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All results concern characteristic 2. Two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of the latter new, are offered. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures, so we classified all simple finite-dimensional Lie superalgebras modulo non-existing at the moment classification of simple finite-dimensional Lie algebras. This result concerns Lie superalgebras considered naively, as vector spaces. To obtain classification of simple Lie superalgebras in the category of supervarieties, one should list the non-isomorphic deforms (results of deformations) with odd parameter. This problem is open bar several examples described in arXiv~0807.3054. For Lie algebras, in addition to the known ---"classical" --- restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and of Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: $(2|4)$- and $(2|2)$-structures, one more analog --- a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras.

For each of the exceptional (not entering infinite series) finite-dimensional modular Lie superalgebras with indecomposable Cartan matrix, we give the explicit list of its roots, and the corresponding Chevalley basis, for one of its inequivalent Cartan matrices, namely the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots (this number, called the defect of the Lie superalgebra, is important in the representation theory). Our main tools: Grozman’s Mathematica-based code SuperLie, Python, and A. Lebedev’s help.

1) The differential equation considered in terms of exterior differential forms, as \'E.Cartan did, singles out a differential ideal in the supercommutative superalgebra of differential forms, hence an affine supervariety. In view of this observation, it is evident that every differential equation has a supersymmetry (perhaps trivial). Superymmetries of which (systems of) classical differential equations are missed yet? 2) Why criteria of formal integrability of differential equations are never used in practice?

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.

As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2. We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or $0|1$, or $1|1$; it is equal to $1|1$ if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic distinct from 2, see arXiv:1806.05505). We give examples of NISes on deformations (with both even and odd parameters) of several simple finite-dimensional Lie superalgebras in characteristic 2. We also recall examples of multiple NISes on simple Lie algebras over non-closed fields., 17 pages, updated version of OWP-2020-02

The inverses of indecomposable Cartan matrices are computed for finite-dimensional Lie algebras and Lie superalgebras over fields of any characteristic, and for hyperbolic (almost affine) complex Lie (super)algebras. We discovered three yet inexplicable new phenomena, of which (a) and (b) concern hyperbolic (almost affine) complex Lie (super)algebras, except for the 5 Lie superalgebras whose Cartan matrices have 0 on the main diagonal: (a) several of the inverses of Cartan matrices have all their elements negative (not just non-positive, as they should be according to an a priori characterization due to Zhang Hechun); (b) the 0s only occur on the main diagonals of the inverses; (c) the determinants of inequivalent Cartan matrices of the simple Lie (super)algebra may differ (in any characteristic). We interpret most of the results of Wei Yangjiang and Zou Yi Ming, Inverses of Cartan matrices of Lie algebras and Lie superalgebras, Linear Alg. Appl., 521 (2017) 283--298 as inverses of the Gram matrices of non-degenerate invariant symmetric bilinear forms on the (super)algebras considered, not of Cartan matrices, and give more adequate references. In particular, the inverses of Cartan matrices of simple Lie algebras were already published, starting with Dynkin's paper in 1952, see also Table 2 in Springer's book by Onishchik and Vinberg (1990)., We reproduce definition of root spaces from arXiv:0710.5149 and arXiv:0906.1860. Version 2 contains inadvertently forgotten subsection 8.3 and accordingly edited Introduction and Abstract

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form $B$ is called a nis-(super)algebra. The double extension $\mathfrak{g}$ of a nis-(super)algebra $\mathfrak{a}$ is the result of simultaneous adding to $\mathfrak{a}$ a central element and a derivation so that $\mathfrak{g}$ is a nis-algebra. Loop algebras with values in simple complex Lie algebras are most known among the Lie (super)algebras suitable to be doubly extended. In characteristic 2 the notion of double extension acquires specific features. Restricted Lie (super)algebras are among the most interesting modular Lie superalgebras. In characteristic 2, using Grozman's Mathematica-based package SuperLie, we list double extensions of restricted Lie superalgebras preserving the non-degenerate closed 2-forms with constant coefficients. The results are proved for the number of indeterminates ranging from 4 to 7 - sufficient to conjecture the pattern for larger numbers. Considering multigradings allowed us to accelerate computations up to 100 times., 18 pages

