scholarly journals Existence and dimensionality of simple weight modules for quantum enveloping algebras

1993 ◽  
Vol 48 (1) ◽  
pp. 35-40
Author(s):  
Zhiyong Shi
Keyword(s):  
Necessary Conditions ◽  
Simple Module ◽  
Weight Space ◽  
Semisimple Lie Algebra ◽  
Simple Modules ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Sufficient And Necessary Conditions

We give sufficient and necessary conditions for simple modules of the quantum group or the quantum enveloping algebra Uq(g) to have weight space decompositions, where g is a semisimple Lie algebra and q is a nonzero complex number. We show that(i) if q is a root of unity, any simple module of Uq(g) is finite dimensional, and hence is a weight module;(ii) if q is generic, that is, not a root of unity, then there are simple modules of Uq(g) which do not have weight space decompositions.Also the group of units of Uq(g) is found.

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 BGG CATEGORY FOR THE QUANTUM SCHRÖDINGER ALGEBRA

2020 ◽  
pp. 1-14
Author(s):  
GENQIANG LIU ◽  
YANG LI
Keyword(s):  
Complex Number ◽  
Full Subcategory ◽  
Weyl Algebra ◽  
Central Charge ◽  
Enveloping Algebra ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Quantum Weyl Algebra ◽  
Schrödinger Algebra

Abstract In 1996, a q-deformation of the universal enveloping algebra of the Schrödinger Lie algebra was introduced in Dobrev et al. [J. Phys. A 29 (1996) 5909–5918.]. This algebra is called the quantum Schrödinger algebra. In this paper, we study the Bernstein-Gelfand-Gelfand (BGG) category $\mathcal{O}$ for the quantum Schrödinger algebra $U_q(\mathfrak{s})$ , where q is a nonzero complex number which is not a root of unity. If the central charge $\dot z\neq 0$ , using the module $B_{\dot z}$ over the quantum Weyl algebra $H_q$ , we show that there is an equivalence between the full subcategory $\mathcal{O}[\dot Z]$ consisting of modules with the central charge $\dot z$ and the BGG category $\mathcal{O}^{(\mathfrak{sl}_2)}$ for the quantum group $U_q(\mathfrak{sl}_2)$ . In the case that $\dot z = 0$ , we study the subcategory $\mathcal{A}$ consisting of finite dimensional $U_q(\mathfrak{s})$ -modules of type 1 with zero action of Z. We directly construct an equivalence functor from $\mathcal{A}$ to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional $U_q(\mathfrak{s})$ -modules is wild.

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 CONFORMAL $\mathcal{s}$$\mathcal{l}$2 ENVELOPING ALGEBRAS AS AMBISKEW POLYNOMIAL RINGS

2002 ◽  
Vol 45 (1) ◽  
pp. 91-115 ◽  
Author(s):  
Martin O’Neill
Keyword(s):  
Characteristic Zero ◽  
Subject Classification ◽  
Polynomial Rings ◽  
Irreducible Representations ◽  
Prime Ideals ◽  
Closed Field ◽  
Enveloping Algebras ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Parameter Deformation

AbstractWe study a three parameter deformation $\mathcal{U}_{abc}$ of $\mathcal{U}(\mathfrak{sl}_2)$ introduced by Le Bruyn in 1995. Working over an arbitrary algebraically closed field of characteristic zero, we determine the centres, the finite-dimensional irreducible representations, and, when the parameter $a$ is not a non-trivial root of unity, the prime ideals of those $\mathcal{U}_{abc}$, with $ac\neq0$, which are conformal as ambiskew polynomial rings.AMS 2000 Mathematics subject classification: Primary 16W35; 17B37. Secondary 16S36; 16S80

Structure of Modules Induced from Simple Modules with Minimal Annihilator

2004 ◽  
Vol 56 (2) ◽  
pp. 293-309 ◽  
Author(s):  
Oleksandr Khomenko ◽  
Volodymyr Mazorchuk
Keyword(s):  
Simple Module ◽  
Verma Modules ◽  
Simple Complex ◽  
Parabolic Subalgebra ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Levi Factor ◽  
Generalized Verma Modules ◽  
Primitive Ideals ◽  
Irreducibility Criterion

AbstractWe study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.

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 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.

scholarly journals Primeness of the enveloping algebra of Hamiltonian superalgebras

1997 ◽  
Vol 56 (3) ◽  
pp. 483-488 ◽  
Author(s):  
Mark C. Wilson
Keyword(s):  
Bilinear Form ◽  
Characteristic Zero ◽  
Lie Superalgebra ◽  
Sufficient Condition ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

In 1990 Allen Bell presented a sufficient condition for the primeness of the universal enveloping algebra of a Lie superalgebra. Let Q be a nonsingular bilinear form on a finite-dimensional vector space over a field of characteristic zero. In this paper we show that Bell's criterion applies to the Hamiltonian Cartan type superalgebras determined by Q, and hence that their enveloping algebras are semiprimitive.

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 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.

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