Semigroup graded algebras and graded PI-exponent

Abstract

Let S be a semigroup. We study the structure of graded-simple S-graded algebras A and the exponential rate PIexp^S-gr(A):= lim_n→∞ $\sqrt[n]{{c_n^{S - gr}\left( A \right)}}$ of growth of codimensions c _n ^S-gr (A) of their graded polynomial identities. This is of great interest since such algebras can have non-integer PIexp^S-gr(A) despite being finite dimensional and associative. In addition, such algebras can have a non-trivial Jacobson radical J(A). All this is in strong contrast with the case when S is a group since in the group case J(A) is trivial, PIexp^S-gr(A) is always integer and, if the base field is algebraically closed, then PIexp^S-gr(A) equals dimA. Without any restrictions on the base field F, we classify graded-simple S-graded algebras A for a class of semigroups S which is complementary to the class of groups. We explicitly describe the structure of J(A) showing that J(A) is built up of pieces of a maximal S-graded semisimple subalgebra of A which turns out to be simple. When F is algebraically closed, we get an upper bound for ${\overline {\lim } _{n \to \infty }}\sqrt[n]{{c_n^{S - gr}\left( A \right)}}$. If A/J(A) ≈ M ₂(F) and S is a right zero band, we show that this upper bound is sharp and PIexp^S-gr(A) indeed exists. In particular, we present an infinite family of graded-simple algebras A with arbitrarily large non-integer PIexp^S-gr(A).