Cartan described some of the finite dimensional simple Lie algebras and three of the four series of simple infinite dimensional vectorial Lie algebras with polynomial coefficients as prolongs, which now bear his name. The rest of the simple Lie algebras of these two types (finite dimensional and vectorial) are, if the depth of their grading is greater than 1, results of generalized Cartan–Tanaka–Shchepochkina (CTS) prolongs. Here we are looking for new examples of simple finite dimensional modular Lie (super)algebras in characteristic 2 obtained as Cartan prolongs. We consider pairs (an (ortho-)orthogonal Lie (super)algebra or its derived algebra, its irreducible module) and compute the Cartan prolongs of such pairs. The derived algebras of these prolongs are simple Lie (super)algebras. We point out several amazing phenomena in characteristic 2: a supersymmetry of representations of certain Lie algebras, latent or hidden over complex numbers, becomes manifest; the adjoint representation of some simple Lie...

The classification of simple finite dimensional modular Lie algebras over algebraically closed fields of characteristic p > 3 (described by the generalized Kostrikin–Shafarevich conjecture) being completed due to Block, Wilson, Premet and Strade (with contributions from other researchers) the next major classification problems are those of simple finite dimensional modular Lie algebras over fields of characteristic 3 and 2. For the latter, the Kochetkov–Leites conjecture involved classification of Lie superalgebras and their inhomogeneous with respect to parity subalgebras, called Volichenko algebras. In characteristic 2, we consider the result of application of the functor forgetting the superstructure to the simple serial vectorial Lie algebras known to us and their Volichenko subalgebras.

Felix Alexandrovich Berezin was born on April 25, 1931 and perished on July 14, 1980 (or several days before) under obscure circumstances while serving — during his summer holidays — as an auxiliary worker for a geological party near the Kolyma river. Wikipedia currently lists several researchers as discoverers of what we now call supersymmetry. Interestingly, Berezin is not even mentioned in this Wikipedia article. However, none of the authors mentioned so far as “pioneers of supersymmetry” qualify as such: Before Wess and Zumino have explained the meaning and importance of supersymmetry none of the earlier discoverers did fully appreciate the meaning and value of their own works, and some of them do not realize its full meaning even now judging by their later works. Contrariwise, Berezin definitely was the first to realize the real value of his discoveries, the fact that he stands on the threshold of the new branch of mathematics, currently referred to as “supermathematics” and guess its possible worthiness for physics, first consciously demonstrated by Wess and Zumino. In early 1960s, Berezin obtained a parallel description of the second quantization for Boze and Fermi particles summarized in his book [1] (where, among other things, Berezin introduces the Berezin integral, and gives a parallel description of the spinor and oscillator representations of, respectively, infinite dimensional orthogonal o(2∞) and symplectic Lie algebras sp(2∞), later rediscovered many times; for an overview of Berezin’s works, see the “scientific part” of [5]). In May 2007, World Scientific published the book “FELIX BEREZIN: Life and Death of the Mastermind of Supermathematics” edited by M. Shifman, [8]. The book consists of contributions of Berezin’s friends, colleagues and students; recollections of Berezin as a person, and some scientific reviews of his results and ideas, their role in mathematical physics and their developments. The core of the book is the skillfully written heart-breaking story of Berezin’s widow, Elena Karpel; and the two Berezin’s letters — to the Rector of Moscow State University and the Moscow Mathematical Society — on the situation in these institutions and in mathematical community in the Soviet Union in general (the problems pointed at in these letters did not lose their timeliness today).

We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac--Moody (super)algebras are the most known examples., 42 pages. Definitions of certain Lie (super)algebras follow previous works of some of the authors

Several improvements of the Kostrikin-Shafarevich method conjecturally producing all simple finite-dimensional Lie algebras over algebraically closed fields of any positive characteristic were rece ...

