Pavel Grozman

Invariant operators on supermanifolds and Standard Models

Here we continue to list the differential operators invariant with respect to the 15 exceptional simple Lie superalgebras 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 Grozmans Mathematica-based SuperLie package. When degeneracy conditions are violated (absence of singular vectors) the corresponding module of tensor fields is irreducible. We consider generalized tensors corresponding to fibers of countable infinite dimension, not only finite dimensional ones. We also verified some of the earlier findings.

Structures of G(2) Type and Nonintegrable Distributions in Characteristic p

Lately, the following were observed: (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. The importance of nonintegrable distributions in the above classifications were observed and illustrated by 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 are new; one more, rather mysterious, is partly elucidated

Lie superalgebras of supermatrices of complex size. Their generalizations and related integrable systems

We distinguish a class of simple filtered Lie algebras \(L{U_\mathfrak{g}}(\lambda )\) of polynomial growth whose associated graded Lie algebras are not simple. We describe presentations of such algebras. A contraction sends \(L{U_\mathfrak{g}}(\lambda )\) into algebras of the same class studied by Donin, Gurevich and Shnider; \(L{U_\mathfrak{g}}(\lambda )\) are quantizations of the DGS algebras.

Lie superalgebra structures in H*(g;g)

Let g=vect(M) be the Lie (super)algebra of vector fields on any connected (super)manifold M; let “-” be the change of parity functor, Ci and Hi the space of i-chains and i-cohomology. The Nijenhuis bracket makes 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 L* and for any Lie superalgebra g. We use a Mathematica-based package SuperLie (already proven useful in various problems) to explicitly describe the algebras l* for some simple finite dimensional Lie superalgebras g and their “relatives” - the nontrivial central extensions or derivation algebras of the considered simple ones.

Realization of Lie algebras and superalgebras in terms of creation and annihilation operators: I

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.

An Unconventional Supergravity

We introduce and completely describe the analogues of the Riemann curvature tensor for the curved supergrassmannian of the passing through the origin (0|2)-dimensional subsupermanifolds in the (0|4)-dimensional supermanifold with the preserved volume form. The underlying manifold of this supergrassmannian is the conventional Penrose’s complexified and compactified version of the Minkowski space, i.e. the Grassmannian of 2-dimensional subspaces in the 4-dimensional space.

On Bilinear Invariant Differential Operators Acting on Tensor Fields on the Symplectic Manifold

Abstract Let M be an n-dimensional manifold, V the space of a representation ρ : GL(n) → GL(V). Locally, let T (V ) be the space of sections of the tensor bundle with fiber V over a sufficiently small open set U ⊂ M, in other words, T (V ) is the space of tensor fields of type V on M on which the group Diff(M) of diffeomorphisms of M naturally acts. Elsewhere, the author classified the Diff(M)-invariant differential operators D : T (V 1) ⊗ T (V 2) → T (V 3) for irreducible fibers with lowest weight. Here the result is generalized to bilinear operators invariant with respect to the group Diffω(M) of symplectomorphisms of the symplectic manifold (M, ω). We classify all first order invariant operators; the list of other operators is conjectural. Among the new operators we mention a 2nd order one which determins an “algebra” structure on the space of metrics (symmetric forms) on M.

The Shapovalov Determinant for the Poisson Superalgebras

Abstract Among simple ℤ-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C 2: these are the Lie superalgebra of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra of special Hamiltonian fields in 2k odd indeterminates, and the Kac–Moody version of . Using C 2 we compute N. Shapovalov determinant for and , and for the Poisson superalgebras associated with . A. Shapovalov described irreducible finite dimensional representations of and ; we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over and

The highest weight representations of the contact Lie superalgebra on 1|6-dimensional supercircle

AbstractAmong simple