On the classification of primitive ideals in the enveloping algebra of a semisimple lie algebra

1983 ◽  
pp. 30-76 ◽  
Author(s):  
A. Joseph
Keyword(s):  
Lie Algebra ◽  
Semisimple Lie Algebra ◽  
Enveloping Algebra ◽  
Primitive Ideals

scholarly journals Chern characters, reduced ranks and -modules on the flag variety

1994 ◽  
Vol 37 (3) ◽  
pp. 477-482 ◽  
Author(s):  
T. J. Hodges ◽  
M. P. Holland
Keyword(s):  
Lie Algebra ◽  
Euler Characteristic ◽  
Maximal Ideal ◽  
Chern Character ◽  
Flag Variety ◽  
Central Character ◽  
Geometric Description ◽  
Semisimple Lie Algebra ◽  
Enveloping Algebra ◽  
Chern Characters

Let D be the factor of the enveloping algebra of a semisimple Lie algebra by its minimal primitive ideal with trival central character. We give a geometric description of the Chern character ch: K0(D)→HC0(D) and the state (of the maximal ideal m) s: K0(D)→K0(D/m) = ℤ in terms of the Euler characteristic χ:K0()→ℤ, where is the associated flag variety.

scholarly journals Primitive ideals of the quantised enveloping algebra of a complex lie algebra

1994 ◽  
Vol 49 (1) ◽  
pp. 81-84 ◽  
Author(s):  
Sei-Qwon Oh
Keyword(s):  
Lie Algebra ◽  
Enveloping Algebra ◽  
Primitive Ideals

In this paper, we characterise all primitive ideals of the quantised enveloping algebra Uq[sl(2, ℂ)] of the complex Lie algebra sl(2, ℂ) and show how they are similar to those of U[sl(2, ℂ)], the enveloping algebra of sl(2, ℂ).

scholarly journals Rings with involution whose symmetric elements are central

1980 ◽  
Vol 3 (2) ◽  
pp. 247-253
Author(s):  
Taw Pin Lim
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Semisimple Lie Algebra ◽  
Direct Sum ◽  
Rings With Involution ◽  
Torsion Free

In a ringRwith involution whose symmetric elementsSare central, the skew-symmetric elementsKform a Lie algebra over the commutative ringS. The classification of such rings which are2-torsion free is equivalent to the classification of Lie algebrasKoverSequipped with a bilinear formfthat is symmetric, invariant and satisfies[[x,y],z]=f(y,z)x−f(z,x)y. IfSis a field of char≠2,f≠0anddimK>1thenKis a semisimple Lie algebra if and only iffis nondegenerate. Moreover, the derived algebraK′is either the pure quaternions overSor a direct sum of mutually orthogonal abelian Lie ideals ofdim≤2.

Voronoi Domains and Dual Cells in the Generalized Kaleidoscope with Applications to Root and Weight Lattices

1995 ◽  
Vol 47 (3) ◽  
pp. 573-605 ◽  
Author(s):  
R. V. Moody ◽  
J. Patera
Keyword(s):  
General Theory ◽  
Lie Algebra ◽  
Reflection Groups ◽  
Semisimple Lie Algebra ◽  
Voronoi Cells ◽  
Special Cases

AbstractWe give a uniform description, in terms of Coxeter diagrams, of the Voronoi domains of the root and weight lattices of any semisimple Lie algebra. This description provides a classification not only of all the facets of these Voronoi domains but simultaneously a classification of their dual or Delaunay cells and their facets. It is based on a much more general theory that we develop here providing the same sort of information in the setting of chamber geometries defined by arbitrary reflection groups. These generalized kaleidoscopes include the classical spherical, Euclidean, and hyperbolic kaleidoscopes as special cases. We prove that under certain conditions the Delaunay cells are Voronoi cells for the vertices of the Voronoi complex. This leads to the description in terms of Wythoff polytopes of the Voronoi cells of the weight lattices.

scholarly journals CLASSIFICATION OF PM QUIVER HOPF ALGEBRAS

2007 ◽  
Vol 06 (06) ◽  
pp. 919-950 ◽  
Author(s):  
SHOUCHUAN ZHANG ◽  
YAO-ZHONG ZHANG ◽  
HUI-XIANG CHEN
Keyword(s):  
Abelian Group ◽  
Hopf Algebras ◽  
Finite Abelian Group ◽  
Complex Field ◽  
Ground Field ◽  
Semisimple Lie Algebra ◽  
Enveloping Algebra ◽  
Complex Semisimple ◽  
Nichols Algebras

We describe certain quiver Hopf algebras by parameters. This leads to the classification of multiple Taft algebras as well as pointed Yetter–Drinfeld modules and their corresponding Nichols algebras. In particular, when the ground-field k is the complex field and G is a finite abelian group, we classify quiver Hopf algebras over G, multiple Taft algebras over G and Nichols algebras in [Formula: see text]. We show that the quantum enveloping algebra of a complex semisimple Lie algebra is a quotient of a semi-path Hopf algebra.

scholarly journals Affinoid Dixmier Modules and the Deformed Dixmier-Moeglin Equivalence

2021 ◽  
Author(s):  
Adam Jones
Keyword(s):  
Representation Theory ◽  
Lie Groups ◽  
Lie Algebra ◽  
Algebraic Structure ◽  
Verma Module ◽  
Finitely Generated ◽  
Enveloping Algebra ◽  
Primitive Ideals

AbstractThe affinoid enveloping algebra $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K of a free, finitely generated $\mathbb {Z}_{p}$ ℤ p -Lie algebra ${\mathscr{L}}$ L has proven to be useful within the representation theory of compact p-adic Lie groups, and we aim to further understand its algebraic structure. To this end, we define the notion of a Dixmier module over $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K , a generalisation of the Verma module, and we prove that when ${\mathscr{L}}$ L is nilpotent, all primitive ideals of $\widehat {U({\mathscr{L}})}_{K}$ U ( L ) ̂ K can be described in terms of annihilator ideals of Dixmier modules. Using this, we take steps towards proving that this algebra satisfies a version of the classical Dixmier-Moeglin equivalence.

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