Relations Between Algebras and Their Subalgebras of Invariants

Abstract

We prove that, if H is a finite-dimensional semisimple Hopf algebra, and A is an FCR H-module algebra over an algebraically closed field, then A is a PI-algebra, provided the subalgebra of invariants is a PI-algebra. We also show that if A is an affine algebra with an action of a finite group G by automorphisms, the subalgebra of the fixed points A^G is in the center of A, and the characteristic of the ground field is either zero or relatively prime to the order of G, then A^G is affine. Analogous results are proved for graded algebras and H-module algebras over a semisimple triangular Hopf algebra over a field of characteristic zero. We prove also that, if A is an H-module algebra with an identity element, and H is either a semisimple group algebra or its dual, then, if A is semiprimitive (semiprime), then so is A^H.

Key Words:

• Mathematics
• Quantum algebra

Mathematics Subject Classification:

• 16W30
• 16R10

ACKNOWLEDGMENT

I thank my parents and sister for moral support.

Notes

Communicated by H.-J. Schneider.