Search results for "Polynomial identity"
showing 10 items of 57 documents
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…
Computing the ℤ2-Cocharacter of 3 × 3 Matrices of Odd Degree
2013
Let F be a field of characteristic 0 and A = M 2, 1(F) the algebra of 3 × 3 matrices over F endowed with the only non trivial ℤ2-grading. Aver'yanov in [1] determined a set of generators for the T 2-ideal of graded identities of A. Here we study the identities in variables of homogeneous degree 1 via the representation theory of the symmetric group, and we determine the decomposition of the corresponding character into irreducibles.
Group graded algebras and almost polynomial growth
2011
Let F be a field of characteristic 0, G a finite abelian group and A a G-graded algebra. We prove that A generates a variety of G-graded algebras of almost polynomial growth if and only if A has the same graded identities as one of the following algebras: (1) FCp, the group algebra of a cyclic group of order p, where p is a prime number and p||G|; (2) UT2G(F), the algebra of 2×2 upper triangular matrices over F endowed with an elementary G-grading; (3) E, the infinite dimensional Grassmann algebra with trivial G-grading; (4) in case 2||G|, EZ2, the Grassmann algebra with canonical Z2-grading.
Computing with Rational Symmetric Functions and Applications to Invariant Theory and PI-algebras
2012
The research of the first named author was partially supported by INdAM. The research of the second, third, and fourth named authors was partially supported by Grant for Bilateral Scientific Cooperation between Bulgaria and Ukraine. The research of the fifth named author was partially supported by NSF Grant DMS-1016086.
Some Numerical Invariants of Multilinear Identities
2017
We consider non-necessarily associative algebras over a field of characteristic zero and their polynomial identities. Here we describe most of the results obtained in recent years on two numerical sequences that can be attached to the multilinear identities satisfied by an algebra: the sequence of codimensions and the sequence of colengths.
Polynomial Identities of Algebras of Small Dimension
2009
It is well known that given an associative algebra or a Lie algebra A, its codimension sequence c n (A) is either polynomially bounded or grows at least as fast as 2 n . In [2] we proved that for a finite dimensional (in general nonassociative) algebra A, dim A = d, the sequence c n (A) is also polynomially bounded or c n (A) ≥ a n asymptotically, for some real number a > 1 which might be less than 2. Nevertheless, for d = 2, we may take a = 2. Here we prove that for d = 3 the same conclusion holds. We also construct a five-dimensional algebra A with c n (A) < 2 n .
Understanding star-fundamental algebras
2021
Star-fundamental algebras are special finite dimensional algebras with involution ∗ * over an algebraically closed field of characteristic zero defined in terms of multialternating ∗ * -polynomials. We prove that the upper-block matrix algebras with involution introduced in Di Vincenzo and La Scala [J. Algebra 317 (2007), pp. 642–657] are star-fundamental. Moreover, any finite dimensional algebra with involution contains a subalgebra mapping homomorphically onto one of such algebras. We also give a characterization of star-fundamental algebras through the representation theory of the symmetric group.
Graded polynomial identities and codimensions: Computing the exponential growth
2010
Abstract Let G be a finite abelian group and A a G-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied by A. We study the asymptotic behavior of c n G ( A ) , n = 1 , 2 , … , the sequence of graded codimensions of A and we prove that if A satisfies an ordinary polynomial identity, lim n → ∞ c n G ( A ) n exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple G × Z 2 -graded algebra related to A.
Polynomial identities with involution, superinvolutions and the Grassmann envelope
2017
Let A be an algebra with involution ∗ over a field of characteristic zero. We prove that in case A satisfies a non-trivial ∗-identity, then A has the same ∗-identities as the Grassmann envelope of a finite dimensional superalgebra with superinvolution. As a consequence we give a positive answer to the Specht problem for algebras with involution, i.e., any T-ideal of identities of an algebra with involution is finitely generated as a T-ideal.
Asymptotics for the Amitsur's Capelli - Type Polynomials and Verbally Prime PI-Algebras
2006
We consider associativePI-algebras over a field of characteristic zero. The main goal of the paper is to prove that the codimensions of a verbally prime algebra [11] are asymptotically equal to the codimensions of theT-ideal generated by some Amitsur's Capelli-type polynomialsEM,L* [1]. We recall that two sequencesan,bnare asymptotically equal, and we writean≃bn,if and only if limn→∞(an/bn)=1.In this paper we prove that\(c_n \left( {M_k \left( G \right)} \right) \simeq c_n \left( {E_{k^2 ,k^2 }^ * } \right) and c_n \left( {M_{k,l} \left( G \right)} \right) \simeq c_n \left( {E_{k^2 + l^2 ,2kl}^ * } \right) \)% MathType!End!2!1!, whereG is the Grassmann algebra. These results extend to all v…