Search results for "Polynomial identitie"
showing 10 items of 19 documents
Polynomial identities on superalgebras and exponential growth
2003
Abstract Let A be a finitely generated superalgebra over a field F of characteristic 0. To the graded polynomial identities of A one associates a numerical sequence {cnsup(A)}n⩾1 called the sequence of graded codimensions of A. In case A satisfies an ordinary polynomial identity, such sequence is exponentially bounded and we capture its exponential growth by proving that for any such algebra lim n→∞ c n sup (A) n exists and is a non-negative integer; we denote such integer by supexp(A) and we give an effective way for computing it. As an application, we construct eight superalgebras Ai, i=1,…,8, characterizing the identities of any finitely generated superalgebra A with supexp(A)>2 in the f…
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…
Asymptotics for Graded Capelli Polynomials
2014
The finite dimensional simple superalgebras play an important role in the theory of PI-algebras in characteristic zero. The main goal of this paper is to characterize the T 2-ideal of graded identities of any such algebra by considering the growth of the corresponding supervariety. We consider the T 2-ideal Γ M+1,L+1 generated by the graded Capelli polynomials C a p M+1[Y,X] and C a p L+1[Z,X] alternanting on M+1 even variables and L+1 odd variables, respectively. We prove that the graded codimensions of a simple finite dimensional superalgebra are asymptotically equal to the graded codimensions of the T 2-ideal Γ M+1,L+1, for some fixed natural numbers M and L. In particular csupn(Γk2+l2+1…
Codimension and colength sequences of algebras and growth phenomena
2015
We consider non necessarily associative algebras over a field of characteristic zero and their polynomial identities. Here we describe some of the results obtained in recent years on the sequence of codimensions and the sequence of colengths of an algebra.
Polynomial growth and star-varieties
2016
Abstract Let V be a variety of associative algebras with involution over a field F of characteristic zero and let c n ⁎ ( V ) , n = 1 , 2 , … , be its ⁎-codimension sequence. Such a sequence is polynomially bounded if and only if V does not contain the commutative algebra F ⊕ F , endowed with the exchange involution, and M, a suitable 4-dimensional subalgebra of the algebra of 4 × 4 upper triangular matrices. Such algebras generate the only varieties of ⁎-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all subvarieties of the ⁎-varieties of almost polynomial growth by gi…
Trace Identities on Diagonal Matrix Algebras
2020
Let Dn be the algebra of n × n diagonal matrices. On such an algebra it is possible to define very many trace functions. The purpose of this paper is to present several results concerning trace identities satisfied by this kind of algebras.
Polynomial identities for the Jordan algebra of upper triangular matrices of order 2
2012
Abstract The associative algebras U T n ( K ) of the upper triangular matrices of order n play an important role in PI theory. Recently it was suggested that the Jordan algebra U J 2 ( K ) obtained by U T 2 ( K ) has an extremal behaviour with respect to its codimension growth. In this paper we study the polynomial identities of U J 2 ( K ) . We describe a basis of the identities of U J 2 ( K ) when the field K is infinite and of characteristic different from 2 and from 3. Moreover we give a description of all possible gradings on U J 2 ( K ) by the cyclic group Z 2 of order 2, and in each of the three gradings we find bases of the corresponding graded identities. Note that in the graded ca…
Ordinary and graded cocharacter of the Jordan algebra of 2x2 upper triangular matrices
2014
Abstract Let F be a field of characteristic zero and U J 2 ( F ) be the Jordan algebra of 2 × 2 upper triangular matrices over F . In this paper we give a complete description of the space of multilinear graded and ordinary identities in the language of Young diagrams through the representation theory of a Young subgroup of S n . For every Z 2 -grading of U J 2 ( F ) we compute the multiplicities in the graded cocharacter sequence and furthermore we compute the ordinary cocharacter.
Proper identities, Lie identities and exponential codimension growth
2008
Abstract The exponent exp ( A ) of a PI-algebra A in characteristic zero is an integer and measures the exponential rate of growth of the sequence of codimensions of A [A. Giambruno, M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998) 145–155; A. Giambruno, M. Zaicev, Exponential codimension growth of P.I. algebras: An exact estimate, Adv. Math. 142 (1999) 221–243]. In this paper we study the exponential rate of growth of the sequences of proper codimensions and Lie codimensions of an associative PI-algebra. We prove that the corresponding proper exponent exists for all PI-algebras, except for some algebras of exponent two strictly related to t…
Trace identities and almost polynomial growth
2021
In this paper we study algebras with trace and their trace polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: $D_2$, the algebra of $2\times 2$ diagonal matrices and $C_2$, the algebra of $2 \times 2$ matrices generated by $e_{11}+e_{22}$ and $e_{12}$. We describe all possible traces on these algebras and we study the corresponding trace codimensions. Moreover we characterize the varieties with trace of polynomial growth generated by a finite dimensional algebra. As a consequence, we see that the growth of a variety with trace is either polynomial or exponential.