; \textsf{hep-th/0209168}.
\bibitem[R]{roger}
C. Roger, \emph{Cohomology of current Lie algebras},
Deformation Theory of Algebras and Structures and Applications, Kluwer, 1988, 357--374.
\bibitem[S]{sos2}
\emph{Seminar on Supersymmetries. Vol. 2. Algebra and Analysis: Additional Chapters}
(ed. D. Leites), MCCME, 2011 (in Russian).
\bibitem[SP]{shep-post}
I. Shchepochkina and G. Post,
\emph{Explicit bracket in an exceptional simple Lie superalgebra $\mathfrak{cvect}(0|3)_*$},
Intern. J. Algebra Comput. \textbf{8} (1998), 479--495; \textsf{arXiv:physics/9703022}.
\bibitem[W]{wagemann} F. Wagemann,
\emph{Explicit formulae for cocycles of holomorphic vector fields with values in
$\lambda$ densities},
J. Lie Theory \textbf{11} (2001), 173--184; \textsf{arXiv:math-ph/0003035}.
\bibitem[Ze]{zeitlin} V. Zeitlin,
\emph{Finite-mode analogs of 2D ideal hydrodynamics: Coadjoint orbits and local
canonical structure}, Physica D \textbf{49} (1991), 353--362.
\bibitem[Zu1]{me-tams}
P. Zusmanovich, \emph{Central extensions of current algebras},
Trans. Amer. Math. Soc. \textbf{334} (1992), 143--152;
Erratum and addendum: \textbf{362} (2010), 5601--5603; \textsf{arXiv:0812.2625}.
\bibitem[Zu2]{me-deformations}
\bysame{}, \emph{Deformations of $W_1(n)\otimes A$ and modular semisimple Lie
algebras with a solvable maximal subalgebra}, J. Algebra \textbf{268} (2003), 603--635;
\textsf{arXiv:math/0204004}.
\bibitem[Zu3]{zus-sf} \bysame{},
\emph{Low-dimensional cohomology of current Lie algebras and structure functions
associated with loop manifolds},
Lin. Algebra Appl. \textbf{407} (2005), 71--104; \textsf{arXiv:math/0302334}.
\end{thebibliography}
\section*{Addendum August 5, 2018}
This was mainly written long time ago. It seems that all the questions described
here largely remain relevant, though there may be some advancement concerning
some particular superalgebras. Below is the list of references which may contain
such advancements, though I did not attempt to go into details thoroughly.
\renewcommand{\refname}{}
\begin{thebibliography}{BGLL}
\bibitem[BGLL]{bgll} S. Bouarroudj, P. Grozman, A. Lebedev, and D. Leites,
\emph{Derivations and central extensions of symmetric modular Lie algebras and
superalgebras}; arXiv:1307.1858.
\bibitem[BKLS]{bkls} \bysame, A. Krutov, D. Leites, and I. Shchepochkina,
\emph{Non-degenerate invariant (super)symmetric bilinear forms on simple Lie (super)algebras},
Algebras Repr. Theory, to appear; arXiv:1806.05505.
\bibitem[C]{1} C.H. Conley,
\emph{Conformal symbols and the action of contact vector fields over the superline},
J. Reine Angew. Math. \textbf{633} (2009), 115--163; arXiv:0712.1780.
\bibitem[P]{2} E. Poletaeva,
\emph{The first cohomology of the superconformal algebra $K(1|4)$},
J. Nonlin. Math. Phys. \textbf{19} (2012), no.~3, 1250020.
\end{thebibliography}
\end{document}
% end of leites.tex