Sequence Of Codimensions Research Articles

On the asymptotic behaviour of the graded-star-codimension sequence of upper triangular matrices

We study the algebra of upper triangular matrices endowed with a group grading and a homogeneous involution over an infinite field. We compute the asymptotic behaviour of its (graded) star-codimension sequence. It turns out that the asymptotic growth of the sequence is independent of the grading and the involution under consideration, depending solely on the size of the matrix algebra. This independence of the group grading also applies to the graded codimension sequence of the associative algebra of upper triangular matrices.Comment: arXiv admin note: text overlap with arXiv:2402.02671

On star-homogeneous-graded polynomial identities of upper triangular matrices over an arbitrary field

We study the graded polynomial identities with a homogeneous involution on the algebra of upper triangular matrices endowed with a fine group grading. We compute their polynomial identities and a basis of the relatively free algebra, considering an arbitrary base field. We obtain the asymptotic behaviour of the codimension sequence when the characteristic of the base field is zero. As a consequence, we compute the exponent and the second exponent of the same algebra endowed with any group grading and any homogeneous involution.

Free bicommutative superalgebras

We introduce the variety Bsup of bicommutative superalgebras over an arbitrary field of characteristic different from 2. The variety consists of all nonassociative Z2-graded algebras satisfying the polynomial super-identities of super- left- and right-commutativityx(yz)=(−1)x‾y‾y(xz) and (xy)z=(−1)y‾z‾(xz)y, where u‾∈{0,1} is the parity of the homogeneous element u.We present an explicit construction of the free bicommutative superalgebras, find their bases as vector spaces and show that they share many properties typical for ordinary bicommutative algebras and super-commutative associative superalgebras. In particular, in the case of free algebras of finite rank we compute the Hilbert series and find explicitly its coefficients. As a consequence we give a formula for the codimension sequence. We establish an analogue of the classical Hilbert Basissatz for two-sided ideals. We see that the Gröbner-Shirshov bases of these ideals are finite, the Gelfand-Kirillov dimensions of finitely generated bicommutative superalgebras are nonnegative integers and the Hilbert series of finitely generated graded bicommutative superalgebras are rational functions. Concerning problems studied in the theory of varieties of algebraic systems, we prove that the variety of bicommutative superalgebras satisfies the Specht property. In the case of characteristic 0 we compute the sequence of cocharacters.

Graded identities with involution for the algebra of upper triangular matrices

Let F be a field of characteristic zero and let m≥2 be an integer. In this paper, we prove that if a group grading on UTm(F) admits a graded involution then this grading is a coarsening of a Z⌊m2⌋-grading on UTm(F) and the graded involution is equivalent to the reflection or symplectic involution on UTm(F), this grading is called the finest grading on UTm(F). Furthermore, if m≤4 the algebra UTm(F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z⌊m2⌋,⁎)-identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z⌊m2⌋,⁎)-codimensions is determined. As a consequence, we prove that for any G-grading on UTm(F) and any graded involution the (G,⁎)-exponent is m. Finally, we study the algebra UT3(F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z-grading and the Z2-grading induced by (0,1,0). We determine a basis for the (Z2,⁎)-identities and we compute the codimension sequence for the (Z2,⁎)-graded identities for UT3(F).

Differential codimensions and exponential growth

Let A be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra L by derivations, and let cnL(A), n≥1, be its differential codimension sequence. Such sequence is exponentially bounded and expL⁡(A)=limn→∞⁡cnL(A)n is an integer that can be computed, called differential PI-exponent of A.In this paper we prove that for any Lie algebra L, expL⁡(A) coincides with exp⁡(A), the ordinary PI-exponent of A. Furthermore, in case L is a solvable Lie algebra, we apply such result to classify varieties of L-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth.

