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

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.

Codimension growth and minimal superalgebras

A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope G(A) of a finite dimensional superalgebra A. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field. The importance of such algebras is readily proved: A is a minimal superalgebra if and only if the ideal of identities of G(A) is a product…

Symmetric identities in graded algebras

Let P k be the symmetric polynomial of degree k i.e., the full linearization of the polynomial x k . Let G be a cancellation semigroup with 1 and R a G-graded ring with finite support of order n. We prove that if R 1 satisfies $ P_k \equiv 0 $ then R satisfies $ P_{kn} \equiv 0 $ .

Finite-dimensional non-associative algebras and codimension growth

AbstractLet A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded.Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One…

Polynomial identities on superalgebras and exponential growth

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…

Group-graded algebras with polynomial identity

LetG be a finite group and letR=Σg∈GRg be any associative algebra over a field such that the subspacesRg satisfyRgRh⊆Rgh. We prove that ifR1 satisfies a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the order ofG. This result implies the following: ifH is a finite-dimensional semisimple commutative Hopfalgebra andR is anyH-module algebra withRH satisfying a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the dimension ofH.

Identities of sums of commutative subalgebras

SiaR un'algebra associativa tale cheR=A+B conA, B sottoalgebre commutative. Si dimostra cheR soddisfa l'identita polinomiale [[x,y],[z,t]]≡0 e che, seV e la varieta determinata da questa identita,V e la piu piccola varieta contenente tutte le algebre somma di sottoalgebre commutative. Si determina inoltre la struttura delle algebre libere diV.

POLYNOMIAL IDENTITIES ON SUPERALGEBRAS AND ALMOST POLYNOMIAL GROWTH

Let A be a superalgebra over a field of characteristic zero. In this paper we investigate the graded polynomial identities of A through the asymptotic behavior of a numerical sequence called the sequence of graded codimensions of A. Our main result says that such sequence is polynomially bounded if and only if the variety of superalgebras generated by A does not contain a list of five superalgebras consisting of a 2-dimensional algebra, the infinite dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with trivial and nontrivial gradings. Our main tool is the representation theory of the symmetric group.

Star-group identities and groups of units

Analogous to *-identities in rings with involution we define *-identities in groups. Suppose that G is a torsion group with involution * and that F is an infinite field with char F ≠ 2. Extend * linearly to FG. We prove that the unit group \({\mathcal{U}}\) of FG satisfies a *-identity if and only if the symmetric elements \({\mathcal{U}^+}\) satisfy a group identity.

On minimal ∗-identities of matrices∗

Let Mn (F) be the algebra of n×n matrices (n≥2) over a field F of characteristic different from 2 and let ∗ be an involution in Mn (F) In case ∗ is the transpose involution, we construct a multilinear ∗ polynomial identify of Mn (F) of degree 2n−1, P 2n−1(k 1, s 2, … s 2n−1) in one skew variable and the remaining symmetric variables of minimal degree among all ∗-polynomial identities of this type. We also prove that any other multilinear ∗-polynomial identity of Mn (F) of this type of degree 2n−1 is a scalar multiple of P2n−1 . In case ∗ is the symplectic involution in Mn (F), we construct a ∗-polynomial identity of Mn (F) of degree 2n−1 in skew variables T2n−1 (k 1,…,k 2n−1) and we prove t…

A characterization of fundamental algebras through S-characters

Abstract Fundamental algebras play an important role in the theory of algebras with polynomial identities in characteristic zero. They are defined in terms of multialternating polynomials non vanishing on them. Here we give a characterization of fundamental algebras in terms of representations of symmetric groups obtaining this way an equivalent definition. As an application we determine when a finitely generated Grassmann algebra is fundamental.

Polynomial codimension growth and the Specht problem

Abstract We construct a continuous family of algebras over a field of characteristic zero with slow codimension growth bounded by a polynomial of degree 4. This is achieved by building, for any real number α ∈ ( 0 , 1 ) a commutative nonassociative algebra A α whose codimension sequence c n ( A α ) , n = 1 , 2 , …  , is polynomially bounded and lim ⁡ log n ⁡ c n ( A α ) = 3 + α . As an application we are able to construct a new example of a variety with an infinite basis of identities.

