Search results for "Involution"

Varieties of Algebras with Superinvolution of Almost Polynomial Growth

2015

Let A be an associative algebra with superinvolution ∗ over a field of characteristic zero and let $c_{n}^{\ast }(A)$ be its sequence of corresponding ∗-codimensions. In case A is finite dimensional, we prove that such sequence is polynomially bounded if and only if the variety generated by A does not contain three explicitly described algebras with superinvolution. As a consequence we find out that no intermediate growth of the ∗-codimensions between polynomial and exponential is allowed.

Asymptotics for Capelli polynomials with involution

2021

Let F be the free associative algebra with involution ∗ over a field F of characteristic zero. We study the asymptotic behavior of the sequence of ∗- codimensions of the T-∗-ideal Γ∗ M+1,L+1 of F generated by the ∗-Capelli polynomials Cap∗ M+1[Y, X] and Cap∗ L+1[Z, X] alternanting on M + 1 symmetric variables and L + 1 skew variables, respectively. It is well known that, if F is an algebraic closed field of characteristic zero, every finite dimensional ∗-simple algebra is isomorphic to one of the following algebras: · (Mk(F ), t) the algebra of k × k matrices with the transpose involution; · (M2m(F ), s) the algebra of 2m × 2m matrices with the symplectic involution; · (Mh(F ) ⊕ Mh(F )op, e…

ON THE ASYMPTOTICS OF CAPELLI POLYNOMIALS

2021

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…

*-Graded Capelli polynomials and their asymptotics

2022

Let [Formula: see text] be the free *-superalgebra over a field [Formula: see text] of characteristic zero and let [Formula: see text] be the [Formula: see text]-ideal generated by the set of the *-graded Capelli polynomials [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] alternating on [Formula: see text] symmetric variables of homogeneous degree zero, on [Formula: see text] skew variables of homogeneous degree zero, on [Formula: see text] symmetric variables of homogeneous degree one and on [Formula: see text] skew variables of homogeneous degree one, respectively. We study the asymptotic behavior of the sequence of *-graded codimensions of [Formula: se…

Graded involutions on upper-triangular matrix algebras

2009

Capelli identities on algebras with involution or graded involution

2022

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 for $ast$-Capelli identities

Let Fbe the free associative algebra with involution ∗ over a field of characteristic zero. If L and M are two natural numbers let Γ∗_M+1,L+1 denote theT∗-idealofFgenerated by the∗-capellipolynomialsCap+M+1,Cap−L+1 alternanting on M+1 symmetric variables and L+1skew variables,respectively.It is well known that, if F is an algebraic closed field, every finite dimensional ∗-simple algebra is isomorphic to one of the following algebras (see [4], [2]):· (Mk(F),t) with the transpose involution; · (M2m(F),s) with the symplectic involution; · (Mk(F)⊕Mk(F)op,∗) with the exchange involution. The aim of this talk is to show a relation among the asymptotics of the∗-codimensions of the finite dimensional ∗…

Nitration of cathepsin D enhances its proteolytic activity during mammary gland remodelling after lactation

2009

Proteomic studies in the mammary gland of control lactating and weaned rats have shown that there is an increased pattern of nitrated proteins during weaning when compared with controls. Here we report the novel finding that cathepsin D is nitrated during weaning. The expression and protein levels of this enzyme are increased after 8 h of litter removal and this up-regulation declines 5 days after weaning. However, there is a marked delay in cathepsin D activity since it does not increase until 2 days post-weaning and remains high thereafter. In order to find out whether nitration of cathepsin D regulates its activity, iNOS (inducible nitric oxide synthase)−/− mice were used. The expression…

Induction of mitochondrial xanthine oxidase activity during apoptosis in the rat mammary gland

2006

Oxidative stress is an important signal for apoptosis to start. So far the mitochondrial respiratory chain has been considered as the major, if not the only, cause of such stress. Here we report that this is not the case. Xanthine oxidase, a O2(-) and H2O2 generating enzyme which is important in causing significant oxidative stress in the cytosol, is also present in the mitochondrial fraction of rat mammary gland. After weaning, during the involution of the mammary gland, massive apoptosis occurs. Mitochondrial xanthine oxidase activity increases and high mitochondrial H2O2 production takes place. Inhibition of xanthine oxidase activity by allopurinol, a specific inhibitor of xanthine oxida…

Some results on ∗-minimal algebras with involution

2009

Let $(A, *)$ be an associative algebra with involution over a field $F$ of characteristic zero, $T_*(A)$ the ideal of $*$-polynomial identities of $A$ and $c_n(A, *),$ $n=1, 2, \ldots$, the corresponding sequence of $*$-codimensions. Recall that $c_n(A, *)$ is the dimension of the space of multilinear polynomials in $n$ variables in the corresponding relatively free algebra with involution of countable rank. \par When $A$ is a finite dimensional algebra, Giambruno and Zaicev [J. Algebra 222 (1999), no. 2, 471–484; MR1734235 (2000i:16046)] proved that the limit $$\exp(A, *)=\lim_{n\to \infty}\sqrt[n]{c_n(A, *)}$$ exists and is an integer called the $*$-exponent of $A.$ \par Among finite dime…