Search results for " Involution"

Varieties of algebras with pseudoinvolution and polynomial growth

2017

Let A be an associative algebra with pseudoinvolution (Formula presented.) over an algebraically closed field of characteristic zero and let (Formula presented.) be its sequence of (Formula presented.) -codimensions. We shall prove that such a sequence is polynomially bounded if and only if the variety generated by A does not contain five explicitly described algebras with pseudoinvolution. As a consequence, we shall classify the varieties of algebras with pseudoinvolution of almost polynomial growth, i.e. varieties of exponential growth such that any proper subvariety has polynomial growth and, along the way, we shall give also the classification of their subvarieties. Finally, we shall de…

On the ∗-cocharacter sequence of 3×3 matrices

2000

Abstract Let M 3 (F) be the algebra of 3×3 matrices with involution * over a field F of characteristic zero. We study the ∗ -polynomial identities of M 3 (F) , where ∗=t is the transpose involution, through the representation theory of the hyperoctahedral group B n . After decomposing the space of multilinear ∗ -polynomial identities of degree n under the B n -action, we determine which irreducible B n -modules appear with non-zero multiplicity. In symbols, we write the nth ∗ -cocharacter χ n (M 3 (F),*)=∑ r=0 n ∑ λ⊢r,h(λ)⩽6 μ⊢n−r,h(μ)⩽3 m λ,μ χ λ,μ , where λ and μ are partitions of r and n−r , respectively, χ λ,μ is the irreducible B n -character associated to the pair (λ,μ) and m λ,μ ⩾0 i…

Star-polynomial identities: computing the exponential growth of the codimensions

2017

Abstract Can one compute the exponential rate of growth of the ⁎-codimensions of a PI-algebra with involution ⁎ over a field of characteristic zero? It was shown in [2] that any such algebra A has the same ⁎-identities as the Grassmann envelope of a finite dimensional superalgebra with superinvolution B. Here, by exploiting this result we are able to provide an exact estimate of the exponential rate of growth e x p ⁎ ( A ) of any PI-algebra A with involution. It turns out that e x p ⁎ ( A ) is an integer and, in case the base field is algebraically closed, it coincides with the dimension of an admissible subalgebra of maximal dimension of B.

Polynomial codimension growth of algebras with involutions and superinvolutions

2017

Abstract Let A be an associative algebra over a field F of characteristic zero endowed with a graded involution or a superinvolution ⁎ and let c n ⁎ ( A ) be its sequence of ⁎-codimensions. In [4] , [12] it was proved that if A is finite dimensional such sequence is polynomially bounded if and only if A generates a variety not containing a finite number of ⁎-algebras: the group algebra of Z 2 and a 4-dimensional subalgebra of the 4 × 4 upper triangular matrices with suitable graded involutions or superinvolutions. In this paper we focus our attention on such algebras since they are the only finite dimensional ⁎-algebras, up to T 2 ⁎ -equivalence, generating varieties of almost polynomial gr…

Identities of *-superalgebras and almost polynomial growth

2015

We study the growth of the codimensions of a *-superalgebra over a field of characteristic zero. We classify the ideals of identities of finite dimensional algebras whose corresponding codimensions are of almost polynomial growth. It turns out that these are the ideals of identities of two algebras with distinct involutions and gradings. Along the way, we also classify the finite dimensional simple *-superalgebras over an algebraically closed field of characteristic zero.

2004

With advancing age, thymic efficiency shows progressive decline due to thymic involution allowing impaired cell-mediated immunity and the appearance of age-related diseases. The intrinsic cause of thymic involution is still undefined. Chronic inflammation and high glucocorticoids (GCs) may be involved. However, transgenic mice, with increased GC sensitivity and over expression of GC receptors, display delayed age-associated thymic involution. This fact suggests that other substances may affect thymic involution. Among them, both isoforms of metallothioneins (MTs) I+II and III are the major candidates because their increments leads to organ atrophy in constant stress and are induced by IL-6,…

Superalgebras with Involution or Superinvolution and Almost Polynomial Growth of the Codimensions

2018

Let A be a superalgebra with graded involution or superinvolution ∗ and let $c_{n}^{*}(A)$, n = 1,2,…, be its sequence of ∗-codimensions. In case A is finite dimensional, in Giambruno et al. (Algebr. Represent. Theory 19(3), 599–611 2016, Linear Multilinear Algebra 64(3), 484–501 2016) it was proved that such a sequence is polynomially bounded if and only if the variety generated by A does not contain the group algebra of $\mathbb {Z}_{2}$ and a 4-dimensional subalgebra of the 4 × 4 upper-triangular matrices with suitable graded involutions or superinvolutions. In this paper we study the general case of ∗-superalgebras satisfying a polynomial identity. As a consequence we classify the varie…

Codimensions of star-algebras and low exponential growth

2020

In this paper we prove that if A is any algebra with involution * satisfying a non-trivial polynomial identity, then its sequence of *-codimensions is eventually non-decreasing. Furthermore, by making use of the *-exponent we reconstruct the only two *-algebras, up to T*-equivalence, generating varieties of almost polynomial growth. As a third result we characterize the varieties of algebras with involution whose exponential growth is bounded by 2.

Classifying Algebras with Graded Involutions or Superinvolutions with Multiplicities of their Cocharacter Bounded by One

2020

Let A be superalgebra over a field of characteristic zero and let ∗ be either a graded involution or a superinvolution defined on A. In this paper we characterize the ∗-algebras whose ∗-cocharacter has multiplicities bounded by one, showing a set of ∗-polynomial identities satisfied by such algebras.

Group identities on symmetric units

2009

Abstract Let F be an infinite field of characteristic different from 2, G a group and ∗ an involution of G extended by linearity to an involution of the group algebra FG. Here we completely characterize the torsion groups G for which the ∗-symmetric units of FG satisfy a group identity. When ∗ is the classical involution induced from g → g − 1 , g ∈ G , this result was obtained in [A. Giambruno, S.K. Sehgal, A. Valenti, Symmetric units and group identities, Manuscripta Math. 96 (1998) 443–461].