Group algebras of torsion groups and Lie nilpotence

Letbe an involution of a group algebra FG induced by an involution of the group G. For char F 0 2, we classify the torsion groups G with no elements of order 2 whose Lie al- gebra of � -skew elements is nilpotent.

Codimension growth of central polynomials of Lie algebras

Abstract Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero and let I be the T-ideal of polynomial identities of the adjoint representation of L. We prove that the number of multilinear central polynomials in n variables, linearly independent modulo I, grows exponentially like ( dim ⁡ L ) n {(\dim L)^{n}} .

Modular representation theory and pi-algebras

(1988). Modular representation theory and pi-algebras. Communications in Algebra: Vol. 16, No. 10, pp. 2043-2067.

Codimensions of star-algebras and low exponential growth

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.

Group identities on units of rings

ON THE COLENGTH OF A VARIETY OF LIE ALGEBRAS

We study the variety of Lie algebras defined by the identity [Formula: see text] over a field of characteristic zero. We prove that, as in the associative case, in the nth cocharacter χn of this variety, every irreducible Sn-character appears with polynomially bounded multiplicity (not greater than n2). Anyway, surprisingly enough, we also show that the colength of this variety, i.e. the total number of irreducibles appearing in χn is asymptotically equal to [Formula: see text].

A characterization of algebras with polynomial growth of the codimensions

Sturmian words and overexponential codimension growth

Abstract Let A be a non necessarily associative algebra over a field of characteristic zero satisfying a non-trivial polynomial identity. If A is a finite dimensional algebra or an associative algebra, it is known that the sequence c n ( A ) , n = 1 , 2 , … , of codimensions of A is exponentially bounded. If A is an infinite dimensional non associative algebra such sequence can have overexponential growth. Such phenomenon is present also in the case of Lie or Jordan algebras. In all known examples the smallest overexponential growth of c n ( A ) is ( n ! ) 1 2 . Here we construct a family of algebras whose codimension sequence grows like ( n ! ) α , for any real number α with 0 α 1 .

On the identities of the Grassmann algebras in characteristicp>0

In this note we exhibit bases of the polynomial identities satisfied by the Grassmann algebras over a field of positive characteristic. This allows us to answer the following question of Kemer: Does the infinite dimensional Grassmann algebra with 1, over an infinite fieldK of characteristic 3, satisfy all identities of the algebraM 2(K) of all 2×2 matrices overK? We give a negative answer to this question. Further, we show that certain finite dimensional Grassmann algebras do give a positive answer to Kemer's question. All this allows us to obtain some information about the identities satisfied by the algebraM 2(K) over an infinite fieldK of positive odd characteristic, and to conjecture ba…

Central Units, Class Sums and Characters of the Symmetric Group

In the search for central units of a group algebra, we look at the class sums of the group algebra of the symmetric group S n in characteristic zero, and we show that they are units in very special instances.

Multialternating graded polynomials and growth of polynomial identities

Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we construct multialternating graded polynomials of arbitrarily large degree non vanishing on A. As a consequence we compute the exponential rate of growth of the sequence of graded codimensions of an arbitrary G-graded algebra satisfying an ordinary polynomial identity. In particular we show it is an integer. The result was proviously known in case G is abelian.

Central idempotents and units in rational group algebras of alternating groups

Let ℚAn be the group algebra of the alternating group over the rationals. By exploiting the theory of Young tableaux, we give an explicit description of the minimal central idempotents of ℚAn. As an application we construct finitely many generators for a subgroup of finite index in the centre of the group of units of ℚAn.

Automorphisms of the integral group ring of the hyperoctahedral group

The purpose of this paper is to verify a conjecture of Zassenhaus [3] for hyperoctahedral groups by proving that every normalized automorphism () of ZG can be written in the form () = Tu 0 I where I is an automorphism of ZG obtained by extending an automorphism of G linearly to ZG and u is a unit of (JJG. A similar result was proved for symmetric groups by Peterson in [2]; the reader should consult [3] or the survey [4] for other results of this kind. 1989

Central Polynomials of Algebras and Their Growth

A polynomial in noncommutative variables taking central values in an algebra A is called a central polynomial of A. For instance the algebra of k × k matrices has central polynomials. For general algebras the existence of central polynomials is not granted. Nevertheless if an algebra has such polynomials, how can one measure how many are there?

