Polynomial growth of the codimensions: a characterization. (English) Zbl 1200.17003
Let \(A\) be a not necessarily associative algebra over a field of characteristic zero. For associative algebras satisfying a nontrivial polynomial identity, A. Regev [Isr. J. Math. 11, 131–152 (1972; Zbl 0249.16007)] proved that the sequence of codimensions is exponentially bounded, but this same conclusion does not hold for Lie algebras [I. B. Volichenko, Sib. Mat. Zh. 25, No.3 (145), 40–54 (1984; Zbl 0575.17006)]. It is well-known that for both associative and Lie algebras the sequence of codimensions cannot have intermediate growth (between polynomial and exponential). The case of polynomial growth has been considered in this paper. The authors characterize the \(T\)-ideal of identities of \(A\) in case the corresponding sequence of codimensions is polynomially bounded.
Reviewer: Hamid Usefi (Toronto)
MSC:
| 17A50 | Free nonassociative algebras |
| 16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |
| 16P90 | Growth rate, Gelfand-Kirillov dimension |
| 20C30 | Representations of finite symmetric groups |
References:
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