Structure of Zariski-closed algebras

Authors:
Alexei Belov-Kanel, Louis Rowen and Uzi Vishne

Journal:
Trans. Amer. Math. Soc. **362** (2010), 4695-4734

MSC (2010):
Primary 16G99

DOI:
https://doi.org/10.1090/S0002-9947-10-04993-7

Published electronically:
April 28, 2010

MathSciNet review:
2645047

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The objective of this paper is to describe the structure of Zariski-closed algebras, which provide a useful generalization to finite dimensional algebras in the study of representable algebras over finite fields. Our results include a version of Wedderburn's principal theorem as well as a more explicit description using representations, in terms of ``gluing'' in Wedderburn components. Finally, we construct ``generic'' Zariski-closed algebras, whose description is considerably more complicated than the description of generic algebra of finite dimensional algebras.

Special attention is given to infinite dimensional algebras over finite fields.

**[B1]**Belov, A. Ya.,*Algebras with polynomial identities: Representations and combinatorical methods*, Doctor of Science Dissertation, Moscow (2002).**[B2]**Belov, A.,*Counterexamples to the Specht problem*, Sb. Math.**191**(2000), pp. 329-340. MR**1773251 (2001g:16043)****[BR]**Belov, A.Ya. and Rowen, L.H. ``Computational aspects of polynomial identities''. Research Notes in Mathematics, 9. AK Peters, Ltd., Wellesley, MA, 2005. MR**2124127 (2006b:16001)****[GZ]**Giambruno, A. and Zaicev, M.,*Minimal varieties of algebras of exponential growth*, Adv. Math.**174**(2003), pp. 310-323. MR**1963697 (2004b:16028)****[H]**Humphreys, J.,*Linear algebraic groups*, Springer Lecture Notes in Mathematics**21**Springer, New York, 1975. MR**0396773 (53:633)****[KMT]**Kambayashi, T.; Miyanishi, M. and Takeuchi, M., ``Unipotent algebraic groups'', Lecture Notes in Mathematics,**414**, Springer-Verlag, Berlin-New York, 1974. MR**0376696 (51:12871)****[L]**Lewin, J.,*A matrix representation for associative algebras. I and II*, Trans. Amer. Math. Soc.**188**(2) (1974), 293-317; ibid. MR**0338081 (49:2848)****[M]**Miyanishi, M.,*Questions of rationality of solvable algebraic groups over non-perfect fields*, Annali Mat. Pura Appl.**61**(4), (1963) 97-120. MR**0158891 (28:2113)****[P]**Putcha, M.,*Linear algebraic monoids*, London Math. Soc. Lecture Notes**133**, Cambridge University Press, 1988. MR**964690 (90a:20003)****[R1]**Rowen, L.H.,*Polynomial Identities in Ring Theory*, Academic Press, Inc., Pure and Applied Math.,**84**, New York, 1980. MR**576061 (82a:16021)****[R2]**-,*Ring Theory I*, Pure and Applied Math,**127**, Academic Press., Inc., New York, 1988. MR**1095047 (94e:16001)****[R3]**-,*Graduate Algebra: A noncommutative view*, Graduate Series in Math., Amer. Math. Soc., Providence, RI, 2008. MR**2462400 (2009k:16001)****[T]**Tits, J., ``Lectures on algebraic groups'', Dept. of Math., Yale Univ. - New Haven, 1966/67.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
16G99

Retrieve articles in all journals with MSC (2010): 16G99

Additional Information

**Alexei Belov-Kanel**

Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

Email:
belova@macs.biu.ac.il

**Louis Rowen**

Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

Email:
rowen@macs.biu.ac.il

**Uzi Vishne**

Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

Email:
vishne@macs.biu.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-10-04993-7

Received by editor(s):
September 22, 2008

Published electronically:
April 28, 2010

Additional Notes:
This research was supported by the Israel Science Foundation, grant #1178/06.

Article copyright:
© Copyright 2010
American Mathematical Society

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