Polynomial identities on superalgebras: Classifying linear growth

Abstract We classify, up to PI-equivalence, the superalgebras over a field of characteristic zero whose sequence of codimensions is linearly bounded. As a consequence we determine the linear functions describing the graded codimensions of a superalgebra.

On the Codimension Growth of Finite-Dimensional Lie Algebras

Abstract We study the exponential growth of the codimensions cn(L) of a finite-dimensional Lie algebra L over a field of characteristic zero. We show that if the solvable radical of L is nilpotent then lim n → ∞ c n ( L ) exists and is an integer.

Understanding star-fundamental algebras

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.

Simple and semisimple Lie algebras and codimension growth

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

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…

Some Numerical Invariants of Multilinear Identities

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.

Rings with algebraic n-engel elements

(1994). Rings with algebraic n-engel elements. Communications in Algebra: Vol. 22, No. 5, pp. 1685-1701.

On codimension growth of finite-dimensional Lie superalgebras

Asymptotics for the multiplicities in the cocharacters of some PI-algebras

We consider associative PI-algebras over a field of characteristic zero. We study the asymptotic behavior of the sequence of multiplicities of the cocharacters for some significant classes of algebras. We also give a characterization of finitely generated algebras for which this behavior is linear or quadratic.

On ∗-polynomial identities for n × n matrices

Rappresentazioni di gruppi simmetrici ed identita' polinomiali

Si studiano le identita polinomiali di un'algebra (ovvero:T-ideali dell'algebra libera) utilizzando la teoria delle rappresentazioni dei gruppi simmetrici. Si espongono risultati validi su un campo di caratteristica arbitraria.

Cocharacters of Bilinear Mappings and Graded Matrices

Let Mk(F) be the algebra of k ×k matrices over a field F of characteristic 0. If G is any group, we endow Mk(F) with the elementary grading induced by the k-tuple (1,...,1,g) where g ∈ G, g2 ≠ 1. Then the graded identities of Mk(F) depending only on variables of homogeneous degree g and g − 1 are obtained by a natural translation of the identities of bilinear mappings (see Bahturin and Drensky, Linear Algebra Appl 369:95–112, 2003). Here we study such identities by means of the representation theory of the symmetric group. We act with two copies of the symmetric group on a space of multilinear graded polynomials of homogeneous degree g and g − 1 and we find an explicit decomposition of the …

Codimension growth of two-dimensional non-associative algebras

Let F be a field of characteristic zero and let A be a two-dimensional non-associative algebra over F. We prove that the sequence c n (A), n =1,2,..., of codimensions of A is either bounded by n + 1 or grows exponentially as 2 n . We also construct a family of two-dimensional algebras indexed by rational numbers with distinct T-ideals of polynomial identities and whose codimension sequence is n + 1, n > 2.

Polynomial identities with involution, superinvolutions and the Grassmann envelope

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.

On permutations of class sums of alternating groups

We prove a result concerning the class sums of the alternating group An; as a consequence we deduce that if θ is a normalized automorphism of the integral group ring then there exists such that is the identity on , where Sn:is the symmetric group and is the center of

F-algebraic extensions of rings

Identities of *-superalgebras and almost polynomial growth

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.

