Classification of linearly compact simple Jordan and generalized Poisson superalgebras

The notion of an anti-commutative (resp. commutative) rigid superalgebra is a natural generalisation of the notion of a Lie (resp. Jordan) superalgebra. Intuitively rigidity means that small deformations of the product under the structural group produce an isomorphic algebra. In this paper we classify all linearly compact simple anti-commutative (resp. commutative) rigid superalgebras. Beyond Lie (resp. Jordan) superalgebras the complete list includes four series and twenty t...

In this paper we look into the structure of finite-dimensional graded superalgebras of various types such as associative, Lie and Jordan over an algebraically closed field of characteristic zero.

We classify simple linearly compact n-Lie superalgebras with n>2 over a field F of characteristic 0. The classification is based on a bijective correspondence between non-abelian n-Lie superalgebras and transitive Z-graded Lie superalgebras of the form L=\oplus_{j=-1}^{n-1} L_j, such that L_{-1}=g, where dim L_{n-1}=1, L_{-1} and L_{n-1} generate L, and [L_j, L_{n-j-1}] =0 for all j, thereby reducing it to the known classification of simple linearly compact Lie superalgebras ...

Motivated by the theory of unitary representations of finite dimensional Lie supergroups, we describe those Lie superalgebras which have a faithful finite dimensional unitary representation. We call these Lie superalgebras unitary. This is achieved by describing the classification of real finite dimensional compact simple Lie superalgebras, and analyzing, in a rather elementary and direct way, the decomposition of reductive Lie superalgebras ($\g$ is a semisimple $\g_{\bar 0}...

In this article we begin the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree $\geq 3$ we show that any finite-dimensional representation is completely reducible and, depending on the superalgebra, quasiassociative or Jordan. Then we study representations of superalgebras $D_t(\alpha,\beta,\gamma)$ and $K_3(\alpha, \beta, \gamma)$ and prove the Kronecker factorization theorem for superalgebras $D_t(\alpha,\beta,\...

I.P. Shestakov constructed an example of a unital simple special Jordan superalgebra over the field of real numbers. It turned out to be a subsuperalgebra of the Jordan superalgebra of vector type, but not isomorphic to a superalgebra of this type. Moreover, its superalgebra of fractions is isomorphic to a Jordan superalgebra of vector type. Author constructed a similar example of a Jordan superalgebra over a field of characteristic~0 in which the equation $t^2+1=0$. In this ...

We describe non-trivial $\delta$-derivations of semisimple finite-dimensional Jordan algebras over an algebraically closed field of characteristic not 2, and of simple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic 0. For these classes of algebras and superalgebras, non-zero $\delta$-derivations are shown to be missing for $\delta\neq 0,{1}{2},1$, and we give a complete account of ${1}{2}$-derivations.

A way to construct (conjecturally all) simple finite dimensional modular Lie (super)algebras over algebraically closed fields of characteristic not 2 is offered. In characteristic 2, the method is supposed to give only simple Lie (super)algebras graded by integers and only some of the non-graded ones). The conjecture is backed up with the latest results computationally most difficult of which are obtained with the help of Grozman's software package SuperLie.

We compare a number of different definitions of structure algebras and TKK constructions for Jordan (super)algebras appearing in the literature. We demonstrate that, for unital superalgebras, all the definitions of the structure algebra and the TKK constructions fall apart into two cases. Moreover, one can be obtained as the Lie superalgebra of superderivations of the other. We also show that, for non-unital superalgebras, more definitions become non-equivalent. As an applica...

In this paper, we develop a method to obtain the algebraic classification of noncommutative Jordan algebras from the classification of Jordan algebras of the same dimension. We use this method to obtain the algebraic classification of complex $3$-dimensional noncommutative Jordan algebras. As a byproduct, we obtain the classification of complex $3$-dimensional Kokoris, standard, generic Poisson, and generic Poisson--Jordan algebras; and also complex $4$-dimensional nilpotent ...