Bulletin of the London Mathematical Society



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STABLE JACOBSON RADICALS AND SEMIPRIME SMASH PRODUCTS


V. LINCHENKO a1, S. MONTGOMERY a2 1 and L. W. SMALL a3 1
a1 Yerakhtur, Shilovsky District, Ryazansky Region, Russia 391534
a2 University of Southern California, Los Angeles, CA 90089-2532, USA, smontgom@math.usc.edu
a3 University of California at San Diego, La Jolla, CA 92093, USA, lwsmall@ucsd.edu

Article author query
linchenko v   [Google Scholar] 
montgomery s   [Google Scholar] 
small lw   [Google Scholar] 
 

Abstract

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)

Maths Classification

16W30; 16N20; 16R99; 16S40.



Footnotes

1 The second author was supported by NSF grant DMS-0100461, and the third author by NSA grant MDA 904-02-1-0004.