"An algorithm for automorphisms of infinite dimensional Grassmann algebras"

2021 ◽  
Vol 30 (2) ◽  
pp. 121-128
Author(s):  
NAZAN AKDOĞAN ◽  
Keyword(s):  
Automorphism Group ◽  
Grassmann Algebra ◽  
Generating Set ◽  
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."

scholarly journals THE CENTRAL POLYNOMIALS OF THE INFINITE-DIMENSIONAL UNITARY AND NONUNITARY GRASSMANN ALGEBRAS

2010 ◽  
Vol 09 (05) ◽  
pp. 687-704 ◽  
Author(s):  
C. BEKH-OCHIR ◽  
S. A. RANKIN
Keyword(s):  
Grassmann Algebra ◽  
Infinite Field ◽  
Characteristic P ◽  
Infinite Dimensional ◽  
Grassmann Algebras ◽  
Unitary Algebra ◽  
Algebra Over A Field

We describe the T-space of central polynomials for both the unitary and the nonunitary infinite-dimensional Grassmann algebra over a field of characteristic p≠2 (infinite field in the case of the unitary algebra).

scholarly journals Symmetric polynomials in the variety generated by Grassmann algebras

2021 ◽  
Author(s):  
Nazan Akdoğan ◽  
Şehmus Fındık
Keyword(s):  
Symmetric Group ◽  
Free Algebra ◽  
Module Structure ◽  
Symmetric Polynomials ◽  
Associative Algebras ◽  
Generating Set ◽  
Commutator Ideal ◽  
Infinite Dimensional ◽  
Grassmann Algebras ◽  
Polynomial Formula

Let [Formula: see text] denote the variety generated by infinite-dimensional Grassmann algebras, i.e. the collection of all unitary associative algebras satisfying the identity [Formula: see text], where [Formula: see text]. Consider the free algebra [Formula: see text] in [Formula: see text] generated by [Formula: see text]. We call a polynomial [Formula: see text] symmetric if it is preserved under the action of the symmetric group [Formula: see text] on generators, i.e. [Formula: see text] for each permutation [Formula: see text]. The set of symmetric polynomials forms the subalgebra [Formula: see text] of invariants of the group [Formula: see text] in [Formula: see text]. The commutator ideal [Formula: see text] of the algebra [Formula: see text] has a natural left [Formula: see text]-module structure, and [Formula: see text] is a left [Formula: see text]-module. We give a finite free generating set for the [Formula: see text]-module [Formula: see text].

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

2016 ◽  
Vol 26 (06) ◽  
pp. 1125-1140 ◽  
Author(s):  
Lucio Centrone ◽  
Viviane Ribeiro Tomaz da Silva
Keyword(s):  
Positive Characteristic ◽  
Finite Abelian Group ◽  
Grassmann Algebra ◽  
Graded Algebra ◽  
Infinite Field ◽  
Infinite Dimensional ◽  
Graded Identities ◽  
Graded Polynomial Identities ◽  
Specific Formula ◽  
Gk Dimension

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].

scholarly journals A GENERATING SET FOR THE AUTOMORPHISM GROUP OF A GRAPH PRODUCT OF ABELIAN GROUPS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250003 ◽  
Author(s):  
L. J. CORREDOR ◽  
M. A. GUTIERREZ
Keyword(s):  
Automorphism Group ◽  
London Math ◽  
Abelian Groups ◽  
Graph Products ◽  
Graph Product ◽  
Finitely Generated ◽  
Generating Set ◽  
Labeled Graph

We find a set of generators for the automorphism group Aut G of a graph product G of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup Aut ⋆ G defined in [Automorphisms of graph products of abelian groups, to appear in Groups, Geometry and Dynamics]. We follow closely the plan of M. Laurence's paper [A generating set for the automorphism group of a graph group, J. London Math. Soc. (2)52(2) (1995) 318–334].

ℤ2-Graded Gelfand–Kirillov dimension of the Grassmann algebra

2014 ◽  
Vol 24 (03) ◽  
pp. 365-374 ◽  
Author(s):  
Lucio Centrone
Keyword(s):  
Grassmann Algebra ◽  
Infinite Dimensional ◽  
Gk Dimension ◽  
Kirillov Dimension

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

The automorphism groups of foliations with transverse linear connection

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Nina Zhukova ◽  
Anna Dolgonosova
Keyword(s):  
Automorphism Group ◽  
Lie Group ◽  
Automorphism Groups ◽  
Linear Connection ◽  
The Other ◽  
Riemannian Foliations ◽  
Infinite Dimensional ◽  
Infinite Dimensional Lie Group

AbstractThe category of foliations is considered. In this category morphisms are differentiable maps sending leaves of one foliation into leaves of the other foliation. We prove that the automorphism group of a foliation with transverse linear connection is an infinite-dimensional Lie group modeled on LF-spaces. This result extends the corresponding result of Macias-Virgós and Sanmartín Carbón for Riemannian foliations. In particular, our result is valid for Lorentzian and pseudo-Riemannian foliations.

Graded identities for algebras with elementary gradings over an infinite field

2022 ◽  
pp. 1-21
Author(s):  
Diogo Diniz ◽  
Claudemir Fidelis ◽  
Plamen Koshlukov
Keyword(s):  
Tensor Product ◽  
Associative Algebra ◽  
Positive Characteristic ◽  
Grassmann Algebra ◽  
Alternative Proof ◽  
Infinite Field ◽  
Infinite Dimensional ◽  
Graded Identities ◽  
Graded Polynomial Identities ◽  
Field Of Positive Characteristic

Abstract Let $F$ 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.

Infinite Dimensional DeWitt Supergroups and their Bodies

2014 ◽  
Vol 57 (2) ◽  
pp. 283-288 ◽  
Author(s):  
Ronald Fulp
Keyword(s):  
Banach Space ◽  
Lie Algebra ◽  
Lie Group ◽  
Grassmann Algebra ◽  
The Body ◽  
Exponential Mapping ◽  
Infinite Dimensional ◽  
Group Operation ◽  
Banach Lie Group ◽  
Quotient Manifold

AbstractFor Dewitt super groups G modeled via an underlying finitely generated Grassmann algebra it is well known that when there exists a body group BG compatible with the group operation on G, then, generically, the kernel K of the body homomorphism is nilpotent. This is not true when the underlying Grassmann algebra is infinitely generated. We show that it is quasi-nilpotent in the sense that as a Banach Lie group its Lie algebra κ has the property that for each a ∊ κ ada has a zero spectrum. We also show that the exponential mapping from κ to K is surjective and that K is a quotient manifold of the Banach space κ via a lattice in κ.

scholarly journals THE CENTRAL POLYNOMIALS OF THE FINITE-DIMENSIONAL UNITARY AND NONUNITARY GRASSMANN ALGEBRAS

2010 ◽  
Vol 03 (02) ◽  
pp. 235-249 ◽  
Author(s):  
C. Bekh-Ochir ◽  
S. A. Rankin
Keyword(s):  
Grassmann Algebra ◽  
Infinite Field ◽  
Characteristic P ◽  
Finite Dimensional ◽  
Grassmann Algebras ◽  
Unitary Algebra ◽  
Algebra Over A Field

We describe the T-space of central polynomials for both the unitary and the nonunitary finite dimensional Grassmann algebra over a field of characteristic p ≠ 2 (infinite field in the case of the unitary algebra).

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