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

On The *-polynomial Identities Of M1,1(e)

In this paper we consider the algebra M1,1(E) endowed with the involution * induced by the transposition superinvolution of the superalgebra M1,1(F) of 2×2-matrices over the field F. We study the *-polynomial identities for this algebra in the case of characteristic zero. We describe a finite set generating the ideal of its *-identities. We also consider Mn(E), the algebra of n×n matrices over the Grassmann algebra E. We prove that for a large class of involutions defined on it any *-polynomial identity is indeed a polynomial identity. A similar result holds for the verbally prime algebra Mk,l(E). © 2010 Elsevier B.V.2153262275Bahturin, Y., Giambruno, A., Group gradings on associative algebras with involution (2008) Canad. Math. Bull., 51, pp. 182-194Bahturin, Y., Shestakov, I., Zaicev, M., Gradings on simple Jordan and Lie algebras (2005) J. Algebra, 283, pp. 849-868Bahturin, Y., Tvalavadze, M., Tvalavadze, T., Group gradings on superinvolution simple superalgebras (2009) Linear Algebra Appl., 31, pp. 1054-1069Bahturin, Y., Zaicev, M., Involutions on graded matrix algebras (2007) J. Algebra, 315, pp. 527-540Berele, A., Generic verbally prime algebras and their GK-dimensions (1993) Commun. Algebra, 21 (5), pp. 1487-1504Colombo, J., Koshlukov, P., Identities with involution for the matrix algebra of order two over an infinite field of characteristic p (2005) Israel J. Math., 146, pp. 337-356Di Vincenzo, O.M., Koshlukov, P., La Scala, R., Involutions for upper triangular matrix algebra (2006) Adv. Appl. Math., 37, pp. 541-568Drensky, V., Giambruno, A., Cocharacters, codimensions and Hilbert series of the polynomial identities for 2×2 matrices with involution (1994) Canad. J. Math, 46, pp. 718-733Gómez-Ambrosi, C., Shestakov, I.P., On the Lie structure of the skew elements of a simple superalgebra with superinvolution (1998) J. Algebra, 208, pp. 43-71Kac, V.G., Lie superalgebras (1977) Adv. Math., 26, pp. 8-96Kemer, A.R., Varieties and Z2-graded algebras (1985) Math. USSR Izv., 25, pp. 359-374Kemer, A.R., Ideals of identities of associative algebras (1991) AMS Trans. Math. Monogr., 87Knus, M.A., Merkurjev, A., Rost, M., Tignol, J.-P., The book of involutions (1998) Amer. Math. Soc. Colloquium Publ., 44. , AMS, Providence, RILevchenko, D., Finite basis property of identities with involution of a second-order matrix algebra (1982) Serdica, 8 (1), pp. 42-56. , (in Russian)Marcoux, L., Sourour, A., Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras (1999) Linear Algebra Appl., 288, pp. 89-104Racine, M.L., Primitive superalgebras with superinvolution (1998) J. Algebra, 206, pp. 588-614Rowen, L.H., (1980) Polynomial Identities in Ring Theory, , Academic Press, New Yor

*-Polynomial identities of a nonsymmetric *-minimal algebra

Let F be an infinite field of characteristic different from 2. We study the ∗-polynomial identities of the ∗-minimal algebra R = UT∗(F ⊕ F, F). We describe the generators of the *-polynomial identities of R and a linear basis of the relatively free algebra of R. When char.F = 0, these results allow us to provide a complete list of polynomials generating irreducible GL × GL-modules decomposing the proper part of the relatively free algebra of R. Finally, the ∗-codimension sequence of R is explicitly computed

Some results on ∗\ast-minimal algebras with involution

Let $(A,\ast)$ be a $\ast$-PI algebra with involution over a field of characteristic zero and let $c_m(A,\ast)$ denote its $m$-th $\ast$-codimension. Giambruno and Zaicev, in [\textit{Involution codimension of finite dimensional algebras and exponential growth}, J. Algebra \textbf{222} (1999), 471--484], proved that, if $A$ is finite dimensional, there exists the $\lim_{m \to +\infty} \sqrt[m]{ c_m(A,\ast)}$, and it is an integer, which is called the \emph{$\ast$-exponent} of $A$. As a consequence of the presence of this invariant, it is natural to introduce the concept of \emph{$\ast$-minimal algebra}. Our goal in this paper is to move some steps towards a complete classification of $\ast$-minimal algebras

Robinson-Schensted-Knuth correspondence and Weak Polynomial Identities of M1,1(E)M_{1,1}(E)

In this paper, it is proved that the ideal I_w of the weak polynomial identities of the superalgebra M_1,1(E) is generated by the proper polynomials [x_1, x_2, x_3] and [x_2, x_1 ][x_3, x_1 ][x_4, x_1]. This is proved for any infinite field F of characteristic different from 2. Precisely, if B is the subalgebra of the proper polynomials of F, we determine a basis and the dimension of any multihomogeneous component of the quotient algebra B/(B ∩ I_w ). We also compute the Hilbert series of this algebra. One of the main tools of this paper is a variant we found of the Robinson–Schensted–Knuth correspondence defined for single semistandard tableaux of double shape

On the *-minimality of algebras with involution

In [Di Vincenzo, O.M., La Scala, R.: \emph{Minimal algebras with respect to their $\ast$-exponent}, J. Algebra 317 (2007), 642-657] given a $n$-tuple $(A_1,\ldots ,A_n)$ of finite dimensional $\ast$-simple algebras over a field of characteristic zero, a block-triangular matrix algebra with involution, denoted by $R:=UT_{\ast}(A_1, \ldots , A_n)$, was introduced and it was proved that any finite dimensional algebra with involution which is minimal with respect to its $\ast$-exponent is $\ast$-PI equivalent to $R$ for a suitable choice of the algebras $A_i$. Motivated by a conjecture stated in the same paper, here we show that $R$ is $\ast$-minimal when either it is $\ast$\emph{-symmetric} or $n=2$