Infinite Dimensional Grassmann Algebra Research Articles

ℤ-gradings on the Grassmann algebra over infinite fields: Graded identities and central polynomials

Let E be the infinite-dimensional Grassmann algebra over an infinite field F of characteristic different from 2. The main purpose of this paper is to describe the [Formula: see text]-ideal of the graded polynomial identities and the [Formula: see text]-space of the central polynomials of E equipped with its [Formula: see text] and 3-induced [Formula: see text]-gradings. Therefore, we generalize the results of [A. Brandão, C. Fidelis and A. Guimarães, [Formula: see text]-gradings of full support on the Grassmann algebra, J. Algebra 601 (2022) 332–353; C. Fidelis, A. Guimarães and P. Koshlukov, A note on [Formula: see text]-gradings on the Grassmann algebra and elementary number theory, Linear Multilinear Algebra 71(7) (2023) 1244–1264] in the PI theory context.

Renormalized Von Neumann entropy with application to entanglement in genuine infinite dimensional systems

A renormalized version of the von Neumann quantum entropy (which is finite and continuous in general, infinite dimensional case) which obeys several of the natural physical demands (as expected for a “good” measure of entanglement in the case of general quantum states describing bipartite and infinite-dimensional systems) is proposed. The renormalized quantum entropy is defined by the explicit use of the Fredholm determinants theory. To prove the main results on continuity and finiteness of the introduced renormalization, the fundamental Grothendick approach, which is based on the infinite dimensional Grassmann algebra theory, is applied. Several features of majorization theory are preserved under the introduced renormalization as it is proved in this paper. This fact allows to extend most of the known (mainly, in the context of two-partite, finite-dimensional quantum systems) results of the LOCC comparison theory to the case of genuine infinite-dimensional, two-partite quantum systems.

Formula omitted]-gradings of full support on the Grassmann algebra

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic zero. In this paper we investigate the structures of Z-gradings on E of full support. Using methods of elementary number theory, we describe the Z-graded polynomial identities for the so-called 2-induced Z-gradings on E of full support. As a consequence of this fact we provide examples of Z-gradings on E which are PI-equivalent but not Z-isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent, but non-isomorphic as graded algebras. We also present the notion of central Z-gradings on E and we show that their Z-graded polynomial identities are closely related to the Z2-graded polynomial identities of E.

Graded identities of Mn(E) and their generalizations over infinite fields

Let G be a group and F an infinite field. Assume that A is a finite dimensional F-algebra with an elementary G-grading. In this paper, we study the graded identities satisfied by the tensor product grading on the F-algebra A⊗C, where C is an H-graded colour β-commutative algebra. More precisely, under a technical condition, we provide a basis for the TG-ideal of graded polynomial identities of A⊗C, up to graded monomial identities. Furthermore, the F-algebra of upper block-triangular matrices UT(d1,…,dn), as well as the matrix algebra Mn(F), with an elementary grading such that the neutral component corresponds to its diagonal, are studied. As a consequence of our results, a basis for the graded identities, up to graded monomial identities of degrees ≤2d−1, for Md(E) and Mq(F)⊗UT(d1,…,dn), with a tensor product grading, is exhibited. In this latter case, d=d1+…+dn. Here E denotes the infinite dimensional Grassmann algebra with its natural Z2-grading, and the grading on Mq(F) is Pauli grading. The results presented in this paper generalize results from [14] and from other papers which were obtained for fields of characteristic zero.

Codimension growth for polynomial identities of representations of Lie algebras

AbstractLet K be a field of characteristic zero. We study the asymptotic behavior of the codimensions for polynomial identities of representations of Lie algebras, also called weak identities. These identities are related to pairs of the form where A is an associative enveloping algebra for the Lie algebra L. We obtain a characterization of ideals of weak identities with polynomial growth of the codimensions in terms of their cocharacter sequence. Recall that such a characterization was obtained by Kemer in [12] for associative algebras and by Benediktovich and Zalesskii in [2] for Lie algebras. We prove that the pairs , and generate varieties of pairs of almost polynomial growth. Here E denotes the infinite dimensional Grassmann algebra with 1. Also is the associative subalgebra of M2 (the 2 × 2 matrices over the field K) consisting of upper triangular matrices and is the Lie subalgebra of of the traceless matrices.

