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The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the $1|n$-dimensional supermanifold (superstring) $\mathcal{M}$, we classify analogs of Bol operators invariant under the simple maximal subalgebra $\mathfrak{h}$ of the same rank as its simple ambient superalgebra $\mathfrak{g}$ of vector fields on $\mathcal{M}$ and containing all elements of negative degree of $\mathfrak{g}$ in a $\mathbb{Z}$-grading. We also consider the Lie superalgebras of vector fields $\mathfrak{g}$ preserving a contact structure on the superstring $\mathcal{M}$. We have discovered many new operators., Comment: 23 pages

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

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

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

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/

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

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

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

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

The analogs of Chevalley generators are offered for simple (and close to them) Z-graded complex Lie algebras and Lie superalgebras of polynomial growth without Cartan matrix. We show how to derive the defining relations between these generators and explicitly write them for a "most natural" ("distinguished" in terms of Penkov and Serganova) system of simple roots. The results are given mainly for Lie superalgebras whose component of degree zero is a Lie algebra (other cases being left to the reader). Observe presentations of exceptional Lie superalgebras and Lie superalgebras of hamiltonian vector fields. Now we can, at last, q-quantize the Lie Lie superalgebras of hamiltonian vector fields and Poisson superalgebras., 13p., Latex (this is an expanded version of the SQS'99 talk)