Of four types of Kaplansky algebras, type-2 and type-4 algebras have previously unobserved Z/2-gradings: nonlinear in roots. A method assigning a simple Lie superalgebra to every Z/2-graded simple Lie algebra in characteristic 2 is illustrated by seven new series. Type-2 algebras and one of the two type-4 algebras are demystified as nontrivial deforms (the results of deformations) of the alternate Hamiltonian algebras. The type-1 Kaplansky algebra is recognized as the derived of the nonalternate version of the Hamiltonian Lie algebra, the one that preserves a tensorial 2-form. Deforms corresponding to nontrivial cohomology classes can be isomorphic to the initial algebra, e.g., we confirm Grishkov's implicit claim and explicitly describe the Jurman algebra as such a semitrivial deform of the derived of the alternate Hamiltonian Lie algebra. This paper helps to sharpen the formulation of a conjecture describing all simple finite-dimensional Lie algebras over any algebraically closed field of nonzero characteristic and supports a conjecture of Dzhumadildaev and Kostrikin stating that all simple finite-dimensional modular Lie algebras are either of standard type or deforms thereof. In characteristic 2, we give sufficient conditions for the known deformations to be semitrivial.

In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that mathematicians study are real or (almost) complex ones, while Minkowski superspaces are completely different objects. They are what we call almost real-complex supermanifolds, i.e., real supermanifolds with a non-integrable distribution, the collection of subspaces of the tangent space, and in every subspace a complex structure is given. An almost complex structure on a real supermanifold can be given by an even or odd operator; it is complex (without "always") if the suitable superization of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we define the circumcised analog of the Nijenhuis tensor. We compute it for the Minkowski superspaces and superstrings. The space of values of the circumcised Nijenhuis tensor splits into (indecomposable, generally) components whose irreducible constituents are similar to those of Riemann or Penrose tensors. The Nijenhuis tensor vanishes identically only on superstrings of superdimension 1|1 and, besides, the superstring is endowed with a contact structure. We also prove that all real forms of complex Grassmann algebras are isomorphic although singled out by manifestly different anti-involutions., Exposition of the same results as in v.1 is more lucid. Reference to related recent work by Witten is added

Let a given finite dimensional simple Lie superalgebra g possess an even invariant non-degenerate supersymmetric bilinear form. We show how to recover the quadratic Casimir element for the Kac-Moody superalgebra related to the loop superalgebra with values in g from the quadratic Casimir element for g. Our main tool here is an explicit Wick normal form of the even quadratic Casimir operator for the Kac--Moody superalgebra associated with g; this Wick normal form is of independent interest. If g possesses an odd invariant supersymmetric bilinear form we compute the cubic Casimir element. In addition to the cases of Lie superalgebras g(A) with Cartan matrix A for which the answer was known, we consider the Poisson Lie superalgebra poi(0|n) and the related Kac--Moody superalgebra., Comment: Format settings changed to widen bottom margins

On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form omega can be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian tensor and d omega. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically ""flat"": it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet's theorems describing these cohomologies. Using Premet's theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue.

Recently, Grozman and Leites returned to the original Cartan's description of Lie algebras to interpret the Melikyan algebras (for p, 10 pages; no figures

We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectorial Lie superalgebras. We consider odd parameters of deformations. For all 15 Weisfeiler gradings of the 5 exceptional families, and one Weisfeiler grading for each of 2 serial simple complex Lie superalgebras (with 2 exceptional subseries), we describe their characteristic-2 analogs - new simple Lie algebras. Descriptions of several of these analogs, and of their desuperizations, are far from obvious. One of the exceptional simple vectorial Lie algebras is a previously unknown deform (the result of a deformation) of the characteristic-2 version of the Lie algebra of divergence-free vector fields; this is a new simple Lie algebra with no analogs in characteristics distinct from 2. In characteristic 2, every simple Lie superalgebra can be obtained from a simple Lie algebra by one of the two methods described in arXiv:1407.1695. Most of the simple Lie superalgebras thus obtained from simple Lie algebras we describe here are new., see pdf-file with index at https://www.emis.de/journals/SIGMA/2020/089/

