Abstract. 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 superalgebra 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.

The notion of defining relations is well-defined for a nilpotent Lie (super)algebra. One of the ways to present a simple Lie algebra is, therefore, by splitting it into the direct sum of a maximal diagonalizing (commutative) subalgeba and 2 nilpotent subalgebras (positive and negative). The relations obtained for finite dimensional Lie algebras are neat; they are called Serre relations and can be encoded via an integer symmetrizable matrix, the Cartan matrix, which, in turn, can be encoded by means of a graph, the Dynkin diagram. The complete set of relations for Lie algebras with an arbitrary Cartan matrix is unknown.We completely describe presentations of Lie superalgebras with Cartan matrix if they are simple Zgraded 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.Our results are applicable to the Lie algebras with the same Cartan matrices as the Lie superalgebras considered.1991 Mathematics Subject Classification. 17A70, 17B01, 17B70.

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For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we explicitly give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan matrix in characteristic 2 (and -in the arXiv version of the paper -in other characteristics for completeness of the picture). In the modular and super cases, we define notions of Chevalley generators and Cartan matrix, and an auxiliary notion of the Dynkin diagram. The relations of simple 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 (indigenous for characteristic 2) are also given. To D.B. Fuchs on the occasion of his 70th birthday

ABSTRACT. 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 greater than 3; (3) importance of nonintegrable distributions in observations (1) and (2).We add to interrelation of (1)- (3) an explicit description of several exceptional simple Lie algebras for p=2, 3 (Brown, Ermolaev, Frank, and Skryabin algebras, and analogs of Melikyan 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), sp(10) and the Brown algebra br(3). 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; at least two families of simple Lie algebras found in the process are new.

Here we continue to list the differential operators invariant with respect to the 15 exceptional simple Lie superalgebras g of polynomial vector fields. A part of the list (for operators acting on tensors with finite dimensional fibers) was earlier obtained in 2 of the 15 cases by Kochetkov and in one more instance by Kac and Rudakov. Broadhurst and Kac conjectured that some of these structures pertain to the Standard Models of elementary particles and the Grand Unified Theories. So, GUT, if any exists, will be formulated in terms of operators we found, or their r-nary analogs to be found. Calculations are performed with the aid of Grozman's Mathematica-based SuperLie package. When degeneracy conditions are violated (absence of singular vectors) the corresponding module of tensor fields is irreducible. We also verified some of the earlier findings.

We distinguish a class of simple filtered Lie algebras LU g (λ) of polynomial growth with increasing filtration and whose associated graded Lie algebras are not simple. We describe presentations of such algebras. The Lie algebras LU g (λ), where λ runs over the projective space of dimension equal to the rank of g, are quantizations of the Lie algebras of functions on the orbits of the coadjoint representation of g.The Lie algebra gl(λ) of matrices of complex size is the simplest example; it is LU sl(2) (λ). The dynamical systems associated with it in the space of pseudodifferential operators in the same way as the KdV hierarchy is associated with sl(n) are those studied by Gelfand-Dickey and Khesin-Malikov. For g = sl(2) we get generalizations of gl(λ) and the corresponding dynamical systems, in particular, their superized versions. The algebras LU sl(2) (λ) posess a trace and an invariant symmetric bilinear form, hence, with these Lie algebras associated are analogs of the Yang-Baxter equation, KdV, etc.Our presentation of LU s (λ) for a simple s is related to presentation of s in terms of a certain pair of generators. For s = sl(n) there are just 9 such relations.

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