scholarly journals Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra

1990 ◽  
Vol 3 (1) ◽  
pp. 257-257 ◽  
Author(s):  
George Lusztig
Keyword(s):  
Hopf Algebras ◽  
Universal Enveloping Algebra ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Quantized Universal Enveloping 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.

On the Hopf algebra dual of an enveloping algebra

1982 ◽  
Vol 91 (2) ◽  
pp. 215-224 ◽  
Author(s):  
Stephen Donkin
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebraic Groups ◽  
Field Extension ◽  
Smash Product ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Polycyclic Groups ◽  
Algebra Over A Field

In (1) it is claimed that the main results of that paper have applications to the representation theory of algebraic groups, of polycyclic groups and of Lie algebras. An application to algebraic groups is given in Corollary 6·4 of (1), the applications to polycyclic groups are given in (2), the purpose of this work is to deal with the outstanding case of enveloping algebras. To make use of the results of (1), in this context, we show that the Hopf algebra dual of the enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is quasi-affine (see § 1·5). This is done by an easy field extension argument and a generalization, to the Hopf algebra dual of the smash product of Hopf algebras, of Proposition 1·6·3 of (2) on the dual of the group algebra of a semidirect product of groups. Since this paper is aimed at those readers interested in enveloping algebras, the Hopf theoretic aspects are dealt with at a fairly leisurely pace.

scholarly journals Assembling Crystals of Type A

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Vladimir I. Danilov ◽  
Alexander V. Karzanov ◽  
Gleb A. Koshevoy
Keyword(s):  
Directed Graphs ◽  
Type A ◽  
Universal Enveloping Algebra ◽  
Combinatorial Structure ◽  
Efficient Procedure ◽  
Enveloping Algebra ◽  
Quantized Universal Enveloping Algebra

Regular An-crystals are certain edge-colored directed graphs, which are related to representations of the quantized universal enveloping algebra Uq(𝔰𝔩n+1). For such a crystal K with colors 1,2,…,n, we consider its maximal connected subcrystals with colors 1,…,n-1 and with colors 2,…,n and characterize the interlacing structure for all pairs of these subcrystals. This enables us to give a recursive description of the combinatorial structure of K via subcrystals and develop an efficient procedure of assembling K.

scholarly journals Primary decomposition in enveloping algebras

1981 ◽  
Vol 24 (2) ◽  
pp. 83-85 ◽  
Author(s):  
K. A. Brown ◽  
T. H. Lenagan
Keyword(s):  
Lie Algebra ◽  
Universal Enveloping Algebra ◽  
The Other ◽  
Primary Decomposition ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Factor Ring ◽  
Finite Dimensional ◽  
Other Hand

Recently, the first author and, independently, A. V. Jategaonkar have shown that every factor ring of U(g), the universal enveloping algebra of a finite dimensional complex Lie algebra, has a primary decomposition if g is solvable and almost algebraic. On the other hand, a suitable factor ring of U(SL(2, ℂ) fails to have a primary decomposition (1).

scholarly journals Classification theorem on irreducible representations of theq-deformed algebraU′q(son)

2005 ◽  
Vol 2005 (2) ◽  
pp. 225-262 ◽  
Author(s):  
N. Z. Iorgov ◽  
A. U. Klimyk
Keyword(s):  
Lie Algebra ◽  
Complete Classification ◽  
Universal Enveloping Algebra ◽  
Irreducible Representations ◽  
Classical Type ◽  
Enveloping Algebra ◽  
Root Of Unity ◽  
Complete Reducibility ◽  
Finite Dimensional

The aim of this paper is to give a complete classification of irreducible finite-dimensional representations of the nonstandardq-deformationU′q(son)(which does not coincide with the Drinfel'd-Jimbo quantum algebraUq(son)) of the universal enveloping algebraU(son(ℂ))of the Lie algebrason(ℂ)whenqis not a root of unity. These representations are exhausted by irreducible representations of the classical type and of the nonclassical type. The theorem on complete reducibility of finite-dimensional representations ofU′q(son)is proved.

scholarly journals COASSOCIATIVE LIE ALGEBRAS

2013 ◽  
Vol 55 (A) ◽  
pp. 195-215 ◽  
Author(s):  
D.-G. WANG ◽  
J. J. ZHANG ◽  
G. ZHUANG
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Compatibility Condition ◽  
Hopf Algebras ◽  
Universal Enveloping Algebra ◽  
Enveloping Algebra ◽  
Coalgebra Structure ◽  
Kirillov Dimension

AbstractA coassociative Lie algebra is a Lie algebra equipped with a coassociative coalgebra structure satisfying a compatibility condition. The enveloping algebra of a coassociative Lie algebra can be viewed as a coalgebraic deformation of the usual universal enveloping algebra of a Lie algebra. This new enveloping algebra provides interesting examples of non-commutative and non-cocommutative Hopf algebras and leads to the classification of connected Hopf algebras of Gelfand–Kirillov dimension four in Wang et al. (Trans. Amer. Math. Soc., to appear).

THE HEISENBERG FAMILY FOR THE PRINCIPAL GRADING OF $U_q(\hat{\cal G})$

2009 ◽  
Vol 23 (30) ◽  
pp. 5649-5656
Author(s):  
A. ZUEVSKY
Keyword(s):  
Universal Enveloping Algebra ◽  
Moody Algebra ◽  
Enveloping Algebra ◽  
Existence Conditions ◽  
Quantized Universal Enveloping Algebra

We describe existence conditions and explicitly construct elements for a Heisenberg family in the principal grading of the quantized universal enveloping algebra [Formula: see text] of an affine Kac–Moody algebra [Formula: see text] in the Drinfeld formulation.

Formally-radical Functions in Elements of a Nilpotent Lie Algebra and Noncommutative Localizations

2010 ◽  
Vol 17 (spec01) ◽  
pp. 749-788 ◽  
Author(s):  
Anar Dosi
Keyword(s):  
Lie Algebra ◽  
Holomorphic Functions ◽  
Universal Enveloping Algebra ◽  
Nilpotent Lie Algebra ◽  
Character Space ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Taylor Spectrum ◽  
Algebra Of Operators

In the present paper, we introduce the sheaf 𝔗𝔤 of germs of non-commutative holomorphic functions in elements of a finite-dimensional nilpotent Lie algebra 𝔤, which is a sheaf of non-commutative Fréchet algebras over the character space of 𝔤. We prove that 𝔗𝔤(D) is a localization over the universal enveloping algebra [Formula: see text] whenever D is a polydisk, which in turn allows to describe the Taylor spectrum of a supernilpotent Lie algebra of operators in terms of the transversality.

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