scholarly journals A superdimension formula for 𝔤𝔩(m|n)-modules

2016 ◽  
Vol 15 (05) ◽  
pp. 1650080
Author(s):  
Michael Chmutov ◽  
Rachel Karpman ◽  
Shifra Reif
Keyword(s):  
Simple Formula ◽  
Simple Module ◽  
Character Formula ◽  
Algebraic Proof ◽  
Maximal Degree ◽  
Finite Dimensional ◽  
Associated Variety

We give a formula for the superdimension of a finite-dimensional simple [Formula: see text]-module using the Su–Zhang character formula. This formula coincides with the superdimension formulas proven by Weissauer and Heidersdorf–Weissauer. As a corollary, we obtain a simple algebraic proof of a conjecture of Kac–Wakimoto for [Formula: see text], namely, a simple module has nonzero superdimension if and only if it has maximal degree of atypicality. This conjecture was proven originally by Serganova using the Duflo–Serganova associated variety.

CHARACTERS OF STRONGLY GENERIC IRREDUCIBLE LIE SUPERALGEBRA REPRESENTATIONS

1998 ◽  
Vol 09 (03) ◽  
pp. 331-366 ◽  
Author(s):  
IVAN PENKOV
Keyword(s):  
Lie Superalgebra ◽  
Character Formula ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Simple Component

An explicit character formula is established for any strongly generic finite-dimensional irreducible [Formula: see text]-module, [Formula: see text] being an arbitrary finite-dimensional complex Lie superalgebra. This character formula had been conjectured earlier by Vera Serganova and the author for any generic irreducible finite-dimensional [Formula: see text]-module, i.e. such that its highest weight is far enough from the walls of the Weyl chambers. The condition of strong genericity, under which the conjecture is proved in this paper, is slightly stronger than genericity, but if in particular no simple component of [Formula: see text] is isomorphic to psq(n) for n ≥ 3 or to H(2k + 1) for k ≥ 2, strong genericity is equivalent to genericity.

scholarly journals Simple Finite-Dimensional Modules and Monomial Bases from the Gelfand-Testlin Patterns

2021 ◽  
pp. 60-65
Author(s):  
Amadou Keita
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Simple Module ◽  
Explicit Construction ◽  
Research Area ◽  
The Past ◽  
Finite Dimensional ◽  
Dimensional Module

One of the most important classes of Lie algebras is sl_n, which are the n×n matrices with trace 0. The representation theory for sl_n has been an interesting research area for the past hundred years and in it, the simple finite-dimensional modules have become very important. They were classified and Gelfand and Tsetlin actually gave an explicit construction of a basis for every simple finite-dimensional module. This paper extends their work by providing theorems and proofs and constructs monomial bases of the simple module.

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 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.

scholarly journals ON A FORMULA FOR THE PI-EXPONENT OF LIE ALGEBRAS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350069 ◽  
Author(s):  
A. S. GORDIENKO
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Simple Formula ◽  
Natural Generalization ◽  
Rational Action ◽  
Semisimple Lie Algebra ◽  
Arbitrary Group ◽  
Solvable Radical ◽  
Finite Dimensional ◽  
Algebra Over A Field

We prove that one of the conditions in Zaicev's formula for the PI-exponent and in its natural generalization for the Hopf PI-exponent, can be weakened. Using the modification of the formula, we prove that if a finite-dimensional semisimple Lie algebra acts by derivations on a finite-dimensional Lie algebra over a field of characteristic 0, then the differential PI-exponent coincides with the ordinary one. Analogously, the exponent of polynomial G-identities of a finite-dimensional Lie algebra with a rational action of a connected reductive affine algebraic group G by automorphisms, coincides with the ordinary PI-exponent. In addition, we provide a simple formula for the Hopf PI-exponent and prove the existence of the Hopf PI-exponent itself for H-module Lie algebras whose solvable radical is nilpotent, assuming only the H-invariance of the radical, i.e. under weaker assumptions on the H-action, than in the general case. As a consequence, we show that the analog of Amitsur's conjecture holds for G-codimensions of all finite-dimensional Lie G-algebras whose solvable radical is nilpotent, for an arbitrary group G.

