Codimension growth and minimal superalgebras
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- by A. Giambruno and M. Zaicev
- Trans. Amer. Math. Soc. 355 (2003), 5091-5117
- DOI: https://doi.org/10.1090/S0002-9947-03-03360-9
- Published electronically: July 24, 2003
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Abstract:
A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope $G(A)$ of a finite dimensional superalgebra $A$. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field. The importance of such algebras is readily proved: $A$ is a minimal superalgebra if and only if the ideal of identities of $G(A)$ is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties $\mathcal {V}$ such that $\exp ({\mathcal {V}})=d\ge 2$ and $\exp (\mathcal {U})<d$ for all proper subvarieties ${\mathcal {U}}$ of ${\mathcal {V}}$. This proves in the positive a conjecture of Drensky (1988). As a corollary we obtain that there is only a finite number of minimal varieties for any given exponent. A classification of minimal varieties of finite basic rank was proved by the authors (2003). As an application we give an effective way for computing the exponent of a T-ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras.References
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Bibliographic Information
- A. Giambruno
- Affiliation: Dipartimento di Matematica ed Applicazioni, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy
- MR Author ID: 73185
- ORCID: 0000-0002-3422-2539
- Email: agiambr@unipa.it
- M. Zaicev
- Affiliation: Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119992 Russia
- MR Author ID: 256798
- Email: zaicev@mech.math.msu.su
- Received by editor(s): June 12, 2002
- Received by editor(s) in revised form: March 20, 2003
- Published electronically: July 24, 2003
- Additional Notes: The first author was supported in part by MIUR of Italy.
The second author was partially supported by RFBR, grants 02-01-00219 and 00-15-96128. - © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 5091-5117
- MSC (2000): Primary 16R10; Secondary 16P90
- DOI: https://doi.org/10.1090/S0002-9947-03-03360-9
- MathSciNet review: 1997596