arXiv subjects

Pavel Grozman

Publications and source records attributed to Pavel Grozman.

26 records · Page 2Linked to original sources

arXiv2002

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. The result provides with yet another counterexample to Coleman-Mandula's theorem.

hep-th↗

arXiv2002

Defining relations for the exceptional Lie superalgebras of vector fields pertaining to The Standard Model

We list defining relations for the four of the five exceptional simple Lie superalgebras some of which, as David Broadhurst conjectured and Kac demonstrated, may pertain to The Standard Model or Grand unified theories of elementary particles. For the fifth superalgebra the result is not final: there might be infinitely many relations. Contrariwise, for the same Lie superalgebra with Laurent polynomials as coefficients there are only finitely many relations.

math-ph↗

arXiv2002

Defining relations for classical Lie superalgebras without Cartan matrices

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.

math.RT↗

arXiv2002

Generalizations of the Lie superalgebras of supermatrices of complex size and related topics

A class of simple filtered Lie algebras of polynomial growth with increasing filtration is distinguished and presentations of these algebras are explicitely described for the simplest examples. Lie (super)algebras of this class appear in relation with Calogero--Sutherland model, high-spin supergravity, etc.; they are associated with the associative algebras of twisted differential operators on the big Schubert cell of the flag varieties. The Lie algebra of matrices of complex size introduced by Feigin is a simplest example of our algebras. Usually, they posess a trace and an invariant symmetric bilinear form; hence, analogs of dynamical systems such as Yang-Baxter, KdV, Leznov--Saveliev, etc. are associated with them. In particular, in the space of pseudodifferential operators there are analogs of the KdV hierarchies associated with sl(n) for n complex in the same way as the KdV hierarchy is associated with sl(n) for n integer are those studied by Gelfand--Dickey and Khesin--Malikov. We briefly describe such dynamical systems, and generalizations of the classical orthogonal polynomials.

math.RT↗

arXiv2001

The Shapovalov determinant for the Poisson superalgebras

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 k^L(1|6) of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov described irreducible finite dimensional representations of po(0|n) and sh(0|n); we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over po(0|2k) and sh(0|2k).

math.QA↗

arXiv2001

On bilinear invariant differential operators acting on tensor fields on the symplectic manifold

Let $M$ be an $n$-dimensional manifold, $V$ the space of a representation $ρ: GL(n)\longrightarrow GL(V)$. Locally, let $T(V)$ be the space of sections of the tensor bundle with fiber $V$ over a sufficiently small open set $U\subset M$, 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 $D: T(V_{1})\otimes T(V_{2})\longrightarrow T(V_{3})$ for irreducible fibers with lowest weight. Here the result is generalized to bilinear operators invariant with respect to the group $\Diff_ω(M)$ 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 $M$.

math.SG↗

arXiv1997

Lie superalgebras of string theories

We define and describe simple complex Lie superalgbras of vector fields on "supercircles" - simple stringy superalgebras. There are four series of such algebras and four exceptional stringy superalgebras. The 13 of the simple stringy Lie superalgebras are distinguished: only they have nontrivial central extensions; since two of the distinguish algebras have 3 nontrivial central extensions each, there are exactly 16 superizations of the Liouville action, Schroedinger equation, KdV hierarchy, etc. We also present the three nontrivial cocycles on the N=4 extended Neveu-Schwarz and Ramond superalgebras in terms of primary fields and describe the "classical" stringy superalgebras close to the simple ones. One of these stringy superalgebras is a Kac-Moody superalgebra G(A) with a nonsymmetrizable Cartan matrix A. Unlike the Kac-Moody superalgebras of polynomial growth with symmetrizable Cartan matrix, it can not be interpreted as a central extension of a twisted loop algebra.The stringy superalgebras are often referred to as superconformal ones. We discuss how superconformal stringy superalgebras really are.

hep-th↗

arXiv1997

Defining Relations for Lie Superalgebras with Cartan matrix

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. Our results are applicable to the Lie algebras with the same Cartan matrices as the Lie superalgebras considered.

hep-th↗