Representations of simple noncommutative Jordan superalgebras I

Abstract

In this article we begin the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree ≥ 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 (α, β, γ) and K 3 (α, β, γ) and prove the Kronecker factorization theorem for superalgebras D t (α, β, γ). In the last section we use a new approach to study noncommutative Jordan representations of simple Jordan superalgebras.

Non-associative algebraic structures: classification and structure

“…Simple noncommutative Jordan superalgebras were described by Pozhidaev and Shestakov in [225,226]. Representations of simple noncommutative superalgebras were described by Popov [223]. Nowadays, the study of properties of simple non-associative algebras and superalgebras has aroused a strong interest.…”

On Malcev algebras that contain the three dimensional simple central Lie algebra

“…Similarly, C. Martinez and E. Zelmanov [14] obtained a Kronecker Factorization Theorem for the exceptional ten dimensional Kac superalgebra K 10 . Also, Y. Popov [17] studied the representations of simple finite-dimensional noncommutative Jordan superalgebras and proved some analogues of the Kronecker factorization theorem for such superalgebras.…”