Superalgebras with superinvolution or graded involution with colengths sequence bounded by 3

Let A be a superalgebra with superinvolution or graded involution over a field of characteristic zero and let χn1,⋯,n4(A), n1 + ⋯ + n4 = n, be the (n1,⋯,n4)-cocharacter of A. The ∗-colengths sequence, ln∗(A), n = 1, 2,⋯, is the sum of the multiplicities in the decomposition of the (n1,⋯,n4)-cocharacter χn1,⋯,n4(A), for all n = n1 + ⋯ + n4 ≥ 1. The main purpose of this paper is to classify the superalgebras with superinvolution with ∗-colengths sequence bounded by three. Moreover, we shall extend to the general case, the analogous result proved by do Nascimento and Vieira in [Superalgebras with graded involution and star-graded colength bounded by 3, Linear Multilinear Algebra 67(10) (2019) 1999-2020] for finite-dimensional superalgebras with graded involution.