Simple right alternative superalgebras (Q6608062)

The problem of classifying simple algebras is one of the central ones in the structural theory of algebras and superalgebras. So, there are many results about the classification of simple Lie and Jordan superalgebras. In 1964, \textit{C. T. C. Wall} [J. Reine Angew. Math. 213, 187--199 (1964; Zbl 0125.01904)] described the structure of simple associative superalgebras; simple alternative superalgebras of arbitrary characteristic are described by \textit{I. P. Shestakov} [Algebra Logika 36, No. 6, 675--716 (1997; Zbl 0904.17025); translation in Algebra Logic 36, No. 6, 389--412 (1997)]. The main supervariety generalizing the supervarieties of associative and alternative superalgebras is the variety of right alternative superalgebras. The present paper surveys recently obtained results by authors related to the classification of simple right alternative superalgebras.\N\NSection 1 contains basic definitions and examples of unital superalgebras. Section 2 considers unital superalgebras with bimodule constraints, such that the even part is associative and commutative, and the odd part is an associative bimodule. Section 3 gives a classification of unital superalgebras with restrictions on the even part: (1) the even part is associative and commutative; (2) the even part is semisimple; (3) the even part is the sum of the nil-radical and the ground field. Section 4 deals with singular superalgebras, i.e., superalgebras in which the product of even elements is equal to 0. Section 5 is devoted to the study of derivations and automorphisms of simple superalgebras. The last section lists open questions and unsolved problems of interest.