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Codimension growth and minimal superalgebras.
- Source :
-
Transactions of the American Mathematical Society . Dec2003, Vol. 355 Issue 12, p5091. 27p. - Publication Year :
- 2003
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *SUPERALGEBRAS
*MATRICES (Mathematics)
*NONASSOCIATIVE algebras
*MATHEMATICS
Subjects
Details
- Language :
- English
- ISSN :
- 00029947
- Volume :
- 355
- Issue :
- 12
- Database :
- Academic Search Index
- Journal :
- Transactions of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 10685582
- Full Text :
- https://doi.org/10.1090/S0002-9947-03-03360-9