Two-dimensional Jordan algebras: Their classification and polynomial identities

Let F be an arbitrary field of characteristic different from 2. We classify the two-dimensional power-associative commutative algebras over F. As consequence, we obtain a classification of two-dimensional Jordan algebras over F and prove that there exists, up to isomorphism, a unique two-dimensional nonassociative Jordan algebra. The construction of this algebra can be generalized naturally to produce a Jordan algebra D with an arbitrary dimension. If F is infinite, we determine a finite basis for the polynomial identities of D, as well of all associative Jordan algebras of dimension two. We also determine the codimension sequence of all these algebras, and if the field is of characteristic zero we determine their cocharacter sequence. Hence we conclude that the n-th codimension sequence of a two-dimensional Jordan algebra is bounded by n.

Differential polynomial identities of upper triangular matrices of size three

We determine the differential polynomial identities of 3×3 upper triangular matrices over a base field of characteristic zero, under the action of its full Lie algebra of derivations. We compute the exact differential codimension sequence of the multilinear ones and describe their Sn-structure by means of an explicit decomposition of the Sn-cocharacter of their proper part.

Graded polynomial identities for the Lie algebra of upper triangular matrices of order 3

We compute the graded polynomial identities and its graded codimension sequence for the elementary gradings of the Lie algebra of upper triangular matrices of order 3.

Varieties of algebras with pseudoinvolution: Codimensions, cocharacters and colengths

Let A be a finitely generated superalgebra with pseudoinvolution ⁎ over an algebraically closed field F of characteristic zero. In this paper we develop a theory of polynomial identities for this kind of algebras . In particular, we shall consider three sequences that can be attached to Id ⁎ ( A ) , the T 2 ⁎ -ideal of identities of A : the sequence of ⁎-codimensions c n ⁎ ( A ) , the sequence of ⁎-cocharacter χ 〈 n 〉 ⁎ ( A ) and the ⁎-colength sequence l n ⁎ ( A ) . Our purpose is threefold. First we shall prove that the ⁎-codimension sequence is eventually non-decreasing, i.e., c n ⁎ ( A ) ≤ c n + 1 ⁎ ( A ) , for n large enough. Secondly, we study superalgebras with pseudoinvolution having the multiplicities of their ⁎-cocharacter bounded by a constant. Among them, we characterize the ones with multiplicities bounded by 1. Finally, we classify superalgebras with pseudoinvolution A such that l n ⁎ ( A ) is bounded by 3. In the last section we relate the ⁎-colengths with the polynomial growth of the ⁎-codimensions: we show that l n ⁎ ( A ) is bounded by a constant if and only if c n ⁎ ( A ) grows at most polynomially.

On central polynomials and codimension growth

Let $A$ be an associative algebra over a field of characteristic zero. A central polynomial is a polynomial of the free associative algebra that takes central values of $A.$ In this survey, we present some recent results about the exponential growth of the central codimension sequence and the proper central codimension sequence in the setting of algebras with involution and algebras graded by a finite group.

Codimension Sequences and their Asymptotic Behavior

In this paper, numerical characteristics of identities of non-associative algebras are studied. A short survey of the results of the last 10 years in this area is presented.

Differential Polynomial Identities of Upper Triangular Matrices Under the Action of the Two-Dimensional Metabelian Lie Algebra

We study the differential polynomial identities of the algebra UTm(F) under the derivation action of the two dimensional metabelian Lie algebra, obtaining a generating set of the TL-ideal they constitute. Then we determine the Sn-structure of their proper multilinear spaces and, for the minimal cases m = 2, 3, their exact differential codimension sequence.

Eventually non-decreasing codimensions of $$*$$-identities

Let A be a PI-algebra. If A is an associative algebra, the sequence of codimensions $$c_n(A), n=1,2,\ldots ,$$ of A is asymptotically non-decreasing. For the non-associative case, there are examples of PI-algebras whose sequence of codimensions is not eventually non-decreasing. For a associative PI-algebra A with involution $$*: A\rightarrow A$$ , it was recently shown that its sequence of $$*$$ -codimensions $$c_n^*(A)$$ , $$n=1,2,\ldots $$ , is also asymptotically non-decreasing. In the present paper, we construct a non-associative algebra whose sequence of $$*$$ -codimensions is not eventually non-decreasing.

