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.

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.

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.

THE FOX PROBLEM FOR FREE RESTRICTED LIE ALGEBRAS

2008 ◽  
Vol 18 (02) ◽  
pp. 271-283 ◽  
Author(s):  
HAMID USEFI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Free Groups ◽  
Enveloping Algebra ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras

Let L be a free restricted Lie algebra and R a restricted ideal of L. Denote by u(L) the restricted enveloping algebra of L and by ω(L) the associative ideal of u(L) generated by L. The purpose of this paper is to identify the subalgebra R ∩ ωn(L)ω(R) in terms of R only. This problem is the analogue of the Fox problem for free groups.

GENERALIZED SKLYANIN ALGEBRA

1991 ◽  
Vol 06 (39) ◽  
pp. 3635-3640 ◽  
Author(s):  
YAS-HIRO QUANO ◽  
AKIRA FUJII
Keyword(s):  
Lie Algebra ◽  
Universal Enveloping Algebra ◽  
Baxter Equation ◽  
Enveloping Algebra ◽  
Elliptic Solutions ◽  
Sklyanin Algebra

We propose a [Formula: see text]-Sklyanin algebra to shed new light on elliptic solutions of the Yang–Baxter equation. This object gives a quadratic generalization of the universal enveloping algebra of the Lie algebra [Formula: see text]. We also clarify the meaning of the parameters contained in it.

The depth and LS category of a topological space

2018 ◽  
Vol 123 (2) ◽  
pp. 220-238
Author(s):  
Yves Félix ◽  
Steve Halperin
Keyword(s):  
Topological Space ◽  
Hopf Algebra ◽  
Lie Algebra ◽  
Finite Type ◽  
Universal Enveloping Algebra ◽  
Primitive Elements ◽  
Enveloping Algebra ◽  
Cw Complex ◽  
Lusternik Schnirelmann ◽  
Simply Connected

The depth of an augmented ring $\varepsilon \colon A\to k $ is the least $p$, or ∞, such that \begin {equation*} \Ext _A^p(k , A)\neq 0. \end {equation*} When $X$ is a simply connected finite type CW complex, $H_*(\Omega X;\mathbb {Q})$ is a Hopf algebra and the universal enveloping algebra of the Lie algebra $L_X$ of primitive elements. It is known that $\depth H_*(\Omega X;\mathbb {Q}) \leq \cat X$, the Lusternik-Schnirelmann category of $X$. For any connected CW complex we construct a completion $\widehat {H}(\Omega X)$ of $H_*(\Omega X;\mathbb {Q})$ as a complete Hopf algebra with primitive sub Lie algebra $L_X$, and define $\depth X$ to be the least $p$ or ∞ such that \[ \Ext ^p_{UL_X}(\mathbb {Q}, \widehat {H}(\Omega X))\neq 0. \] Theorem: for any connected CW complex, $\depth X\leq \cat X$.

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