Gradings and graded identities for the upper triangular matrices over an infinite field

Gradings on matrices

Nilpotent varieties and metabelian varieties

We deal with varieties of nonassociative algebras having polynomial growth of codimensions. We describe some results obtained in recent years in the class of left nilpotent algebras of index two. Recently the authors established a correspondence between the growth rates for left nilpotent algebras of index two and the growth rates for commutative or anticommutative metabelian algebras that allows to transfer the results concerning varieties of left nilpotent algebras of index two to varieties of commutative or anticommutative metabelian algebras.

Capelli identities on algebras with involution or graded involution

We present recent results about Capelli polynomials with involution or graded involution and their asymptotics. In the associative case, the asymptotic equality between the codimensions of the T -ideal generated by the Capelli polynomial of rank k2 + 1 and the codimensions of the matrix algebra Mk(F) was proved. This result was extended to superalgebras. Recently, similar results have been determined by the authors in the case of algebras with involution and superalgebras with graded involution.

ON THE ASYMPTOTICS OF CAPELLI POLYNOMIALS

Abstract. We present old and new results about Capelli polynomials, Z2-graded Capelli polynomials and Capelli polynomials with involution and their asymptotics. Let Capm = Pσ2Sm (sgnσ)tσ(1)x1tσ(2) · · · tσ(m−1)xm−1tσ(m) be the m-th Capelli polynomial of rank m. In the ordinary case (see [33]) it was proved the asymptotic equality between the codimensions of the T -ideal generated by the Capelli polynomial Capk2+1 and the codimensions of the matrix algebra Mk(F ). In [9] this result was extended to superalgebras proving that the Z2-graded codimensions of the T2-ideal generated by the Z2-graded Capelli polynomials Cap0 M+1 and Cap1 L+1 for some fixed M, L, are asymptotically equal to the Z2-g…

Group gradings on upper triangular matrices

On the growth of varieties of algebras

Graded involutions on upper-triangular matrix algebras

Group algebras whith symmetric units satisfying a group identity

We study group algebras FG for which the symmetric units under the natural involution: g∗ = g−1 satisfy a group identity. For infinite fields F of characteristic ≠ 2, a classification of torsion groups G whose symmetric units) satisfy a group identity was given by Giambruno-Sehgal-Valenti. We extend this work to non torsion groups.

Multiplicities in the mixed trace cocharacter sequence of two 3x3 matrices