## Algebras with intermediate growth of the codimensions.(English)Zbl 1111.16022

Let $$F$$ be a field of characteristic zero and let $$A$$ be a (not necessarily associative) $$F$$-algebra. By analogy with associative PI-algebras, one introduces the sequence $$c_n(A)$$ of codimensions of the multilinear polynomial identities of $$A$$, the $$S_n$$-cocharacter $$\chi_n(A)=\sum_{\lambda\vdash n}m_\lambda\chi_\lambda$$, and the colength $$l_n(A)=\sum_{\lambda\vdash n}m_\lambda$$. A valuable information on the combinatorial and structural properties of $$A$$ can be obtained from the asymptotic behaviour of $$c_n(A)$$, $$\chi_n(A)$$, $$l_n(A)$$. The authors show that for finite dimensional nonassociative algebras the colength sequence of $$A$$ is polynomially bounded and the codimension sequence cannot have intermediate growth. Then the authors construct a series of examples of nonassociative algebras with intermediate growth of the codimensions. They show that for any real number $$0<\beta<1$$, there exists an algebra $$A$$ whose sequence of codimensions grows like $$n^{n^\beta}$$.

### MSC:

 16R10 $$T$$-ideals, identities, varieties of associative rings and algebras 17A30 Nonassociative algebras satisfying other identities 17B01 Identities, free Lie (super)algebras 16P90 Growth rate, Gelfand-Kirillov dimension
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