scholarly journals Separation of Variables for

2005 ◽  
Vol 48 (4) ◽  
pp. 587-600 ◽  
Author(s):  
Samuel A. Lopes
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Analogous Result ◽  
Separation Of Variables ◽  
Positive Part ◽  
Semisimple Lie Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Complex Semisimple ◽  
Quantized Enveloping Algebra

AbstractLet be the positive part of the quantized enveloping algebra . Using results of Alev–Dumas and Caldero related to the center of , we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra U(g) of a complex semisimple Lie algebra g, and also of an analogous result of Joseph–Letzter for the quantum algebra Ŭq(g). Of greater importance to its representation theory is the fact that is free over a larger polynomial subalgebra N in n variables. Induction from N to provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules.

scholarly journals KRULL DIMENSION OF AFFINOID ENVELOPING ALGEBRAS OF SEMISIMPLE LIE ALGEBRAS

2013 ◽  
Vol 55 (A) ◽  
pp. 7-26
Author(s):  
KONSTANTIN ARDAKOV ◽  
IAN GROJNOWSKI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Krull Dimension ◽  
Semisimple Lie Algebra ◽  
Semisimple Lie Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Complex Semisimple

AbstractUsing Beilinson–Bernstein localisation, we give another proof of Levasseur's theorem on the Krull dimension of the enveloping algebra of a complex semisimple Lie algebra. The proof also extends to the case of affinoid enveloping algebras.

Subalgebras Which Appear in Quantum Iwasawa Decompositions

1997 ◽  
Vol 49 (6) ◽  
pp. 1206-1223 ◽  
Author(s):  
Gail Letzter
Keyword(s):  
Lie Algebra ◽  
Semisimple Lie Algebra ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebra

AbstractLet g be a semisimple Lie algebra. Quantum analogs of the enveloping algebra of the fixed Lie subalgebra are introduced for involutions corresponding to the negative of a diagram automorphism. These subalgebras of the quantized enveloping algebra specialize to their classical counterparts. They are used to form an Iwasawa type decompostition and begin a study of quantum Harish-Chandra modules.

scholarly journals An Algorithm to Compute the Canonical Basis of an Irreducible Module Over a Quantized Enveloping Algebra

2003 ◽  
Vol 6 ◽  
pp. 105-118 ◽  
Author(s):  
Willem A. de Graaf
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Canonical Basis ◽  
Irreducible Module ◽  
Semisimple Lie Algebra ◽  
Canonical Bases ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Quantized Enveloping Algebra

AbstractThe paper describes an algorithm to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for any module that is constructed as a submodule of a tensor product of modules with known canonical bases.

scholarly journals Vanishing resonance and representations of Lie algebras

2015 ◽  
Vol 2015 (706) ◽  
Author(s):  
Stefan Papadima ◽  
Alexander I. Suciu
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Classical Representation ◽  
Semisimple Lie Algebra ◽  
Representations Of Lie Algebras ◽  
Complex Semisimple ◽  
Resonance Varieties

AbstractWe explore a relationship between the classical representation theory of a complex, semisimple Lie algebra 𝔤 and the resonance varieties

scholarly journals Homomorphisms between standard modules over finite-type KLR algebras

2017 ◽  
Vol 153 (3) ◽  
pp. 621-646 ◽  
Author(s):  
Alexander S. Kleshchev ◽  
David J. Steinberg
Keyword(s):  
Representation Theory ◽  
Finite Type ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
Positive Part ◽  
Enveloping Algebra ◽  
Klr Algebras ◽  
Quantized Enveloping Algebra ◽  
Standard Module

Khovanov–Lauda–Rouquier (KLR) algebras of finite Lie type come with families of standard modules, which under the Khovanov–Lauda–Rouquier categorification correspond to PBW bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of a standard module are injective. We present applications to the extensions between standard modules and modular representation theory of KLR algebras.

A Frobenius homomorphism for Lusztig’s quantum groups for arbitrary roots of unity

2016 ◽  
Vol 18 (03) ◽  
pp. 1550040 ◽  
Author(s):  
Simon Lentner
Keyword(s):  
Lie Algebra ◽  
Roots Of Unity ◽  
Semisimple Lie Algebra ◽  
Braided Category ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Nichols Algebras ◽  
Divided Powers

For a finite-dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite-dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite-dimensional Hopf kernel and with the image of the universal enveloping algebra. In this article, we define and completely describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases, the Frobenius–Lusztig kernel is associated to a different Lie algebra than the initial Lie algebra. Moreover, the Frobenius homomorphism often switches short and long roots and/or maps to a braided category.

scholarly journals A note on the zeroth products of Frenkel–Jing operators

2017 ◽  
Vol 16 (03) ◽  
pp. 1750053 ◽  
Author(s):  
Slaven Kožić
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Quantum Vertex ◽  
Algebra Theory

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

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.

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