Bernstein superalgebras and supermodules
This paper is devoted to introduce Bernstein superalgebras and to describe irreducible Bernstein supermodules. Since Bernstein algebras do not form a variety of algebras, the general definition of superalgebras in an arbitrary homogeneous variety of algebras given by \textit{E. I. Zelmanov} and \textit{I.~P. Shestakov} [Izv. Akad. Nauk SSR, Ser. Mat. 54, 676-693 (1990; Zbl 0713.17020)] cannot be directly applied. To overcome this difficulty the authors introduce the new concept of \((U, W)\)-graded Bernstein algebra. A typical exemple of \((U, W)\)-graded Bernstein algebra is given in the following way. Consider a field \(K\) of characteristic different from \(2,\) a Bernstein algebra \(B\) over \(K\) with height \(\omega \) and a nonzero idempotent \(e\) in \(B.\) If \(U_e = \{ x\in \ker \omega : 2ex = x\}\) and \(W_e = \{ x\in \ker \omega : ex = 0\} \) then \(U_e \oplus W_e\) is a \((U_e , W_e)\)-graded Bernstein algebra. Moreover, if \(A\) is a \((U, W)\)-graded Bernstein algebra then \(B = Ke\oplus A\) is a Bernstein algebra with a natural extension of the product of \(A.\) Firstly it is characterized when different graded Bernstein algebras give isomorphic Bernstein algebras. This close link between \((U, W)\)-graded Bernstein algebras and Bernstein algebras is nicely exploited by the authors. Since \((U, W)\)-graded Bernstein algebras form a (graded) variety then \((U, W)\)-graded Bernstein superalgebras and its submodules can be defined in a natural way. In section 3 it is proved that there are no nontrivial semiprime Bernstein superalgebras over a field of characteristic \(\neq 2, 3.\) The last section is devoted to the classification of the irreducible Bernstein superalgebras.
scientific article; zbMATH DE number 1271646
scientific article