⁎-Superalgebras and exponential growth

In this paper, we study the exponential growth of ⁎-graded identities of a finite dimensional ⁎-superalgebra A over a field F of characteristic zero. If a ⁎-superalgebra A satisfies a non-trivial identity, then the sequence {cngri(A)}n≥1 of ⁎-graded codimensions of A is exponentially bounded and so we study the ⁎-graded exponent expgri(A):=limn→∞⁡cngri(A)n of A. We show that expgri(A)=dimF⁡(A) if and only if A is a simple ⁎-superalgebra and F is the symmetric even center of A. Also, we characterize the finite dimensional ⁎-superalgebras such that expgri(A)≤1 by the exclusion of four ⁎-superalgebras from vargri(A) and construct eleven ⁎-superalgebras Ei,i=1,…,11, with the following property: expgri(A)>2 if and only if Ei∈vargri(A), for some i∈{1,…,11}. As a consequence, we characterize the finite dimensional ⁎-superalgebras A such that expgri(A)=2.