Graded identities for algebras with elementary gradings over an infinite field

AbstractLet $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$-grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$, $b\in \mathbb {N}$, we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$, as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$-algebras which are not PI equivalent. Actually, we prove that the $T_{G}$-ideal of the former algebra is contained in the $T$-ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.

$\mathbb{Z}$-graded identities on the infinite-dimensional Grassmann algebra and arithmetic tools: revisited

The main purpose of this paper is to provide a survey of results concerning the $\mathbb{Z}$-gradings on the infinite-dimensional Grassmann algebra $E$ over a field of characteristic zero. First, we provide graded identities and central polynomials for $E$ equipped with fine gradings on $E$ by the semigroup $(\mathbb{Z}^\ast,\times)$. We also describe briefly techniques in order to illustrate some important methods to exhibit graded identities and central polynomials of $E$ for other abelian groups. In particular, over a field of characteristic zero, so-called $2$-induced gradings of full support were considered. In order to obtain these descriptions, we strongly use elementary number theory as a tool, providing an interesting connection between this area and PI-Theory.

"An algorithm for automorphisms of infinite dimensional Grassmann algebras"

"Let G be the infinite dimensional Grassmann algebra. In this study, we determine a subgroup of the automorphism group Aut(G) of the algebra G which is of an importance in the description of the group Aut(G). We give an infinite generating set for this subgroup and suggest an algorithm which shows how to express each automorphism as compositions of generating elements."

Graded A-identities for M 1,1 (E)

Let F be a field of characteristic 0, E be the unitary infinite dimensional Grassmann algebra over F and consider the algebra with its natural -grading. We describe the graded A-identities for and we compute its graded A-codimensions.

On the support of -gradings on the Grassmann algebra

Let E be the infinite dimensional Grassmann algebra over an infinite field of characteristic different from 2. Let be a n-induced -grading on E and let be its support. In this paper, we study the properties of the set as a subset of Assuming that this set is bounded, we describe the graded identities of the respective graded algebra. We also show that is a subgroup of if and only if it is cyclic, and we describe the graded identities in this case.

Identities and central polynomials with involution for the Grassmann algebra

Let E be the infinite dimensional Grassmann algebra over an infinite field of characteristic p different from 2. Given an involution φ on E, denote by Id(E,φ) and C(E,φ) the set of all ⁎-polynomial identities and ⁎-central polynomials of (E,φ) respectively. In this paper we describe Id(E,φ) and C(E,φ). Moreover, we prove that C(E,φ) is not finitely generated as a T(⁎)-space if p>2.

The Nowicki conjecture for relatively free algebras

A linear locally nilpotent derivation of the polynomial algebra K[Xm] in m variables over a field K of characteristic 0 is called a Weitzenböck derivation. It is well known from the classical theorem of Weitzenböck that the algebra of constants K[Xm]δ of a Weitzenböck derivation δ is finitely generated. Assume that δ acts on the polynomial algebra K[X2d] in 2d variables as follows: δ(x2i)=x2i−1, δ(x2i−1)=0, i=1,…,d. The Nowicki conjecture states that the algebra K[X2d]δ is generated by x1,x3.…,x2d−1, and x2i−1x2j−x2ix2j−1, 1≤i<j≤d. The conjecture was proved by several authors based on different techniques. We apply the same idea to two relatively free algebras of rank 2d. We give the infinite set of generators of the algebra of constants in the free metabelian associative algebras F2d(A), and finite set of generators in the free algebra F2d(G) in the variety determined by the identities of the infinite dimensional Grassmann algebra.

