Varieties of bicommutative algebras

Drensky Vesselin

Serdica Mathematical Journal, Tome 45 (2019) no. 2, pp. 167-188.

Résumé

Bicommutative algebras are nonassociative algebras satisfying the polynomial identities of right- and left-commutativity (x1x2)x3 = (x1x3)x2 and x1(x2x3) = x2(x1x3). Let B be the variety of all bicommutative algebras over a field K of characteristic 0 and let F (B) be the free algebra of countable rank in B. We prove that if D is a subvariety of B satisfying a polynomial identity f = 0 of degree k, where 0 ≠ f ∈ F(B), then the codimension sequence c¬n(D), n = 1, 2, . . ., is bounded by a polynomial in n of degree k − 1. Since cn(B) = 2n − 2 for n ≥ 2, and exp(B) = 2, this gives that exp(D) ≤ 1, i.e., B is minimal with respect to the codimension growth. When the field K is algebraically closed there are only three pairwise nonisomorphic two-dimensional bicommutative algebras A which arenonassociative. They are one-generated and with the property dim A2 = 1. We present bases of their polynomial identities and show that one of these algebras generates the whole variety B.

Détail

Keywords: Free bicommutative algebras, varieties of bicommutative algebras, codimension sequence, codimension growth, two-dimensional algebras, 17A30, 17A50, 20C30.