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.

scholarly journals TWO PERMANENTS IN THE UNIVERSAL ENVELOPING ALGEBRAS OF THE SYMPLECTIC LIE ALGEBRAS

2009 ◽  
Vol 20 (03) ◽  
pp. 339-368 ◽  
Author(s):  
MINORU ITOH
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Universal Enveloping Algebra ◽  
Irreducible Representations ◽  
General Linear ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Central Elements ◽  
Orthogonal Lie Algebra

This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra.

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.

Local left invertibility for operator tuples and noncommutative localizations

2008 ◽  
Vol 4 (1) ◽  
pp. 163-191 ◽  
Author(s):  
Anar Dosiev
Keyword(s):  
Lie Algebra ◽  
Normal Growth ◽  
Universal Enveloping Algebra ◽  
Fréchet Space ◽  
Canonical Homomorphism ◽  
Localization Problem ◽  
Nilpotent Lie Algebra ◽  
Operator Approach ◽  
Enveloping Algebra ◽  
Left Invertibility

AbstractIn the paper we propose an operator approach to the noncommutative Taylor localization problem based on the local left invertibility for operator tuples acting on a Fréchet space. We prove that the canonical homomorphism of the universal enveloping algebra of a nilpotent Lie algebra into its Arens-Michael envelope is the Taylor localization whenever has normal growth.

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.

Examples of uniform exponential growth in algebras

2017 ◽  
Vol 16 (12) ◽  
pp. 1750241
Author(s):  
Christopher A. Briggs
Keyword(s):  
Lie Algebra ◽  
Group Algebra ◽  
Exponential Growth ◽  
Universal Enveloping Algebra ◽  
Group Algebras ◽  
Graded Algebra ◽  
Enveloping Algebra ◽  
Restricted Lie Algebra ◽  
Fundamental Ideal ◽  
Uniform Exponential Growth

In this paper, we discuss the concept and examples of algebras of uniform exponential growth. We prove that Golod–Shafarevich algebras and group algebras of Golod–Shafarevich groups are of uniform exponential growth. We prove that uniform exponential growth of the universal enveloping algebra of a Lie algebra [Formula: see text] implies uniform exponential growth of [Formula: see text], and conversely should [Formula: see text] be graded by the natural numbers. We prove that a restricted Lie algebra is of uniform exponential growth if and only if its universal enveloping algebra is. We proceed to give several conditions equivalent to the uniform exponential growth of the graded algebra associated to a group algebra filtered by powers of its fundamental ideal.

ITERATED POWER INTERSECTIONS OF IDEALS IN RINGS OF ITERATED DIFFERENTIAL POLYNOMIALS

2013 ◽  
Vol 12 (07) ◽  
pp. 1350020
Author(s):  
PAVEL PŘÍHODA ◽  
GENA PUNINSKI
Keyword(s):  
Characteristic Zero ◽  
Proper Ideal ◽  
Projective Modules ◽  
Differential Polynomials ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Noetherian Domain ◽  
Solvable Lie Algebra ◽  
Standard Application ◽  
Algebra Over A Field

Let R be an n-iterated ring of differential polynomials over a commutative noetherian domain which is a ℚ-algebra. We will prove that for every proper ideal I of R, the (n + 1)-iterated intersection I(n + 1) of powers of I equals zero. A standard application includes the freeness of non-finitely generated projective modules over such rings. If I is a proper ideal of the universal enveloping algebra of a finite-dimensional solvable Lie algebra over a field of characteristic zero, then we will improve the above estimate by showing that I(2) = 0.

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