Polynomial growth of the codimensions: a characterization

Let A A be a not necessarily associative algebra over a field of characteristic zero. Here we characterize the T-ideal of identities of A A in case the corresponding sequence of codimensions is polynomially bounded.

Zariski Closed Algebras in Varieties of Universal Algebra

Minimal varieties of graded Lie algebras of exponential growth and the special Lie algebra sl2

Symmetric units and group identities

In this paper we study rings R with an involution whose symmetric units satisfy a group identity. An important example is given by FG, the group algebra of a group G over a field F; in fact FG has a natural involution induced by setting g?g −1 for all group elements g∈G. In case of group algebras if F is infinite, charF≠ 2 and G is a torsion group we give a characterization by proving the following: the symmetric units satisfy a group identity if and only if either the group of units satisfies a group identity (and a characterization is known in this case) or char F=p >0 and 1) FG satisfies a polynomial identity, 2) the p-elements of G form a (normal) subgroup P of G and G/P is a Hamiltonia…

Varieties of superalgebras of linear growth

Group algebras whose Lie algebra of skew-symmetric elements is nilpotent

Group Identities on Units of Group Algebras

Abstract Let U be the group of units of the group algebra FG of a group G over a field F . Suppose that either F is infinite or G has an element of infinite order. We characterize groups G so that U satisfies a group identity. Under the assumption that G modulo the torsion elements is nilpotent this gives a complete classification of such groups. For torsion groups this problem has already been settled in recent years.

Group identities on symmetric units

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].

Classifying the Minimal Varieties of Polynomial Growth

Let $\mathcal{V}$ be a variety of associative algebras generated by an algebra with $1$ over a field of characteristic zero. This paper is devoted to the classification of the varieties $\mathcal{V}$ which are minimal of polynomial growth (i.e., their sequence of codimensions growth like $n^k$ but any proper subvariety grows like $n^t$ with $t 4$, the number of minimal varieties is at least $|F|$, the cardinality of the base field and we give a recipe of how to construct them.

Polynomial Identities of Algebras of Small Dimension

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 .

Proper identities, Lie identities and exponential codimension growth

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…

Periodic and Nil Polynomials in Rings

Let R be an associative ring and f(x1,…, xd) a polynomial in noncommuting variables. We say that f is periodic or nil in R if for all r1,…, rd ∈ R we have that f(r1,…, rd) is periodic, respectively nilpotent (recall that a ∈ R is periodic if for some integer ).

Graded polynomial identities and Specht property of the Lie algebrasl2

Abstract Let G be a group. The Lie algebra sl 2 of 2 × 2 traceless matrices over a field K can be endowed up to isomorphism, with three distinct non-trivial G-gradings induced by the groups Z 2 , Z 2 × Z 2 and Z . It has been recently shown (Koshlukov, 2008 [8] ) that for each grading the ideal of G-graded identities has a finite basis. In this paper we prove that when char ( K ) = 0 , the algebra sl 2 endowed with each of the above three gradings has an ideal of graded identities Id G ( sl 2 ) satisfying the Specht property, i.e., every ideal of graded identities containing Id G ( sl 2 ) is finitely based.

Group identities on unit groups of group algebras

Centralizers and Multilinear Polynomials in Non-Commutative Rings

Anomalies on codimension growth of algebras

Abstract This paper deals with the asymptotic behavior of the sequence of codimensions c n ⁢ ( A ) ${c_{n}(A)}$ , n = 1 , 2 , … , ${n=1,2,\ldots,}$ of an algebra A over a field of characteristic zero. It is shown that when such sequence is polynomially bounded, then lim sup n → ∞ ⁡ log n ⁡ c n ⁢ ( A ) ${\limsup_{n\to\infty}\log_{n}c_{n}(A)}$ and lim inf n → ∞ ⁡ log n ⁡ c n ⁢ ( A ) ${\liminf_{n\to\infty}\log_{n}c_{n}(A)}$ can be arbitrarily distant. Also, in case the codimensions are exponentially bounded, we can construct an algebra A such that exp ⁡ ( A ) = 2 ${\exp(A)=2}$ and, for any q ≥ 1 ${q\geq 1}$ , there are infinitely many integers n such that c n ⁢ ( A ) &gt; n q ⁢ 2 n ${c_{n}(A)&…

Central units, class sums and characters of the symmetric group

Polynomial Identities and Asymptotic Methods

Growth of polynomial identities: is the sequence of codimensions eventually non-decreasing?

