scholarly journals On a finite dimensional quasi-simple module

1970 ◽  
Vol 25 (4) ◽  
pp. 801-801 ◽  
Author(s):  
Kwangil Koh ◽  
Jiang Luh
Keyword(s):  
Simple Module ◽  
Finite Dimensional

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 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.

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.

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.

PRESENTATION OF QUANTUM GENERALIZED SCHUR ALGEBRAS

2005 ◽  
Vol 04 (05) ◽  
pp. 567-575
Author(s):  
LIBIN LI
Keyword(s):  
Tensor Product ◽  
Quantum Group ◽  
Simple Module ◽  
Schur Algebras ◽  
Schur Algebra ◽  
Root Of Unity ◽  
Annihilator Ideal ◽  
Finite Dimensional ◽  
Weight Property

We obtain the explicit generators of the annihilator ideal for the tensor product of any finite dimensional simple module over quantum group [Formula: see text], by using the weight property of ideals in [Formula: see text] when q is not a root of unity. As an application, we give a presentation of quantum generalized Schur algebra.

Non-Linear Dynamics of Cardiovascular Variability Signals

1994 ◽  
Vol 33 (01) ◽  
pp. 81-84 ◽  
Author(s):  
S. Cerutti ◽  
S. Guzzetti ◽  
R. Parola ◽  
M.G. Signorini
Keyword(s):  
Dimensional Space ◽  
Severe Heart Failure ◽  
Parametric Models ◽  
Deterministic Systems ◽  
Finite Dimensional ◽  
Return Maps ◽  
Pathological Conditions ◽  
Non Linear ◽  
Space State

Abstract:Long-term regulation of beat-to-beat variability involves several different kinds of controls. A linear approach performed by parametric models enhances the short-term regulation of the autonomic nervous system. Some non-linear long-term regulation can be assessed by the chaotic deterministic approach applied to the beat-to-beat variability of the discrete RR-interval series, extracted from the ECG. For chaotic deterministic systems, trajectories of the state vector describe a strange attractor characterized by a fractal of dimension D. Signals are supposed to be generated by a deterministic and finite dimensional but non-linear dynamic system with trajectories in a multi-dimensional space-state. We estimated the fractal dimension through the Grassberger and Procaccia algorithm and Self-Similarity approaches of the 24-h heart-rate variability (HRV) signal in different physiological and pathological conditions such as severe heart failure, or after heart transplantation. State-space representations through Return Maps are also obtained. Differences between physiological and pathological cases have been assessed and generally a decrease in the system complexity is correlated to pathological conditions.

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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