Crossed products of C∗-algebras C(X,A) and their applications

2016 ◽  
Vol 27 (03) ◽  
pp. 1650029
Author(s):  
Jiajie Hua
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Crossed Products ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
C Algebras ◽  
Finite Dimensional ◽  
Tracial Rank ◽  
Stably Finite ◽  
Unital Simple

Let [Formula: see text] be an infinite compact metric space with finite covering dimension, let [Formula: see text] be a unital separable simple AH-algebra with no dimension growth, and denote by [Formula: see text] the [Formula: see text]-algebra of all continuous functions from [Formula: see text] to [Formula: see text] Suppose that [Formula: see text] is a minimal group action and the induced [Formula: see text]-action on [Formula: see text] is free. Under certain conditions, we show the crossed product [Formula: see text]-algebra [Formula: see text] has rational tracial rank zero and hence is classified by its Elliott invariant. Next, we show the following: Let [Formula: see text] be a Cantor set, let [Formula: see text] be a stably finite unital separable simple [Formula: see text]-algebra which is rationally TA[Formula: see text] where [Formula: see text] is a class of separable unital [Formula: see text]-algebras which is closed under tensoring with finite dimensional [Formula: see text]-algebras and closed under taking unital hereditary sub-[Formula: see text]-algebras, and let [Formula: see text]. Under certain conditions, we conclude that [Formula: see text] is rationally TA[Formula: see text] Finally, we classify the crossed products of certain unital simple [Formula: see text]-algebras by using the crossed products of [Formula: see text].

scholarly journals Dirac Induction for Rational Cherednik Algebras

2018 ◽  
Vol 2020 (17) ◽  
pp. 5155-5214
Author(s):  
Dan Ciubotaru ◽  
Marcelo De Martino
Keyword(s):  
Simple Module ◽  
Dirac Operators ◽  
Reflection Group ◽  
Composition Series ◽  
Cherednik Algebras ◽  
Global Indices ◽  
Finite Dimensional Vector Space ◽  
Complex Reflection Group ◽  
Finite Dimensional ◽  
Rational Cherednik Algebra

Abstract We introduce the local and global indices of Dirac operators for the rational Cherednik algebra $\mathsf{H}_{t,c}(G,\mathfrak{h})$, where $G$ is a complex reflection group acting on a finite-dimensional vector space $\mathfrak{h}$. We investigate precise relations between the (local) Dirac index of a simple module in the category $\mathcal{O}$ of $\mathsf{H}_{t,c}(G,\mathfrak{h})$, the graded $G$-character of the module, the Euler–Poincaré pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for $\mathsf{H}_{t,c}(G,\mathfrak{h})$ constructed from finite-dimensional $G$-modules. We define and compute the index of a Dirac operator on the integral-reflection module and show that the index is, in a sense, independent of the parameter function $c$. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised Dunkl–Opdam operators.

scholarly journals Categorification of the colored 𝔰𝔩3-invariant

2016 ◽  
Vol 25 (07) ◽  
pp. 1650038 ◽  
Author(s):  
Louis-Hadrien Robert
Keyword(s):  
Simple Formula ◽  
Finite Dimensional

We give explicit resolutions of all finite-dimensional simple [Formula: see text]-modules. We use these resolutions to categorify the colored [Formula: see text]-invariant of framed links via a complex of complexes of graded [Formula: see text]-modules.

H-Simple module algebras for eight-dimensional non-semisimple Hopf algebras

2016 ◽  
Vol 15 (07) ◽  
pp. 1650134
Author(s):  
Fengxia Gao ◽  
Shilin Yang
Keyword(s):  
Hopf Algebras ◽  
Characteristic Zero ◽  
Jacobson Radical ◽  
Simple Module ◽  
Simple Algebra ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Module Algebras

Let [Formula: see text] be an algebraically closed field of characteristic zero. For all eight-dimensional non-semisimple Hopf algebras [Formula: see text] which are either pointed or unimodular, we characterrize all finite-dimensional [Formula: see text]-simple module algebras. As a bonus of our approach, it is shown that for any [Formula: see text]-simple algebra, the nilpotent index of the Jacobson radical is at most three.

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