Symmetric polynomials of algebras related with 2 × 2 generic traceless matrices

2021 ◽  
pp. 1-10
Author(s):  
Şehmus Fındık ◽  
Osman Kelekci̇
Keyword(s):  
Symmetric Group ◽  
Lie Algebra ◽  
Characteristic Zero ◽  
Polynomial Algebra ◽  
Symmetric Polynomials ◽  
Finite Sets ◽  
Free Generators ◽  
Polynomial Formula ◽  
Unitary Algebra ◽  
Algebra Over A Field

Let [Formula: see text] and [Formula: see text] be two generic traceless matrices of size [Formula: see text] with entries from a commutative associative polynomial algebra over a field [Formula: see text] of characteristic zero. Consider the associative unitary algebra [Formula: see text], and its Lie subalgebra [Formula: see text] generated by [Formula: see text] and [Formula: see text] over the field [Formula: see text]. It is well known that the center [Formula: see text] of [Formula: see text] is the polynomial algebra generated by the algebraically independent commuting elements [Formula: see text], [Formula: see text], [Formula: see text]. We call a polynomial [Formula: see text] symmetric, if [Formula: see text]. The set of symmetric polynomials is equal to the algebra [Formula: see text] of invariants of symmetric group [Formula: see text]. Similarly, we define the Lie algebra [Formula: see text] of symmetric polynomials in the Lie algebra [Formula: see text]. We give the description of the algebras [Formula: see text] and [Formula: see text], and we provide finite sets of free generators for [Formula: see text], and [Formula: see text] as [Formula: see text]-modules.

scholarly journals ELEMENTARY INVARIANTS FOR CENTRALIZERS OF NILPOTENT MATRICES

2009 ◽  
Vol 86 (1) ◽  
pp. 1-15 ◽  
Author(s):  
JONATHAN BROWN ◽  
JONATHAN BRUNDAN
Keyword(s):  
Lie Algebra ◽  
Characteristic Zero ◽  
Universal Enveloping Algebra ◽  
General Linear ◽  
Enveloping Algebra ◽  
Nilpotent Matrix ◽  
Nilpotent Matrices ◽  
Algebra Over A Field

AbstractWe construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.

ON SOME EMBEDDING OF LIE ALGEBRAS

2012 ◽  
Vol 11 (01) ◽  
pp. 1250017 ◽  
Author(s):  
E. CHIBRIKOV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Free Generators ◽  
Algebra Over A Field

In this paper we prove that every recursively presented Lie algebra over a field which is finite extention of its simple subfield can be embedded in a recursively presented Lie algebra defined by relations which are equalities of (nonassociative) words of generators and α + β = γ(α, β, γ are free generators).

scholarly journals Identities of the left-symmetric Witt algebras

2016 ◽  
Vol 26 (02) ◽  
pp. 435-450 ◽  
Author(s):  
Daniyar Kozybaev ◽  
Ualbai Umirbaev
Keyword(s):  
Lie Algebras ◽  
Characteristic Zero ◽  
Polynomial Algebra ◽  
Witt Algebra ◽  
Symmetric Algebras ◽  
Algebra Over A Field

Let [Formula: see text] be the polynomial algebra over a field [Formula: see text] of characteristic zero in the variables [Formula: see text] and [Formula: see text] be the left-symmetric Witt algebra of all derivations of [Formula: see text] [D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math. 4(3) (2006) 323–357]. We describe all right operator identities of [Formula: see text] and prove that the set of all algebras [Formula: see text], where [Formula: see text], generates the variety of all left-symmetric algebras. We also describe a class of general (not only right operator) identities for [Formula: see text].

scholarly journals FAITHFUL REPRESENTATIONS OF MINIMAL DIMENSION OF CURRENT HEISENBERG LIE ALGEBRAS

2009 ◽  
Vol 20 (11) ◽  
pp. 1347-1362 ◽  
Author(s):  
LEANDRO CAGLIERO ◽  
NADINA ROJAS
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Polynomial Algebra ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Abelian Subalgebras ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Minimal Dimension

Given a Lie algebra 𝔤 over a field of characteristic zero k, let μ(𝔤) = min{dim π : π is a faithful representation of 𝔤}. Let 𝔥m be the Heisenberg Lie algebra of dimension 2m + 1 over k and let k [t] be the polynomial algebra in one variable. Given m ∈ ℕ and p ∈ k [t], let 𝔥m, p = 𝔥m ⊗ k [t]/(p) be the current Lie algebra associated to 𝔥m and k [t]/(p), where (p) is the principal ideal in k [t] generated by p. In this paper we prove that [Formula: see text]. We also prove a result that gives information about the structure of a commuting family of operators on a finite dimensional vector space. From it is derived the well-known theorem of Schur on maximal abelian subalgebras of 𝔤𝔩(n, k ).