Superalgebras: Polynomial identities and asymptotics

To any superalgebra A is attached a numerical sequence cnsup(A), n≥1, called the sequence of supercodimensions of A. In characteristic zero its asymptotics are an invariant of the superidentities satisfied by A. It is well-known that for a PI-superalgebra such sequence is exponentially bounded and expsup(A)=limn→∞⁡cnsup(A)n is an integer that can be explicitly computed. Here we introduce a notion of fundamental superalgebra over a field of characteristic zero. We prove that if A is such an algebra, then C1ntexpsup(A)n≤cnsup(A)≤C2ntexpsup(A)n, where C1>0,C2,t are constants and t is a half integer that can be explicitly written as a linear function of the dimension of the even part of A an…

Groups, Rings and Group Rings

Standard polynomials are characterized by their degree and exponent

Abstract By the Giambruno–Zaicev theorem (Giambruno and Zaicev, 1999) [5] , the exponent exp ( A ) of a p.i. algebra A exists, and is always an integer. In Berele and Regev (2001) [2] it was shown that the exponent exp ( St n ) of the standard polynomial St n of degree n is not smaller than the exponent of any polynomial of degree n. Here it is proved that exp ( St n ) is strictly larger than the exponent of any other polynomial of degree n which is not a multiple of St n .

Exponential Codimension Growth of PI Algebras: An Exact Estimate

Abstract LetAbe an associative PI-algebra over a fieldFof characteristic zero. By studying the exponential behavior of the sequence of codimensions {cn(A)} ofA, we prove thatInv(A)=limn→∞  c n ( A ) always exists and is an integer. We also give an explicit way for computing such integer: letBbe a finite dimensionalZ2-graded algebra whose Grassmann envelopeG(B) satisfies the same identities ofA; thenInv(A)=Inv(G(B))=dim C(0)+dim C(1)whereC(0)+C(1)is a suitableZ2-graded semisimple subalgebra ofB.

Algebras with intermediate growth of the codimensions

AbstractLet F be a field of characteristic zero and let A be an F-algebra. The polynomial identities satisfied by A can be measured through the asymptotic behavior of the sequence of codimensions and the sequence of colengths of A. For finite dimensional algebras we show that the colength sequence of A is polynomially bounded and the codimension sequence cannot have intermediate growth. We then prove that for general nonassociative algebras intermediate growth of the codimensions is allowed. In fact, for any real number 0<β<1, we construct an algebra A whose sequence of codimensions grows like nnβ.

Matrix algebras of polynomial codimension growth

We study associative algebras with unity of polynomial codimension growth. For any fixed degree $k$ we construct associative algebras whose codimension sequence has the largest and the smallest possible polynomial growth of degree $k$. We also explicitly describe the identities and the exponential generating functions of these algebras.

Polynomial Identities and Asymptotic Methods

Polynomial identities and PI-algebras $S_n$-representations Group gradings and group actions Codimension and colength growth Matrix invariants and central polynomials The PI-exponent of an algebra Polynomial growth and low PI-exponent Classifying minimal varieties Computing the exponent of a polynomial $G$-identities and $G\wr S_n$-action Superalgebras, *-algebras and codimension growth Lie algebras and nonassociative algebras The generalized-six-square theorem Bibliography Index.

Lie nilpotence of group rings

Let FG be the group algebra of a group G over a field F. Denote by ∗ the natural involution, (∑fi gi -1. Let S and K denote the set of symmetric and skew symmetric and skew symmetric elements respectively with respect to this involutin. It is proved that if the characteristic of F is zero p≠2 and G has no 2-elements, then the Lie nilpotence of S or K implies the Lie nilpotence of FG.

G-identities on associative algebras

Graded polynomial identities and codimensions: Computing the exponential growth

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.

Star-fundamental algebras: polynomial identities and asymptotics

We introduce the notion of star-fundamental algebra over a field of characteristic zero. We prove that in the framework of the theory of polynomial identities, these algebras are the building blocks of a finite dimensional algebra with involution ∗ * . To any star-algebra A A is attached a numerical sequence c n ∗ ( A ) c_n^*(A) , n ≥ 1 n\ge 1 , called the sequence of ∗ * -codimensions of A A . Its asymptotic is an invariant giving a measure of the ∗ * -polynomial identities satisfied by A A . It is well known that for a PI-algebra such a sequence is exponentially bounded and exp ∗ ⁡ ( A ) = lim n → ∞ c n ∗ ( A ) n \exp ^*(A)=\lim _{n\to \infty }\sqrt [n]{c_n^*(A)} can be explicitly compute…

Multialternating Jordan polynomials and codimension growth of matrix algebras

Abstract Let R be the Jordan algebra of k  ×  k matrices over a field of characteristic zero. We exhibit a noncommutative Jordan polynomial f multialternating on disjoint sets of variables of order k 2 and we prove that f is not a polynomial identity of R . We then study the growth of the polynomial identities of the Jordan algebra R through an analysis of its sequence of Jordan codimensions. By exploiting the basic properties of the polynomial f , we are able to prove that the exponential rate of growth of the sequence of Jordan codimensions of R in precisely k 2 .