For any nonholonomic manifold, i.e., a manifold with nonintegrable distribution, we define an analog of the Riemann curvature tensor and refer to Grozman's package SuperLie with the help of which the tensor had been computed in several cases. Being an analog of the usual curvature tensor, this invariant characterizes (in)stability of any nonholonomic dynamical system, in particular, of markets. Similar invariants give criteria for formal integrability of differential equations whose symmetries are induced by contact transformations similar to Goldshmidt's criteria for formal integrability of differential equations whose symmetries are induced by point transformations. As a byproduct, we obtain an approximate solution of the equation whose integrability is under study. Bibliography: 47 titles.

On a given manifold M, the Nijenhuis bracket makes the superspace of vector-valued differential forms into a Lie superalgebra that can be interpreted as the centralizer of the exterior differential considered as a vector field on the supermanifold associated with the de Rham bundle on M. A similar bracket introduces structures of DG Lie superalgebra in the superspaces of cochains and cohomology with coefficients in the adjoint module for any Lie superalgebra. We use a Mathematica--based package SuperLie (already proven useful in various problems) to explicitly describe these Lie superalgebras for some simple finite dimensional Lie superalgebras and their ``relatives'' (the nontrivial central extensions or derivation algebras of the considered simple ones)., 6 pages, LaTeX

We show that in contrast to $${\mathfrak{p}}o(2n|m)$$ , its quotient modulo center, the Lie superalgebra $${\mathfrak{h}}(2n|m)$$ of Hamiltonian vector fields with polynomial coefficients, has exceptional additional deformations for $$(2n|m) = (2|2)$$ and only for this superdimension. We relate this result to the complete description of deformations of the antibracket (also called the Schouten or Buttin bracket). It turns out that the space in which the deformed Lie algebra (result of quantizing the Poisson algebra) acts coincides with the simplest space in which the Lie algebra of commutation relations acts. This coincidence is not necessary for Lie superalgebras.

We completely describe presentations of Lie superalgebras with Cartan matrix if they are simple ℤ-graded of polynomial growth. Such matrices can be neither integer nor symmetrizable. There are non-Serre relations encountered. In certain cases there are infinitely many relations. For symmetrizable Cartan matrices similar (but not identical) presentations are obtained by Yamane who also considered q-quantized versions of these relations (q-alg 9603015). Yamane did not try to find out which of the relations he offers constitute a minimal set generating all other relations; contrariwise we give minimal sets of defining relations and in various cases our relations look much simpler than Yamane's (though still very complicated in some cases).

For every finite-dimensional nilpotent complex Lie algebra or superalgebran, we offer three algorithms for realizing it in terms of creation and annihilation operators. We use these algorithms to realize Lie algebras with a maximal subalgebra of finite codimension. For a simple finite-dimensionalg whose maximal nilpotent subalgebra isn, this gives its realization in terms of first-order differential operators on the big open cell of the flag manifold corresponding to the negative roots ofg. For several examples, we executed the algorithms using the MATHEMATICA-based package SUPERLie. These realizations form a preparatory step in an explicit construction and description of an interesting new class of simple Lie (super) algebras of polynomial growth, generalizations of the Lie algebra of matrices of complex size.

Indecomposable representations of quivers are in 1–1 correspondence with positive weight vectors of Kac-Moody algebras. The collection of indecomposable representations of the quiver is tame if the quiver corresponds to a Kac-Moody algebra of polynomial growth. What corresponds to positive roots of Lie algebras of polynomial growth different from Kac-Moody algebras? The classification problem for tame representations of quivers associated to Lie superalgebras is a natural step towards the answer to this question. As an aside we announce a classification of simple graded Lie superalgebras of polynomial growth.

