To any superalgebra A is attached a numerical sequence cnsup(A), n≥1, called the sequence of supercodimensions of A. In characteristic zero its asymptotics are an invariant of the superidentities satisfied by A. It is well-known that for a PI-superalgebra such sequence is exponentially bounded and expsup(A)=limn→∞cnsup(A)n is an integer that can be explicitly computed. Here we introduce a notion of fundamental superalgebra over a field of characteristic zero. We prove that if A is such an algebra, then C1ntexpsup(A)n≤cnsup(A)≤C2ntexpsup(A)n, where C1>0,C2,t are constants and t is a half integer that can be explicitly written as a linear function of the dimension of the even part of A and the nilpotency index of the Jacobson radical. We also give a characterization of fundamental superalgebras through the representation theory of the symmetric group. As a consequence we prove that if A is a finitely generated PI-superalgebra, then the inequalities in (1) still hold and t is a half integer. It follows that [Formula presented] exists and is a half integer.
Giambruno A., La Mattina D. (2022). Superalgebras: Polynomial identities and asymptotics. JOURNAL OF ALGEBRA, 604, 614-635 [10.1016/j.jalgebra.2022.04.016].
Superalgebras: Polynomial identities and asymptotics
La Mattina D.
2022-01-01
Abstract
To any superalgebra A is attached a numerical sequence cnsup(A), n≥1, called the sequence of supercodimensions of A. In characteristic zero its asymptotics are an invariant of the superidentities satisfied by A. It is well-known that for a PI-superalgebra such sequence is exponentially bounded and expsup(A)=limn→∞cnsup(A)n is an integer that can be explicitly computed. Here we introduce a notion of fundamental superalgebra over a field of characteristic zero. We prove that if A is such an algebra, then C1ntexpsup(A)n≤cnsup(A)≤C2ntexpsup(A)n, where C1>0,C2,t are constants and t is a half integer that can be explicitly written as a linear function of the dimension of the even part of A and the nilpotency index of the Jacobson radical. We also give a characterization of fundamental superalgebras through the representation theory of the symmetric group. As a consequence we prove that if A is a finitely generated PI-superalgebra, then the inequalities in (1) still hold and t is a half integer. It follows that [Formula presented] exists and is a half integer.File | Dimensione | Formato | |
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