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\(\mathbb Z_2\)-graded codimensions of unital algebras. (English) Zbl 1442.17004
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 multilinear polynomial identities of \(A\) and other related numerical invariants of the polynomial identities.
A. Giambruno et al. [Adv. Appl. Math. 37, No. 3, 360–377 (2006; Zbl 1111.16022)] constructed a series of algebras \(A(K)\), where \(K\) is a sequence of positive integers such that for any real number \(0 <\beta < 1\) the sequence of codimensions \(c_n(A(K))\) grows like \(n^{n^{\beta }}\). Later in [Adv. Math., 217, No. 3, 1027–1052 (2008; Zbl 1133.17001)] they combined this construction with combinatorics of infinite binary words and for any integer \(m\geq 2\) and any periodic or Sturmian word \(w\) with slope \(\alpha\in(0,1)\) they constructed an algebra \(A(m,w)\) such that the exponent \(\exp(A(m,w))=\lim_{n\to\infty}\sqrt[n]{c_n(A(m,w))}\) exists and is equal to \(\Phi(\beta)\), where \(\beta=\frac{1}{m+\alpha}\) and \(\Phi(x)=\frac{1}{x^x(1-x)^{1-x}}\). Later the authors of the paper under review showed in [Commun. Algebra 43, No. 9, 3823–3839 (2015; Zbl 1335.16014)] that for this class of algebras adjoining external unit leads to increasing of PI-exponent precisely by 1.
Now they consider a natural \({\mathbb Z}_2\)-grading of the algebras \(A(m,w)\) and show that the \({\mathbb Z}_2\)-graded codimensions behave in the same way: \(\exp^{\text{gr}}(A(m,w))=\exp(A(m,w))\) and adjoining a unit increases the graded exponent by 1.
MSC:
17A30 Nonassociative algebras satisfying other identities
17A50 Free nonassociative algebras
16P90 Growth rate, Gelfand-Kirillov dimension
17B01 Identities, free Lie (super)algebras
17B70 Graded Lie (super)algebras
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
05A05 Permutations, words, matrices
68R15 Combinatorics on words
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References:
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