Among simple $$\mathbb{Z}$$ -graded Lie superalgebras of polynomial growth there is only one without Cartan matrix but with an invariant nondegenerate supersymmetric bilinear form. This is $$\mathfrak{k}^{\text{L}} (1|6)$$ , the Lie superalgebra of vector fields on the (1|6)-dimensional supercircle preserving the Pfaff equation α = 0, where $$\alpha = dt + \sum\nolimits_{1 \leqslant i \leqslant 3} {(\xi _i {\text{d}}\eta _i + \eta _i {\text{d}}\xi _i )} $$ and where ξ, η are the odd variables, t = exp(iφ) for the angle parameter φ on the circle. For $$\mathfrak{k}^{\text{L}} (1|6)$$ we compute the Casimir element wherefrom we deduce the Shapovalov determinant and the description of the irreducible Verma modules over $$\mathfrak{k}^{\text{L}} (1|6)$$ .

For all almost affine (hyperbolic) Lie superalgebras, the defining relations are computed in terms of their Chevalley generators., Published in the special issue of JNMP in memory of F.A. Berezin

For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan matrix in characteristic 2 (and in other characteristics for completeness of the picture). We correct the currently available in the literature notions of Chevalley generators and Cartan matrix in the modular and super cases, and an auxiliary notion of the Dynkin diagram. In characteristic 2, the defining relations of simple classical Lie algebras of the A, D, E types are not only Serre ones; these non-Serre relations are same for Lie superalgebras with the same Cartan matrix and any distribution of parities of the generators. Presentations of simple orthogonal Lie algebras having no Cartan matrix are also given.., Expounds math/0611391 and math/0611392, it contains new theorems, new material on relations for p=2, and - most important - on div. power (co)homology.To appear in Homol. Homot. Appl

The discrete group generated by reflections of the sphere, or Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lann\'er if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincar\'e series (a.k.a. growth function) is the generating function of the cardinalities of sets of elements of equal length. Solomon established that, for ANY Coxeter group, its Poincar\'e series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave a recurrence formula. The explicit expression of the Poincar\'e series was known for the spherical and Euclidean Coxeter groups, and 3-generated Coxeter groups, and (with mistakes) Lann\'er groups. Here we give a lucid description of the numerator of the Poincar\'e series of any Coxeter group, and denominators for each (quasi-)Lann\'er group, and review the scene. We give an interpretation of some coefficients of the denominator of the Poincar\'e series. The non-real poles behave as in Enestr\"om's theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem's requirements., Comment: 52 pages, 84 figures, 29 tables

We say that an indecomposable Cartan matrix A with entries in the ground field of characteristic 0 is almost affine if the Lie sub(super)algebra determined by it is not finite dimensional or affine but the Lie (super)algebra determined by any submatrix of A, obtained by striking out any row and any column intersecting on the main diagonal, is the sum of finite dimensional or affine Lie (super)algebras. A Lie (super)algebra with Cartan matrix is said to be almost affine if it is not finite dimensional or affine, and all of its Cartan matrices are almost affine. We list all almost affine Lie superalgebras over complex numbers correcting two earlier claims of classification and make available the list of almost affine Lie algebras obtained by Li Wang Lai., Comment: 92 pages

In defining quantum superalgebras, extra relations need to be added to the Serre-like relations. They are obtained for sl q (m, n) and osp q (m, 2n) usingq-oscillator representations.

The analog of the Riemann tensor of the phase spaces of the nonholonomic (with constrained velocities) dynamical systems on manifolds and supermanifolds is proposed. These tensors are needed to perform H. Hertz’s formulation of mechanics (without the notion of force) and to write supergravity equations on any N-extended Minkowski superspace. Our approach provides one also with a method to select coset superspaces of SL(N|4) for the role of the N-extended Minkowski superspaces. For N=1, 2, 4, 8 certain most symmetric examples are considered. The method is applicable as well to M. Vasiliev’s models with N>8.

