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On the growth of the identities of algebras. (English) Zbl 1164.16010
Summary: In the first part we show that for general algebras whose sequence of codimensions is exponentially bounded, any real number \(>1\) can actually appear. In fact we construct, for any real number \(\alpha>1\), an algebra \(A_\alpha\), whose sequence of codimensions grows exponentially and \(\lim_{n\to\infty}(c_n(A_\alpha))^{1/n}=\alpha\).
The second part of the paper concerns intermediate growth. Here we show that in the general case of non-associative algebras, there exist examples of algebras with intermediate growth of the codimensions. To this end for any real number \(0<\beta<1\) we construct an algebra whose sequence of codimensions grows as \(n^{n^\beta}\).
Finally two more results are presented: for any finite dimensional algebra the sequence of codimensions cannot have intermediate growth, and for any two-dimensional algebra either the exponent is equal to 2 or the growth of the codimensions is polynomially bounded by \(n+1\).
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16R30 Trace rings and invariant theory (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)
17A30 Nonassociative algebras satisfying other identities