Codimensions of algebras and growth functions

Abstract Let A be an algebra over a field F of characteristic zero and let c n ( A ) , n = 1 , 2 , … , be its sequence of codimensions. We prove that if c n ( A ) is exponentially bounded, its exponential growth can be any real number >1. This is achieved by constructing, for any real number α > 1 , an F-algebra A α such that lim n → ∞ c n ( A α ) n exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.

On Codimension Growth of Finitely Generated Associative Algebras

Abstract LetAbe a PI-algebra over a fieldF. We study the asymptotic behavior of the sequence of codimensionscn(A) ofA. We show that ifAis finitely generated overFthenInv(A)=limn→∞  c n (A) always exists and is an integer. We also obtain the following characterization of simple algebras:Ais finite dimensional central simple overFif and only ifInv(A)=dim=A.

Central polynomials of associative algebras and their growth

Unitary units and skew elements in group algebras

Let FG be the group algebra of a group G over a field F and let * denote the canonical involution of FG induced by the map g→g −1 ,gG. Let Un(FG)={uFG|uu * =1} be the group of unitary units of FG. In case char F=0, we classify the torsion groups G for which Un(FG) satisfies a group identity not vanishing on 2-elements. Along the way we actually prove that, in characteristic 0, the unitary group Un(FG) does not contain a free group of rank 2 if FG − , the Lie algebra of skew elements of FG, is Lie nilpotent. Motivated by this connection we characterize most groups G for which FG − is Lie nilpotent and char F≠2.

PI-algebras with slow codimension growth

Let $c_n(A),\ n=1,2,\ldots,$ be the sequence of codimensions of an algebra $A$ over a field $F$ of characteristic zero. We classify the algebras $A$ (up to PI-equivalence) in case this sequence is bounded by a linear function. We also show that this property is closely related to the following: if $l_n(A), \ n=1,2,\ldots, $ denotes the sequence of colengths of $A$, counting the number of $S_n$-irreducibles appearing in the $n$-th cocharacter of $A$, then $\lim_{n\to \infty} l_n(A)$ exists and is bounded by $2$.

Differential identities, 2 × 2 upper triangular matrices and varieties of almost polynomial growth

Abstract We study the differential identities of the algebra U T 2 of 2 × 2 upper triangular matrices over a field of characteristic zero. We let the Lie algebra L = Der ( U T 2 ) of derivations of U T 2 (and its universal enveloping algebra) act on it. We study the space of multilinear differential identities in n variables as a module for the symmetric group S n and we determine the decomposition of the corresponding character into irreducibles. If V is the variety of differential algebras generated by U T 2 , we prove that unlike the other cases (ordinary identities, group graded identities) V does not have almost polynomial growth. Nevertheless we exhibit a subvariety U of V having almo…

Group graded algebras and multiplicities bounded by a constant

AbstractLet G be a finite group and A a G-graded algebra over a field of characteristic zero. When A is a PI-algebra, the graded codimensions of A are exponentially bounded and one can study the corresponding graded cocharacters via the representation theory of products of symmetric groups. Here we characterize in two different ways when the corresponding multiplicities are bounded by a constant.

On algebras and superalgebras with linear codimension growth

We present the classification, up to PI-equivalence, of the algebras over a field of characteristic zero whose sequence of codimensions is linearly bounded. We also describe the generalization of this result in the setting of superalgebras and their graded identities. As a consequence we determine all linear functions describing the ordinary codimensions and the graded codimensions of a given algebra.

