Superalgebras with graded involution and star-graded colength bounded by 3

A superalgebra A over a field F of characteristic zero endowed with a graded involution is called a *-superalgebra. Fonseca et al. [Characterizations of *-superalgebras of polynomial growth. Linear Multilinear Algebra. 2016; 64: 1379-1389] and Giambruno et al. [Identities of *-superalgebras and almost polynomial growth. LinearMultilinear Algebra. 2016; 64(3): 484-501] studied the behaviour of the sequence of *-graded codimensions of a finite dimensional *-superalgebra A and introduced a character associated to A denoted by chi((n)) (A). Here we make explicit the decomposition of chi((n)) (A) for some important *-superalgebras A and compute the number l((n)) (A) of irreducibles appearing in that decomposition to form the sequence of *-graded colengths. Finally, we use such decompositions to classify the finite dimensional *-superalgebras A such that the sequence l((n)) (A) is bounded by three.