Identities of graded algebras and codimension growth

Authors:
Yu. A. Bahturin and M. V. Zaicev

Translated by:

Journal:
Trans. Amer. Math. Soc. **356** (2004), 3939-3950

MSC (2000):
Primary 16R10, 16W50

Published electronically:
January 16, 2004

MathSciNet review:
2058512

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a -graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component to that of the whole of , in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where is finite dimensional and has polynomial growth.

**1.**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**2.**Yuri Bahturin, Antonio Giambruno, and David M. Riley,*Group-graded algebras with polynomial identity*, Israel J. Math.**104**(1998), 145–155. MR**1622291**, 10.1007/BF02897062**3.**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**4.**Amitai Regev,*Existence of identities in 𝐴⊗𝐵*, Israel J. Math.**11**(1972), 131–152. MR**0314893****5.**A. Giambruno and M. Zaicev,*On codimension growth of finitely generated associative algebras*, Adv. Math.**140**(1998), no. 2, 145–155. MR**1658530**, 10.1006/aima.1998.1766**6.**A. Giambruno and M. Zaicev,*Exponential codimension growth of PI algebras: an exact estimate*, Adv. Math.**142**(1999), no. 2, 221–243. MR**1680198**, 10.1006/aima.1998.1790**7.**A. Giambruno and M. Zaicev,*Minimal varieties of algebras of exponential growth*, Electron. Res. Announc. Amer. Math. Soc.**6**(2000), 40–44 (electronic). MR**1767635**, 10.1090/S1079-6762-00-00078-0**8.**E. J. Taft,*Invariant Wedderburn factors*, Illinois J. Math.**1**(1957), 565–573. MR**0098124****9.**Yu. A. Bahturin, S. K. Sehgal, and M. V. Zaicev,*Group gradings on associative algebras*, J. Algebra**241**(2001), no. 2, 677–698. MR**1843319**, 10.1006/jabr.2000.8643**10.**Amitai Regev,*Codimensions and trace codimensions of matrices are asymptotically equal*, Israel J. Math.**47**(1984), no. 2-3, 246–250. MR**738172**, 10.1007/BF02760520**11.**A. Malcev,*On the representations of infinite algebras*, Rec. Math. [Mat. Sbornik] N.S.**13 (55)**(1943), 263–286 (Russian, with English summary). MR**0011084**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
16R10,
16W50

Retrieve articles in all journals with MSC (2000): 16R10, 16W50

Additional Information

**Yu. A. Bahturin**

Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1A 5K9 – and – Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119899, Russia

Email:
yuri@math.mun.ca

**M. V. Zaicev**

Affiliation:
Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119899, Russia

Email:
zaicev@mech.math.msu.su

DOI:
http://dx.doi.org/10.1090/S0002-9947-04-03426-9

Received by editor(s):
March 6, 2002

Received by editor(s) in revised form:
May 29, 2003

Published electronically:
January 16, 2004

Additional Notes:
The first author was partially supported by MUN Dean of Science Research Grant #38647

The second author was partially supported by RFBR, grants 99-01-00233 and 00-15-96128

Article copyright:
© Copyright 2004
American Mathematical Society