The Grassmann Algebra and its Differential Identities

Let G be the infinite dimensional Grassmann algebra over an infinite field F of characteristic different from two. In this paper we study the differential identities of G with respect to the action of a finite dimensional Lie algebra L of inner derivations. We explicitly determine a set of generators of the ideal of differential identities of G. Also in case F is of characteristic zero, we study the space of multilinear differential identities in n variables as a module for the symmetric group Sn and we compute the decomposition of the corresponding character into irreducibles. Finally, we prove that unlike the ordinary case the variety of differential algebras with L action generated by G has no almost polynomial growth.

On the graded identities of the Grassmann algebra

We survey the results concerning the graded identities of the infinite dimensional Grassmann algebra.

A note on graded polynomial identities for tensor products by the Grassmann algebra in positive characteristic

Let [Formula: see text] be a finite abelian group. As a consequence of the results of Di Vincenzo and Nardozza, we have that the generators of the [Formula: see text]-ideal of [Formula: see text]-graded identities of a [Formula: see text]-graded algebra in characteristic 0 and the generators of the [Formula: see text]-ideal of [Formula: see text]-graded identities of its tensor product by the infinite-dimensional Grassmann algebra [Formula: see text] endowed with the canonical grading have pairly the same degree. In this paper, we deal with [Formula: see text]-graded identities of [Formula: see text] over an infinite field of characteristic [Formula: see text], where [Formula: see text] is [Formula: see text] endowed with a specific [Formula: see text]-grading. We find identities of degree [Formula: see text] and [Formula: see text] while the maximal degree of a generator of the [Formula: see text]-graded identities of [Formula: see text] is [Formula: see text] if [Formula: see text]. Moreover, we find a basis of the [Formula: see text]-graded identities of [Formula: see text] and also a basis of multihomogeneous polynomials for the relatively free algebra. Finally, we compute the [Formula: see text]-graded Gelfand–Kirillov (GK) dimension of [Formula: see text].

Growth for the Central Polynomials

We study the growth of the central polynomials for the infinite dimensional Grassmann algebra G, and for the algebra Mk(F) of the k × k matrices, both over a field F of characteristic zero.

Almost polynomial growth: Classifying varieties of graded algebras

Let G be a finite group, V a variety of associative G-graded algebras and c (V), n = 1, 2, …, its sequence of graded codimensions. It was recently shown by Valenti that such a sequence is polynomially bounded if and only if V does not contain a finite list of G-graded algebras. The list consists of group algebras of groups of order a prime number, the infinite-dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with suitable gradings. Such algebras generate the only varieties of G-graded 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 G-graded varieties of almost polynomial growth by giving a complete list of finite-dimensional G-graded algebras generating them.

On [formula omitted]-graded identities of [formula omitted] and their growth

Let F be an infinite field of characteristic different from two and E be the infinite dimensional Grassmann algebra over F. We consider the upper triangular matrix algebra UT2(E) with entries in E endowed with the Z2-grading inherited by the natural Z2-grading of E and we study its ideal of Z2-graded polynomial identities (TZ2-ideal) and its relatively free algebra. In particular we show that the set of Z2-graded polynomial identities of UT2(E) does not depend on the characteristic of the field. Moreover we compute the Z2-graded Hilbert series of UT2(E) and its Z2-graded Gelfand–Kirillov dimension.

A primeness property for central polynomials of verbally prime P.I. algebras

Let and be two non-commutative polynomials in disjoint sets of variables. An algebra is verbally prime if whenever is an identity for then either or is also an identity. As an analogue of this property, Regev proved that the verbally prime algebra of matrices over an infinite field has the following primeness property for central polynomials: whenever the product is a central polynomial for then both and are central polynomials. In this paper, we prove that over a field of characteristic zero, Regev’ s result holds for the verbally prime algebras and , where is the infinite-dimensional Grassmann algebra.

ℤ2-Graded Gelfand–Kirillov dimension of the Grassmann algebra

We consider the infinite dimensional Grassmann algebra E over a field F of characteristic 0 or p, where p &gt; 2, and we compute its ℤ2-graded Gelfand–Kirillov (GK) dimension as a ℤ2-graded PI-algebra.