On Shirshov bases of graded algebras

We prove that if the neutral component in a finitely-generated associative algebra graded by a finite group has a Shirshov base, then so does the whole algebra.Comment: 4 pages; v2: minor corrections in English; to appear in Israel J. Mat

Algebraic Geometry over Free Metabelian Lie Algebra I: U-Algebras and Universal Classes

This paper is the first in a series of three, the aim of which is to lay the foundations of algebraic geometry over the free metabelian Lie algebra $F$. In the current paper we introduce the notion of a metabelian Lie $U$-algebra and establish connections between metabelian Lie $U$-algebras and special matrix Lie algebras. We define the $\Delta$-localisation of a metabelian Lie $U$-algebra $A$ and the direct module extension of the Fitting's radical of $A$ and show that these algebras lie in the universal closure of $A$.Comment: 34 page

On a Regev-Seeman conjecture about Z_2-graded tensor products

In the Theory of Polynomial Identities of algebras, superalgebras play a key role, as emphasized by the celebrated Kemer’s results on the structure of T -ideals of the free associative algebra. Kemer succeeded in classifying the T –prime algebras over a field of characteristic zero, and all of them possess a natural superalgebra structure. In a celebrated work Regev proved that the tensor product of PI-algebras is again a PI-algebra, and the so-called Kemer’s Tensor Product Theorem shows that the tensor product of T -prime algebras is again PI-equivalent to a T -prime algebra, explicitly described. When dealing with superalgebras, however, it is possible to define an alternative tensor product, sometimes called super, or graded, or signed tensor product. In a recent paper Regev and Seeman studied graded tensor products, and they proved that the graded tensor product of PI-algebras is again PI, as for the ordinary case. Then natural questions arise: is the graded tensor product of T -prime algebras again T -prime? If so, do the graded and ordinary tensor products of T -prime algebras give the “same results” up to PI-equivalence? Among their results, Regev and Seeman noticed cases for which a graded version of Kemer’s Tensor Product Theorem does hold. More precisely, the re- sulting algebra is still a T -prime algebra, possibly “different” from the “natural” one. Then they conjectured this should be true in general. The present paper positively solves the conjecture. More precisely, we can prove that, in zero characteristic, the graded tensor product of T -prime algebras “is” again T -prime, and we describe the resulting algebra up to PI-equivalence