Actions of pointed Hopf algebras with reduced pi invariants

Authors:
Piotr Grzeszczuk and Malgorzata Hryniewicka

Journal:
Proc. Amer. Math. Soc. **135** (2007), 2381-2389

MSC (2000):
Primary 16R20, 16S40, 16W30

Published electronically:
March 29, 2007

MathSciNet review:
2302559

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an -module algebra, where is a pointed Hopf algebra acting on finitely of dimension . Suppose that for every nonzero -stable left ideal of . It is proved that if satisfies a polynomial identity of degree , then satisfies a polynomial identity of degree provided at least one of the following additional conditions is fulfilled:

- is semiprime and is almost central in ,
- is reduced.

**[AS]**Nicolás Andruskiewitsch and Hans-Jürgen Schneider,*Pointed Hopf algebras*, New directions in Hopf algebras, Math. Sci. Res. Inst. Publ., vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 1–68. MR**1913436**, 10.2977/prims/1199403805**[BaL]**Y. A. Bahturin and V. Linchenko,*Identities of algebras with actions of Hopf algebras*, J. Algebra**202**(1998), no. 2, 634–654. MR**1617671**, 10.1006/jabr.1997.7314**[BaZ]**Y. A. Bahturin and M. V. Zaicev,*Identities of graded algebras*, J. Algebra**205**(1998), no. 1, 1–12. MR**1631298**, 10.1006/jabr.1997.7017**[BeG]**Konstantin I. Beidar and Piotr Grzeszczu,*Actions of Lie algebras on rings without nilpotent elements*, Algebra Colloq.**2**(1995), no. 2, 105–116. MR**1329141****[BeT]**K. I. Beidar and B. Torrecillas,*On actions of Hopf algebras with cocommutative coradical*, J. Pure Appl. Algebra**161**(2001), no. 1-2, 13–30. MR**1834076**, 10.1016/S0022-4049(00)00087-6**[BC]**Jeffrey Bergen and Miriam Cohen,*Actions of commutative Hopf algebras*, Bull. London Math. Soc.**18**(1986), no. 2, 159–164. MR**818820**, 10.1112/blms/18.2.159**[BCF]**Jeffrey Bergen, Miriam Cohen, and Davida Fischman,*Irreducible actions and faithful actions of Hopf algebras*, Israel J. Math.**72**(1990), no. 1-2, 5–18. Hopf algebras. MR**1098978**, 10.1007/BF02764609**[BG]**Jeffrey Bergen and Piotr Grzeszczuk,*Invariants of Lie superalgebras acting on associative algebras*, Israel J. Math.**94**(1996), 403–428. MR**1394584**, 10.1007/BF02762714**[C]**M. Cohen,*Quantum commutativity and central invariants*, Advances in Hopf algebras (Chicago, IL, 1992) Lecture Notes in Pure and Appl. Math., vol. 158, Dekker, New York, 1994, pp. 25–38. MR**1289420****[CW]**Miriam Cohen and Sara Westreich,*Central invariants of 𝐻-module algebras*, Comm. Algebra**21**(1993), no. 8, 2859–2883. MR**1222747**, 10.1080/00927879308824709**[GH]**Piotr Grzeszczuk and Małgorzata Hryniewicka,*Polynomial identities of algebras with actions of pointed Hopf algebras*, J. Algebra**278**(2004), no. 2, 684–703. MR**2071660**, 10.1016/j.jalgebra.2004.03.010**[K1]**V. K. Harčenko,*Generalized identities with automorphisms*, Algebra i Logika**14**(1975), no. 2, 215–237, 241 (Russian). MR**0399153****[K2]**V. K. Harčenko,*Fixed elements of a finite group action on a semiprime ring*, Algebra i Logika**14**(1975), no. 3, 328–344, 369–370 (Russian). MR**0429998****[K3]**V. K. Kharchenko, J. Keller, and S. Rodrigues-Romo,*Prime rings with PI rings of constants*. part B, Israel J. Math.**96**(1996), no. part B, 357–377. MR**1433695**, 10.1007/BF02937311**[M1]**Susan Montgomery,*Hopf algebras and their actions on rings*, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR**1243637****[M2]**S. Montgomery,*Bi-invertible actions of Hopf algebras*, Israel J. Math.**83**(1993), no. 1-2, 45–71. MR**1239716**, 10.1007/BF02764636

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
16R20,
16S40,
16W30

Retrieve articles in all journals with MSC (2000): 16R20, 16S40, 16W30

Additional Information

**Piotr Grzeszczuk**

Affiliation:
Faculty of Computer Science, Technical University of Białystok, Wiejska 45A, 15-351 Białystok, Poland

Email:
piotrgr@pb.bialystok.pl

**Malgorzata Hryniewicka**

Affiliation:
Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, Poland

Email:
margitt@math.uwb.edu.pl

DOI:
https://doi.org/10.1090/S0002-9939-07-08769-2

Received by editor(s):
January 8, 2006

Received by editor(s) in revised form:
April 25, 2006

Published electronically:
March 29, 2007

Additional Notes:
The first author was supported by Polish KBN grant No. 1 P03A 032 27

Communicated by:
Martin Lorenz

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.