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Codimensions of commutators of length 4

Conjectures about the asymptotic behaviour of the codimensions of associative algebras were made by Amitsur and Regev in the 1980s. Amitsur’s conjecture was proved in 1999 by Giambruno and Zaicev [1]. In this article we prove that the identity [x1, x2, x3, x4] ≡ 0 satisfies Regev’s conjecture (this identity is a natural generalization of [x1, x2, x3] ≡ 0, which is a polynomial-identity basis of the Grassmann algebra [2]). For this, a finite-dimensional superalgebra is constructed, with [x1, x2, x3, x4] ≡ 0 being a basis of identities of its Grassmann envelope. The set Id(A) of polynomial identities of A is a T -ideal of the free associative algebra F 〈X〉 on a countable set X = {x1, x2, . . . }.