We continue the study of a generalization of supermanifolds (called here metamanifolds) on which “functions” form a metabelian algebra (one for which [[x, y], z] = 0). The usual superspaces considered as metaspaces and some conventional lagrangeans have a symmetry wider than supersymmetry. Infinitesimal transformations of these metaspaces constitute Volichenko algebras. The Volichenko algebras are natural generalizations of Lie superalgebras. Here we classify simple finite-dimensional complex Volinchenko algebras (under a technical hypothesis). Their list is as discrete as the list of simple Lie superalgebras. The results may be significant for applications to physics in connection with parastatistics.

The infinite-dimensional superalgebras W N , describing arbitrary local supercoordinate transformations in D =1 N -extended superspace, are considered as local W N supercurrent algebras. For N =1 and N =2 these superalgebras admit the central extension, necessary for their quantum realizations. The W 1 supercurrents can be identified with N =2 conformal supercurrents, and the W 2 supercurrent algebra is the extension of the SU(2), N =4 conformal supercurrent algebras.

Over algebraically closed fields of characteristic p>2, prolongations of the simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. Several new simple Lie superalgebras are discovered, serial and exceptional, including superBrown and superMelikyan superalgebras. Simple Lie superalgebras with Cartan matrix of rank 2 are classified., 20 pages, to appear in Funktsional. Anal. i Prilozhen

Exactly 10 of all simple Lie superalgebras of vector fields on 1|N-dimensional supercircles (superstrings) preserving a structure (either nothing, or a volume element, or a contact form) have nontrivial central extensions. The values of the central charges in (projective) spinor-oscillator representations of these stringy superalgebras associated with the adjoint module can be interpreted as critical dimensions of respective superstrings. Apart from the well-known values 26, 10, 2 and 0 corresponding to N = 0, 1, 2 and > 2, respectively, for the contact type stringy superalgebras (Neveu–Schwarz and Ramond types alike), there are two more non-zero values of critical dimension: For the general and divergence-free algebras for N = 2. These dimensions, found here, are i1 and i2, respectively. We also mention related problems.

Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superal- gebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness in characteristic 2, Dynkin diagram, Chevalley generators, and even the notion of Lie superalgebra if the characteristic is equal to 2. Interesting phenomena in characteristic 2: (1) all simple Lie superalgebras with Cartan matrix are obtained from simple Lie algebras with Cartan matrix by declaring several (any) of its Chevalley generators odd; (2) there exist simple Lie superalgebras whose even parts are solvable. The Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams are also listed and their simple subquotients described.

We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's and Hahn's q-polynomials and introduce orthogonal polynomials corresponding to Lie superlagebras. We also describe the real forms of gl(N), quasi-finite modules over gl(N), and conditions for unitarity of the quasi-finite modules. Analogs of tensors over gl(N) are also introduced., Comment: 25 pages, LaTeX

Lately we observe: (1) an upsurge of interest (in particular, triggered by a paper by Atiyah and Witten) to manifolds with G(2)-type structure; (2) classifications are obtained of simple (finite dimensional and graded vectorial) Lie superalgebras over fields of complex and real numbers and of simple finite dimensional Lie algebras over algebraically closed fields of characteristic p>3; (3) importance of nonintegrable distributions in (1) -- (2). We add to interrelation of (1)--(3) an explicit description of several exceptional simple Lie algebras for p=2, 3 (Melikyan algebras; Brown, Ermolaev, Frank, and Skryabin algebras) as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by G(2), O(7), Sp(4) and Sp(10). The description is performed in terms of Cartan-Tanaka-Shchepochkina prolongs and is similar to descriptions of simple Lie superalgebras of vector fields with polynomial coefficients. Our results illustrate usefulness of Shchepochkina's algorithm and SuperLie package; two families of simple Lie algebras found in the process might be new., 33 pages; Formulation of Theorem 3.2.1 corrected; references added; exposition edited

We explicitly describe the defining relations for simple Lie algebra of vector fields with polynomial coefficients and its subalgebras of divergence free, hamiltonian and contact vector fields, and for the Poisson algebra (realized on polynomials). We consider generators of these Lie algebras corresponding to the systems of simple roots associated with the standard grading of these algebras. (These systems of simple roots are distinguished in the sense of Penkov and Serganova., Comment: 11 pages, Latex