Superalgebras and algebras with involution: classifying varieties of quadratic growth

Let be a field of characteristic zero. By a -algebra we mean a superalgebra or an algebra with involution over In the last years, the sequence of -codimensions of a -algebra has been extensively studied. In this paper, we classify varieties generated by unitary -algebras having quadratic growth of -codimensions. As a consequence we obtain that a unitary -algebra with quadratic growth is -equivalent to a finite direct sum of minimal unitary -algebras with at most quadratic growth of the -codimensions. In addition, we explicit all quadratic functions describing the -codimension sequence of a unitary -algebra.

Unitary superalgebras with graded involution or superinvolution of polynomial growth

In this paper we study associative unitary superalgebras with graded involution or superinvolution having polynomial growth of the codimension sequence. The first goal is to prove that, for this kind of algebras , the codimension sequence is a polynomial with rational coefficients. Then we shall construct several superalgebras with graded involution or superinvolution realizing the smallest and the largest value of the leading term of such a polynomial.

Varieties of unitary algebras with small growth of codimensions

In the last years, the sequence of codimensions of PI-algebras has been studied by several authors and the classification of unitary algebras, up to equivalence, with at most cubic codimension growth was given by Giambruno, La Mattina and Petrogradsky in 2007. In this paper, we establish a new approach by studying the possibilities of specific proper codimensions of a unitary algebra with growth [Formula: see text] in order to present a complete list of varieties generated by unitary algebras with polynomial growth [Formula: see text]. Also, we classify, up to PI-equivalence, the unitary algebras with growth [Formula: see text] whose leading coefficient of the polynomial describing the codimension sequence achieves the largest and the smallest possible value.

Minimal varieties of associative algebras and transcendental series

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number [Formula: see text] of minimal varieties of given exponent [Formula: see text] is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit [Formula: see text] exists and can be expressed as the positive solution of an equation [Formula: see text] where [Formula: see text] is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank [Formula: see text]. It follows from classical results on lacunary power series that the generating function of the sequence [Formula: see text], [Formula: see text], is transcendental. With the same approach we construct examples of free graded semigroups [Formula: see text] with the following property. If [Formula: see text] is the number of elements of degree [Formula: see text] of [Formula: see text], then the limit [Formula: see text] exists and is transcendental.

A Characterization of Superalgebras with Pseudoinvolution of Exponent 2

Let A be a superalgebra endowed with a pseudoinvolution ∗ over an algebraically closed field of characteristic zero. If A satisfies an ordinary non-trivial identity, then its graded ∗-codimension sequence $c_{n}^{*}(A)$ , n = 1,2,…, is exponentially bounded (Ioppolo and Martino (Linear Multilinear Algebra 66(11), 2286–2304 2018). In this paper we capture this exponential growth giving a positive answer to the Amitsur’s conjecture for this kind of algebras. More precisely, we shall see that the $\lim _{n \rightarrow \infty } \sqrt [n]{c_{n}^{*}(A)}$ exists and it is an integer, denoted $\exp ^{*}(A)$ and called graded ∗-exponent of A. Moreover, we shall characterize superalgebras with pseudoinvolution according to their graded ∗-exponent.

On multiplicities of cocharacters for algebras with superinvolution

In this paper we deal with finitely generated superalgebras with superinvolution, satisfying a non-trivial identity, whose multiplicities of the cocharacters are bounded by a constant. Along the way, we prove that the codimension sequence of such algebras is polynomially bounded if and only if their colength sequence is bounded by a constant.

Asymptotic codimensions of Mk(E)

We show that the codimension sequence of the algebra of k×k matrices over the Grassmann algebra, cn(Mk(E)), is asymptotic to αn1−k22(2k2)n, where α is an undetermined constant.