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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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