Minimal varieties of algebras of exponential growth. (English) Zbl 1035.16013
Recently, the authors proved the remarkable result that the exponent of a variety \(V\) of associative algebras over a field of characteristic zero is always an integer [A. Giambruno, M. Zaicev, Adv. Math. 142, No. 2, 221-243 (1999; Zbl 0920.16013)]. Namely, let \(c_n(V)\), \(n=1,2,\dots\), be the codimension sequence of a variety \(V\). Then the limit \(\lim_{n\to\infty}\root n\of{c_n(V)}\) exists and is an integer.
Now, the authors classify all minimal varieties of a given exponent and of finite basic rank. As a consequence, they describe the corresponding T-ideals of the free algebra and compute the asymptotics of the relative codimension sequences. They prove that the number of these minimal varieties is finite for any given exponent. Also, some relations between the exponent of a variety and the Gelfand-Kirillov dimension of the respective relatively free algebras are given.
Now, the authors classify all minimal varieties of a given exponent and of finite basic rank. As a consequence, they describe the corresponding T-ideals of the free algebra and compute the asymptotics of the relative codimension sequences. They prove that the number of these minimal varieties is finite for any given exponent. Also, some relations between the exponent of a variety and the Gelfand-Kirillov dimension of the respective relatively free algebras are given.
Reviewer: Victor Petrogradsky (Ulyanovsk)
MSC:
| 16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |
| 16P90 | Growth rate, Gelfand-Kirillov dimension |
Keywords:
varieties of associative algebras; codimension growth; Gelfand-Kirillov dimension; T-ideals; exponents; codimension sequences; relatively free algebrasCitations:
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