The centre of generic algebras of small PI algebras

Verbally prime algebras are important in PI theory. They are well known over a field $K$ of characteristic zero: 0 and $K$ (the trivial ones), $M_n(K)$, $M_n(E)$, $M_{ab}(E)$. Here $K$ is the free associative algebra with free generators $T$, $E$ is the infinite dimensional Grassmann algebra over $K$, $M_n(K)$ and $M_n(E)$ are the $n\times n$ matrices over $K$ and over $E$, respectively. Moreover $M_{ab}(E)$ are certain subalgebras of $M_{a+b}(E)$, defined below. The generic algebras of these algebras have been studied extensively. Procesi gave a very tight description of the generic algebra of $M_n(K)$. The situation is rather unclear for the remaining nontrivial verbally prime algebras. In this paper we study the centre of the generic algebra of $M_{11}(E)$ in two generators. We prove that this centre is a direct sum of the field and a nilpotent ideal (of the generic algebra). We describe the centre of this algebra. As a corollary we obtain that this centre contains nonscalar elements thus we answer a question posed by Berele.Comment: 15 pages. Misprints corrected. Provisionally accepted to publication in Journal of Algebr

JJ-trace identities and invariant theory

We generalize the notion of trace identity to $J$-trace. Our main result is that all $J$-traces of $M_{n,n}$ are consequence of those of degree $\frac12n(n + 3)$. This also gives an indirect description of the queer trace identities of $M_n(E)$

Computing Super Matrix Invariants

In [Trace identities and $\bf {Z}/2\bf {Z}$-graded invariants, {\it Trans. Amer. Math. Soc. \bf309} (1988), 581--589] we generalized the first and second fundamental theorems of invariant theory from the general linear group to the general linear Lie superalgebra. In the current paper we generalize the computations of the the numerical invariants (multiplicities and Poincar\'e series) to the superalgebra case. The results involve either inner products of symmetric functions in two sets of variables, or complex integrals. we generalized the first and second fundamental theorems of invariant theory from the general linear group to the general linear Lie superalgebra. In the current paper we generalize the computations of the the numerical invariants (multiplicities and Poincar\'e series) to the superalgebra case. The results involve either inner products of symmetric functions in two sets of variables, or complex integrals

Some Questions about products of verbally prime T-ideals

2010 Mathematics Subject Classification: 16R10.In [1] we studied identities of finite dimensional incidence algebras and showed how they were gotten by products and intersections of identities of matrices and we left open the question of when two incidence algebras satisfy the same identities, a problem which is still open. In the current paper we re-visit this problem: We describe it, give some partial results and some related problems based on the work of Kemer.* Support by DePaul University Faculty Research Council gratefully acknowledged

Graded polynomial identities, group actions, and exponential growth of Lie algebras

Consider a finite dimensional Lie algebra L with an action of a finite group G over a field of characteristic 0. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial G-identities of L. As a consequence, we prove the analog of Amitsur's conjecture for graded codimensions of any finite dimensional Lie algebra graded by a finite Abelian group.Comment: 26 pages; minor misprints were correcte

On the averages of characteristic polynomials from classical groups

We provide an elementary and self-contained derivation of formulae for products and ratios of characteristic polynomials from classical groups using classical results due to Weyl and Littlewood

Comparison of antioxidant activity in various spirulina containing products and factors affecting it

© 2023. The Author(s).Spirulina is a popular food supplement known for its high antioxidant activity. Several studies have shown that antioxidant activity fluctuates depending on the combination of ingredients in the food. Fresh spirulina is a growing market trend; however, pure spirulina short shelf life is a strong limitation. This study aims to investigate antioxidant activity of various novel commercial fresh spirulina-containing products and the factors affecting it. Antioxidant activity and total phenolic content of each ingredient and binary combinations of spirulina and apple juices, Japanese quince syrup, or cranberry syrup were measured. Synergic, antagonistic, and additive interactions between samples were determined and expressed using the synergy coefficient. FRAP assay showed apparent synergism of spirulina and all the studied ingredients whereas ABTS and Folin–Ciocalteu methods revealed an antagonistic interaction between spirulina and apple juice. Despite the antagonistic interactions, all the products demonstrated at least the same antioxidant activity as pure fresh spirulina and had longer shelf life than, pointing to their commercial potential.publishersversionPeer reviewe

Combinatorial R matrices for a family of crystals : B^{(1)}_n, D^{(1)}_n, A^{(2)}_{2n} and D^{(2)}_{n+1} cases

For coherent families of crystals of affine Lie algebras of type B^{(1)}_n, D^{(1)}_n, A^{(2)}_{2n} and D^{(2)}_{n+1} we describe the combinatorial R matrix using column insertion algorithms for B,C,D Young tableaux.Comment: 39 pages, LaTeX. This is a continuation of the authors' work appeared in "Physical Combinatorics", ed. M.Kashiwara and T.Miwa, Birkha"user, Boston, 200