Palindromes in the free metabelian Lie algebras

2019 ◽  
Vol 29 (05) ◽  
pp. 885-891
Author(s):  
Şehmus Fındık ◽  
Nazar Şahi̇n Öğüşlü
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Reverse Order ◽  
Generating Set ◽  
Linear Basis ◽  
Free Metabelian Lie Algebra ◽  
Definition Of ◽  
Algebra Over A Field

A palindrome, in general, is a word in a fixed alphabet which is preserved when taken in reverse order. Let [Formula: see text] be the free metabelian Lie algebra over a field of characteristic zero generated by [Formula: see text]. We propose the following definition of palindromes in the setting of Lie algebras: An element [Formula: see text] is called a palindrome if it is preserved under the change of generators; i.e. [Formula: see text]. We give a linear basis and an explicit infinite generating set for the Lie subalgebra of palindromes.

The characters of the symmetric inverse semigroup

1957 ◽  
Vol 53 (1) ◽  
pp. 13-18 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Symmetric Group ◽  
Characteristic Zero ◽  
Semisimple Algebra ◽  
Matrix Representations ◽  
Natural Analogues ◽  
Completely Reducible ◽  
Symmetric Inverse Semigroup ◽  
Algebra Over A Field

There are two natural analogues of the symmetric group on n symbols in the theory of semigroups, namely, the set of all mappings of a set of n symbols into itself, and the set of all partial transformations of such a set, with the obvious definitions of multiplication. We are concerned here with the latter system. This is an inverse semigroup, and accordingly we call it the ‘symmetric inverse semigroup’. It gives rise to a semisimple algebra over a field of characteristic zero or a prime greater than n, and its matrix representations over such a field are thus completely reducible.

Uniqueness theorems for left-symmetric enveloping algebras

1996 ◽  
Vol 120 (2) ◽  
pp. 193-206
Author(s):  
J. R. Bolgar
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Automorphism Groups ◽  
Uniqueness Theorems ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Uniqueness Results ◽  
Symmetric Algebras ◽  
Algebra Over A Field

AbstractLet L be a Lie algebra over a field of characteristic zero. We study the uni versai left-symmetric enveloping algebra U(L) introduced Dan Segal in [9]. We prove some uniqueness results for these algebras and determine their automorphism groups, both as left-symmetric algebras and as Lie algebras.

The Second Cohomology of Current Algebras of General Lie Algebras

2008 ◽  
Vol 60 (4) ◽  
pp. 892-922 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Friedrich Wagemann
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Characteristic Zero ◽  
Cohomology Space ◽  
Trivial Module ◽  
Commutative Associative Algebra ◽  
Locally Convex ◽  
Algebra Over A Field

AbstractLet A be a unital commutative associative algebra over a field of characteristic zero, a Lie algebra, and a vector space, considered as a trivial module of the Lie algebra . In this paper, we give a description of the cohomology space in terms of easily accessible data associated with A and . We also discuss the topological situation, where A and are locally convex algebras.

scholarly journals The group of automorphisms of the Lie algebra of derivations of a polynomial algebra

2017 ◽  
Vol 16 (05) ◽  
pp. 1750088 ◽  
Author(s):  
V. V. Bavula
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Short Proof ◽  
Polynomial Algebra ◽  
Group Of Automorphisms ◽  
Cartan Type

A short proof is given of Rudakov’s result (announced in [A. N. Rudakov, Subalgebras and automorphisms of Lie algebras of Cartan type, Funktsional. Anal. i Prilozhen. 20(1) (1986) 83–84]), that the group of automorphisms of the Lie algebra [Formula: see text] of derivations of a polynomial algebra [Formula: see text], over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the polynomial algebra [Formula: see text].

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