STABLE JACOBSON RADICALS AND SEMIPRIME SMASH PRODUCTS
We prove that if H is a finite-dimensional semisimple Hopf algebra acting on a PI-algebra R of characteristic 0, and R is either affine or algebraic over k, then the Jacobson radical of R is H-stable. Under the same hypotheses, we show that the smash product algebra R#H is semiprimitive provided that R is H-semiprime. More generally we show that the ‘finite’ Jacobson radical is H-stable, and that R#H is semiprimitive provided that R is H-semiprimitive and all irreducible representations of R are finite-dimensional. We also consider R#H when R is an FCR-algebra. Finally, we prove a general relationship between stability of the radical and semiprimeness of R#H; in particular if for a given H, any action of H stabilizes the Jacobson radical, then also any action of H stabilizes the prime radical.(Received August 15 2003)
(Revised October 27 2004)
16W30; 16N20; 16R99; 16S40.
1 The second author was supported by NSF grant DMS-0100461, and the third author by NSA grant MDA 904-02-1-0004.