Graded polynomial identities and exponential growth

Let $A$ be a finite dimensional algebra over a field of characteristic zero graded by a finite abelian group $G$. Here we study a growth function related to the graded polynomial identities satisfied by $A$ by computing the exponential rate of growth of the sequence of graded codimensions of $A$. We prove that the $G$-exponent of $A$ exists and is an integer related in an explicit way to the dimension of a suitable semisimple subalgebra of $A$.

Some generalizations of the center of a ring

Si generalizza la nozione di ipercentro introdotta da Herstein in [3] e si trova una forma equivalente alla congettura di Koethe.

Group algebras and Lie nilpotence

Abstract Let ⁎ be an involution of a group algebra FG induced by an involution of the group G. For char F ≠ 2 , we classify the groups G with no 2-elements and with no nonabelian dihedral groups involved whose Lie algebra of ⁎-skew elements is nilpotent.

Group Actions and Asymptotic Behavior of Graded Polynomial Identities

Asymptotics for the standard and the Capelli identities

Let {c n (St k )} and {c n (C k )} be the sequences of codimensions of the T-ideals generated by the standard polynomial of degreek and by thek-th Capelli polynomial, respectively. We study the asymptotic behaviour of these two sequences over a fieldF of characteristic zero. For the standard polynomial, among other results, we show that the following asymptotic equalities hold: $$\begin{gathered} c_n \left( {St_{2k} } \right) \simeq c_n \left( {C_{k^2 + 1} } \right) \simeq c_n \left( {M_k \left( F \right)} \right), \hfill \\ c_n \left( {St_{2k + 1} } \right) \simeq c_n \left( {M_{k \times 2k} \left( F \right) \oplus M_{2k \times k} \left( F \right)} \right), \hfill \\ \end{gathered} $$ wher…

Codimension growth of special simple Jordan algebras

Let $R$ be a special simple Jordan algebra over a field of characteristic zero. We exhibit a noncommutative Jordan polynomial $f$ multialternating on disjoint sets of variables which 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 polynomials $f$, we are able to compute the exponential rate of growth of the sequence of Jordan codimensions of $R$ and prove that it equals the dimension of the Jordan algebra over its center. We also show that for any finite dimensional special Jordan algebra, such exponential rate of growth c…

Super-cocharacters, star-cocharacters and multiplicities bounded by one

Asymptotic growth of codimensions sequences of identities of associative algebras

Asymptotics for multiplicities in the cocharacters of some PI-algebras

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

Free groups and involutions in the unit group of a group algebra

POLYNOMIAL GROWTH OF THE*-CODIMENSIONS AND YOUNG DIAGRAMS

Let A be an algebra with involution * over a field F of characteristic zero and Id(A, *) the ideal of the free algebra with involution of *-identities of A. By means of the representation theory of the hyperoctahedral group Z 2wrS n we give a characterization of Id(A, *) in case the sequence of its *-codimensions is polynomially bounded. We also exhibit an algebra G 2 with the following distinguished property: the sequence of *-codimensions of Id(G 2, *) is not polynomially bounded but the *-codimensions of any T-ideal U properly containing Id(G 2, *) are polynomially bounded.

Lie, Jordan and proper codimensions of associative algebras

Lie properties of symmetric elements in group rings

Abstract Let ∗ be an involution of a group G extended linearly to the group algebra KG . We prove that if G contains no 2-elements and K is a field of characteristic p ≠ 2 , then the ∗-symmetric elements of KG are Lie nilpotent (Lie n -Engel) if and only if KG is Lie nilpotent (Lie n -Engel).

Codimension growth of two-dimensional 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.

Group gradings on associative algebras with involution

Involution Codimensions of Finite Dimensional Algebras and Exponential Growth

Abstract Let F be a field of characteristic zero and let A be a finite dimensional algebra with involution ∗ over F . We study the asymptotic behavior of the sequence of ∗ -codimensions c n ( A , ∗ ) of A and we show that Exp(A, ∗ ) = lim n → ∞ c n ( A , ∗ ) exists and is an integer. We give an explicit way for computing Exp( A , ∗ ) and as a consequence we obtain the following characterization of ∗ -simple algebras: A is ∗ -simple if and only if Exp( A , ∗ ) = dim F A .

On growth of codimensions of Jordan algebras

Poisson identities of enveloping algebras