Group algebras whose units satisfy a group identity

Let F G FG be the group algebra of a torsion group over an infinite field F F . Let U U be the group of units of F G FG . We prove that if U U satisfies a group identity, then F G FG satisfies a polynomial identity. This confirms a conjecture of Brian Hartley.

Standard polynomials and matrices with superinvolutions

Abstract Let M n ( F ) be the algebra of n × n matrices over a field F of characteristic zero. The superinvolutions ⁎ on M n ( F ) were classified by Racine in [12] . They are of two types, the transpose and the orthosymplectic superinvolution. This paper is devoted to the study of ⁎-polynomial identities satisfied by M n ( F ) . The goal is twofold. On one hand, we determine the minimal degree of a standard polynomial vanishing on suitable subsets of symmetric or skew-symmetric matrices for both types of superinvolutions. On the other, in case of M 2 ( F ) , we find generators of the ideal of ⁎-identities and we compute the corresponding sequences of cocharacters and codimensions.

Central polynomials and matrix invariants

LetK be a field, charK=0 andM n (K) the algebra ofn×n matrices overK. If λ=(λ1,…,λ m ) andμ=(μ 1,…,μ m ) are partitions ofn 2 let $$\begin{gathered} F^{\lambda ,\mu } = \sum\limits_{\sigma ,\tau \in S_n 2} {\left( {\operatorname{sgn} \sigma \tau } \right)x_\sigma (1) \cdot \cdot \cdot x_\sigma (\lambda _1 )^{y_\tau } (1)^{ \cdot \cdot \cdot } y_\tau (\mu _1 )^{x\sigma } (\lambda _1 + 1)} \hfill \\ \cdot \cdot \cdot x_\sigma (\lambda _1 + \lambda _2 )^{y_\tau } (\mu _1 ^{ + 1} )^{ \cdot \cdot \cdot y_\tau } (\mu _1 + \mu _2 ) \hfill \\ \cdot \cdot \cdot x_\sigma (\lambda _1 + \cdot \cdot \cdot + \lambda _{\mu - 1} ^{ + 1} ) \hfill \\ \cdot \cdot \cdot x_\sigma (n^2 )^{y_\tau } (\mu _1 ^{ + \…

Non-integrality of the PI-exponent of special Lie algebras

If L is a special Lie algebra over a field of characteristic zero, its sequence of codimensions is exponentially bounded. The PI-exponent measures the exponential rate of growth of such sequence and here we give a first example of a special Lie algebra whose (upper and lower) PI-exponent is non-integer.

Minimal varieties of algebras of exponential growth

Abstract The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer. Through this scale we can now classify all minimal varieties of given exponent and of finite basic rank. As a consequence, we describe the corresponding T-ideals of the free algebra and we compute the asymptotics of the related codimension sequences, verifying in this setting some known conjectures. We also show that the number of these minimal varieties is finite for any given exponent. We finally point out some relations between the exponent of a variety and the Gelfand–Kirillov dimension of the corresponding relatively free algebras of finite rank.

Involution codimensions and trace codimensions of matrices are asymptotically equal

We calculate the asymptotic growth oft n (M p (F),*) andc n (M p (F),*), the trace and ordinary *-codimensions ofp×p matrices with involution. To do this we first calculate the asymptotic growth oft n and then show thatc n ⋍t n .

Degrees of irreducible characters of the symmetric group and exponential growth

We consider sequences of degrees of ordinary irreducible S n S_n - characters. We assume that the corresponding Young diagrams have rows and columns bounded by some linear function of n n with leading coefficient less than one. We show that any such sequence has at least exponential growth and we compute an explicit bound.

Cocharacters of group graded algebras and multiplicities bounded by one

Let G be a finite group and A a G-graded algebra over a field F of characteristic zero. We characterize the (Formula presented.)-ideals (Formula presented.) of graded identities of A such that the multiplicities (Formula presented.) in the graded cocharacter of A are bounded by one. We do so by exhibiting a set of identities of the (Formula presented.)-ideal. As a consequence we characterize the varieties of G-graded algebras whose lattice of subvarieties is distributive.

Varieties of Algebras with Superinvolution of Almost Polynomial Growth

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.