On simple Filippov superalgebras of type A(0,n)

Abstract

It is proved that there exist no simple finite-dimensional Filippov superalgebras of type A(0, n) over an algebraically closed field of characteristic 0.

On simple Filippov superalgebras of type A(m,n)

“…The same problem concerning Filippov superalgebras of type A(m, n) with m = n has recently been solved in [1]. Moreover, the type A(0, n), with n ∈ N, was studied in [2]. The present work represents the final step towards the classification of finite-dimensional simple Filippov superalgebras of type A(m, n) over an algebraically closed field of characteristic zero.…”

“…In this subsection, because of [1] and [2], we assume that n ∈ N \ {1}. We begin with some technical lemmas on irreducible modules of some special types over A(1, n).…”

“…Proof. We can suppose that G = A(m, n) with m = n and m ≥ 2, because we have already proved that there exist no simple Filippov superalgebras of type A(n, n) with n ∈ N, [1], nor of type A(1, n) with n ∈ N 0 \ {1} and of type A(0, n) with n ∈ N, [2]. Assume that V is a finite-dimensional irreducible module over G with the highest weight Λ = (a 1 , .…”

“…The fourth section is devoted to the problem of existence of finite-dimensional simple Filippov superalgebras of type A(m, n) with m = n. We start with the particular case A(1, n) in the first subsection, where, taking into account [1] and [2], it is assumed that n ∈ N \ {1}. The main result of this article (Theorem 4.2) is stated and proved in the second subsection.…”