Representations of simple noncommutative Jordan superalgebras. I. (English) Zbl 1475.17048
Noncommutative Jordan superalgebras are defined by the Jordan (super)identity and the flexibility (super)identity, thus this variety generalize the notion of Jordan superalgebras. The class of noncommutative Jordan superalgebras contains also all alternative and quasiassociative superalgebras.
In the paper under review the author begins to study representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree \(\geq 3\) (where the degree is the maximal number of supplementary orthogonal idempotents) the author shows that any finite-dimensional representation is completely reducible and, depending on the superalgebra, is quasiassociative or Jordan. This is an analogue of McCrimmon’s theorem for noncommutative Jordan algebras. Then the author studies noncommutative Jordan representations of simple noncommutative Jordan superalgebras of degree \(\leq 2\), including \(D_t(\alpha,\beta,\gamma)\), \(K_3(\alpha,\beta,\gamma)\) and all simple supercommutative and quasiassociative superalgebras of degree \(2\).
In the paper under review the author begins to study representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree \(\geq 3\) (where the degree is the maximal number of supplementary orthogonal idempotents) the author shows that any finite-dimensional representation is completely reducible and, depending on the superalgebra, is quasiassociative or Jordan. This is an analogue of McCrimmon’s theorem for noncommutative Jordan algebras. Then the author studies noncommutative Jordan representations of simple noncommutative Jordan superalgebras of degree \(\leq 2\), including \(D_t(\alpha,\beta,\gamma)\), \(K_3(\alpha,\beta,\gamma)\) and all simple supercommutative and quasiassociative superalgebras of degree \(2\).
Reviewer: Maria Trushina (Moskva)
Keywords:
noncommutative Jordan superalgebra; representation theory of nonassociative superalgebras; representations of Jordan superalgebras; Kronecker factorization theoremReferences:
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