ALGEBRAS SATISFYING THE POLYNOMIAL IDENTITY [x₁,x₂][x₃,x₄,x₅]=0

Abstract

Let [Formula: see text] be a field of characteristic zero, and [Formula: see text] the variety of associative unitary algebras defined by the polynomial identity [x1,x2][x3,x4,x5]=0. This variety is one of the several minimal varieties of exponent 3 (and all proper subvarieties are of exponents 1 and 2). We describe asymptotically its proper subvarieties. More precisely, we define certain algebras ℛ2k for any k∈ℕ and show that if [Formula: see text] is a proper subvariety of [Formula: see text], then the T-ideal of its polynomial identities is asymptotically equivalent to the T-ideal of the identities of one of the algebras [Formula: see text], E, ℛ2k or ℛ2k⊕E, for a suitable k∈ℕ. We give also another description relating the T-ideals of the proper subvarieties of [Formula: see text] with the polynomial identities of upper triangular matrices of a suitable size.

Minimal Varieties and Identities of Relatively Free Algebras

“…The reader can find a version of the result in English in [1]. We observe that in [1], Di Vincenzo, Drensky, and Nardozza describe asymptotically the proper subvarieties of 5 .…”

“…We remark that for each p ≥ 2, the polynomial identities of R 2p coincide with the polynomial identities of F 2p−2 A . In particular, since R 2 is PI-equivalent to UT 2 K , the 2 × 2 upper triangular matrix algebra, we can restate the result of [1] as Corollary 16. Any subvariety of 5 is asymptotically equivalent to the variety generated by one among the following algebras: UT 2 K , E, F m A , or F m A ⊕ E, for a suitable m.…”

“…The importance of such polynomials lies in the fact that the identities of an algebra with unit follow from its proper ones (see [2,Proposition 4.3.3]). We denote by P n the vector space of all multilinear polynomials of degree n in K x 1 x n and by n its subspace of the proper multilinear polynomials. If is a variety of algebras, denote P n = P n P n ∩ T and n = n n ∩ T It is well known that P n and n are modules over the symmetric group S n .…”

“…In the description of the subvarieties of 5 in [1], the authors show that any subvariety of 5 is asymptotically equivalent to the variety generated by one among the following algebras: K, E, R p or R p ⊕ E, for a suitable p. Here R p stands for the algebra R p = 11 12 0 22 11 ∈ E 0 12 ∈ p E 0 + p E 1 22 ∈ p E 0 + t p E 1 and p = K t t p . We remark that for each p ≥ 2, the polynomial identities of R 2p coincide with the polynomial identities of F 2p−2 A .…”

“…We denote by K X the free associative unitary algebra freely generated by X. We say that an associative algebra A is an algebra with polynomial identity (PI-algebra, for short) if there exists a nonzero polynomial f = f x 1 x n ∈ K X such that f a 1 a n = 0, for any elements a 1 a n ∈ A. In this case, we say that f is a polynomial identity of A, or simply that A satisfies f .…”