×

zbMATH — the first resource for mathematics

Hopf algebra actions. (English) Zbl 0591.16005
Let H be a finite dimensional Hopf algebra over the field k and let A be an H-module algebra. Then the smash product A#H is a ring extension of A and the ring \(A^ H\) of H-invariants defined by \(A^ H=\{a\in A|\) \(h\cdot a=\epsilon (h)a\}\) is a subring of A. This paper studies the connection between these rings using the key fact that if H is semisimple, then it has a left integral, namely an element x with \(hx=\epsilon (h)x\) for all \(h\in H\), satisfying \(\epsilon\) (x)\(\neq 0.\)
Section 1 obtains a Maschke-type theorem for smash products. Indeed it is shown that if H is semisimple and if \(W\subseteq V\) are A#H-modules, then W is an A-direct summand of V if and only if it is an A#H-direct summand.
Section 2 introduces a Morita context \([B,_ BA_{A\#H},_{A\#H}A_ B,A\#H]\) where \(B=A^ H\) and with [, ] nondegenerate. Then criteria are given for (, ) to be nondegenerate and the context is used to relate \(A^ H\) to A#H. The paper also proposes an interesting question. Namely if H is a finite dimensional semisimple Hopf algebra and if A is semiprime, must A#H also be semiprime? Some partial results are given.
Reviewer: D.S.Passman

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abe, E, Hopf algebras, (1980), Cambridge Univ. Press Cambridge · Zbl 0476.16008
[2] Amitsur, S.A, Rings of quotients and Morita contexts, J. algebra, 17, 273-298, (1971) · Zbl 0221.16014
[3] Bergman, G; Isaacs, I.M, Rings with fixed point free group actions, (), 69-87 · Zbl 0234.16005
[4] \scG. Bergman, Groups acting on rings, group graded rings and beyond, preprint.
[5] Blattner, R.J; Montgomery, S, A duality theorem for Hopf module algebra, J. algebra, 95, 153-172, (1985) · Zbl 0589.16010
[6] Chase, S.U; Harrison, D.K; Rosenberg, A, Galois theory and cohomology of commutative rings, Mem. amer. math. soc., 52, (1965) · Zbl 0143.05902
[7] Cohen, M, A Morita context related to finite automorphism groups of rings, Pacific J. math., 98, 1, 37-54, (1982) · Zbl 0488.16024
[8] Cohen, M; Montgomery, S, Group-graded rings, smash products, and group actions, Trans. amer. math. soc., 282, 237-258, (1984) · Zbl 0533.16001
[9] Cohen, M; Rowen, L.H, Group graded rings, Comm. algebra, 11, 11, 1253-1270, (1983) · Zbl 0522.16001
[10] Fisher, J.W; Montgomery, S, Semiprime skew group rings’, J. algebra, 34, 217-231, (1975)
[11] Larson, R.G; Sweedler, M, An associative orthogonal bilinear form for Hopf algebras, Amer. J. math., 91, (1969) · Zbl 0179.05803
[12] Montgomery, S, Fixed rings of finite automorphism groups of associative rings, () · Zbl 0449.16001
[13] McConnell, J.C; Sweedler, M.E, Simplicity of smash products, (), 251-266 · Zbl 0221.16009
[14] Nicholson, W.K; Watters, J.F, Normal radicals and normal classes of rings, J. algebra, 59, 5-15, (1979) · Zbl 0413.16005
[15] Passman, D.S, The algebraic structure of group rings, (1977), Wiley New York · Zbl 0366.16003
[16] Passman, D.S, Fixed rings and integerality, J. algebra, 68, 510-519, (1981) · Zbl 0455.16016
[17] Passman, D.S, It’s essentially Maschke’s theorem, rocky mountain J. math. (1), 37-54, (1983) · Zbl 0525.16022
[18] Sweedler, M.E, Integrals for Hopf algebras, Ann. of math., 89, 323-335, (1969) · Zbl 0174.06903
[19] Sweedler, M.E, Hopf algebras, (1969), Benjamin New York · Zbl 0194.32901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.