scholarly journals Differential Hopf algebra structures on the universal enveloping algebra of a Lie algebra

1996 ◽  
Vol 37 (1) ◽  
pp. 524-532 ◽  
Author(s):  
N. van den Hijligenberg ◽  
R. Martini
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Universal Enveloping Algebra ◽  
Enveloping Algebra

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, $\depth X\leq \cat X$.

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.

scholarly journals A TRIANGULAR DEFORMATION OF THE TWO-DIMENSIONAL POINCARÉ ALGEBRA

1995 ◽  
Vol 10 (11) ◽  
pp. 873-883 ◽  
Author(s):  
M. KHORRAMI ◽  
A. SHARIATI ◽  
M.R. ABOLHASSANI ◽  
A. AGHAMOHAMMADI
Keyword(s):  
Hopf Algebra ◽  
Lorentz Transformation ◽  
Hopf Algebras ◽  
Universal Enveloping Algebra ◽  
Two Dimensional ◽  
R Matrix ◽  
Enveloping Algebra ◽  
Differential Structure ◽  
Algebra Of Functions ◽  
Poincaré Algebra

Contracting the h-deformation of SL(2, ℝ), we construct a new deformation of two-dimensional Poincaré's algebra, the algebra of functions on its group and its differential structure. It is seen that these dual Hopf algebras are isomorphic to each other. It is also shown that the Hopf algebra is triangular, and its universal R-matrix is also constructed explicitly. We then find a deformation map for the universal enveloping algebra, and at the end, give the deformed mass shells and Lorentz transformation.

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.

Classical Radicals of Invariants of Algebraic Skew Derivations

2022 ◽  
Vol 29 (01) ◽  
pp. 53-66
Author(s):  
Jeffrey Bergen ◽  
Piotr Grzeszczuk
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Jacobson Radical ◽  
Prime Radical ◽  
Nilpotent Radical ◽  
Enveloping Algebra ◽  
Locally Nilpotent ◽  
Nil Radical

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

scholarly journals On the Lie enveloping algebra of a pre-Lie algebra

2008 ◽  
Vol 2 (1) ◽  
pp. 147-167 ◽  
Author(s):  
J.-M. Oudom ◽  
D. Guin
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Rooted Trees ◽  
Enveloping Algebra ◽  
Associative Product ◽  
Planar Rooted Trees ◽  
Symmetric Brace Algebras

AbstractWe construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. It turns S(L) into a Hopf algebra which is isomorphic to the enveloping algebra of LLie. Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees.

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.

Sign in / Sign up

Export Citation Format

Share Document