Splitting Criteria for -Modules Induced from a Parabolic and the Bernstein-Gelfand-Gelfand Resolution of a Finite Dimensional, Irreducible -Module

1980 ◽  
Vol 262 (2) ◽  
pp. 335 ◽  
Author(s):  
Alvany Rocha-Caridi
Keyword(s):  
Irreducible Module ◽  
Finite Dimensional

scholarly journals GENERIC IRREDUCIBLE REPRESENTATIONS OF FINITE-DIMENSIONAL LIE SUPERALGEBRAS

1994 ◽  
Vol 05 (03) ◽  
pp. 389-419 ◽  
Author(s):  
IVAN PENKOV ◽  
VERA SERGANOVA
Keyword(s):  
Irreducible Module ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Irreducible Representations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Necessary And Sufficient

A theory of highest weight modules over an arbitrary finite-dimensional Lie superalgebra is constructed. A necessary and sufficient condition for the finite-dimensionality of such modules is proved. Generic finite-dimensional irreducible representations are defined and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic finite-dimensional irreducible module over any finite-dimensional Lie superalgebra. The conjecture is proved for several classes of Lie superalgebras, in particular for all solvable ones, for all simple ones, and for certain semi-simple ones.

QUANTUM SUPERGROUPS, LINK POLYNOMIALS AND REPRESENTATION OF THE BRAID GENERATOR

1993 ◽  
Vol 05 (02) ◽  
pp. 345-361 ◽  
Author(s):  
J. R. LINKS ◽  
M. D. GOULD ◽  
R. B. ZHANG
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Symmetric Tensor ◽  
Irreducible Module ◽  
Closed Formula ◽  
Cyclic Module ◽  
Tensor Representation ◽  
Finite Dimensional ◽  
Quantum Supergroup ◽  
Rank 2

Unlike the quantum group case, it is shown that the braid generator σ is not always diagonalizable on V ⊗ V, V an irreducible module for a quantum supergroup. Nevertheless a generalization of the Reshetikhin form of the braid generator, obtained previously for quantum groups, is determined corresponding to every finite dimensional standard cyclic module V of a quantum supergroup. This result is applied to obtain a general closed formula for link polynomials arising from standard cyclic modules of a quantum supergroup belonging to a certain class. As explicit examples we determine link polynomials corresponding to the rank 2 symmetric tensor representation of Uq [gl(m|m)] and the defining representation of Uq [osp(2n|2n)].

A New Class of Representations of EALA Coordinated by Quantum Tori in Two Variables

2002 ◽  
Vol 45 (4) ◽  
pp. 672-685 ◽  
Author(s):  
S. Eswara Rao ◽  
Punita Batra
Keyword(s):  
Lie Algebras ◽  
Primitive Root ◽  
Irreducible Module ◽  
Affine Lie Algebras ◽  
Quantum Torus ◽  
New Class ◽  
Root Of Unity ◽  
Quantum Tori ◽  
Finite Dimensional

AbstractWe study the representations of extended affine Lie algebras where q is N-th primitive root of unity (ℂq is the quantum torus in two variables). We first prove that ⊕ for a suitable number of copies is a quotient of . Thus any finite dimensional irreducible module for ⊕ lifts to a representation of . Conversely, we prove that any finite dimensional irreducible module for comes from above. We then construct modules for the extended affine Lie algebras which is integrable and has finite dimensional weight spaces.

Irreducible Modules for Polycyclic Group Algebras

1981 ◽  
Vol 33 (4) ◽  
pp. 901-914 ◽  
Author(s):  
I. M. Musson
Keyword(s):  
Nilpotent Group ◽  
Irreducible Module ◽  
Polycyclic Group ◽  
Group Algebras ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules

If G is a polycyclic group and k an absolute field then every irreducible kG-module is finite dimensional [10], while if k is nonabsolute every irreducible module is finite dimensional if and only if G is abelian-by-finite [3]. However something more can be said about the infinite dimensional irreducible modules. For example P. Hall showed that if G is a finitely generated nilpotent group and V an irreducible kG-module, then the image of kZ in EndkGV is algebraic over k [3]. Here Z = Z(G) denotes the centre of G. It follows that the restriction Vz of V to Z is generated by finite dimensional kZ-modules. In this paper we prove a generalization of this result to polycyclic group algebras.We introduce some terminology.

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.

Representations of ω-Lie algebras and tailed derivations of Lie algebras

2020 ◽  
pp. 1-15
Author(s):  
Runxuan Zhang
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Module ◽  
Nonassociative Algebra ◽  
Complex Field ◽  
Indecomposable Modules ◽  
Finite Dimensional ◽  
Nilpotent Matrices ◽  
One To One

We study the representation theory of finite-dimensional [Formula: see text]-Lie algebras over the complex field. We derive an [Formula: see text]-Lie version of the classical Lie’s theorem, i.e., any finite-dimensional irreducible module of a soluble [Formula: see text]-Lie algebra is 1-dimensional (1D). We also prove that indecomposable modules of some 3D [Formula: see text]-Lie algebras could be parametrized by the complex field and nilpotent matrices. We introduce the notion of a tailed derivation of a nonassociative algebra [Formula: see text] and prove that if [Formula: see text] is a Lie algebra, then there exists a one-to-one correspondence between tailed derivations of [Formula: see text] and 1D [Formula: see text]-extensions of [Formula: see text].

Gröbner–Shirshov pair of irreducible modules over quantized enveloping algebra of type An

2018 ◽  
Vol 17 (02) ◽  
pp. 1850035
Author(s):  
Rabigul Tunyaz ◽  
Abdukadir Obul
Keyword(s):  
Irreducible Module ◽  
Free Module ◽  
Enveloping Algebra ◽  
Type Formula ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Quantized Enveloping Algebra ◽  
Double Free Module

In this paper, first, we give a Gröbner–Shirshov pair of finite-dimensional irreducible module [Formula: see text] over [Formula: see text] the quantized enveloping algebra of type [Formula: see text] by using the double free module method and the known Gröbner–Shirshov basis of [Formula: see text] Then, by specializing a suitable version of [Formula: see text] at [Formula: see text] we get a Gröbner–Shirshov basis of [Formula: see text] and get a Gröbner–Shirshov pair for the finite-dimensional irreducible module [Formula: see text] over [Formula: see text].

scholarly journals On the Gaudin model of type G2

2019 ◽  
Vol 21 (03) ◽  
pp. 1850012
Author(s):  
Kang Lu ◽  
Evgeny Mukhin
Keyword(s):  
Tensor Product ◽  
Bethe Ansatz ◽  
Irreducible Module ◽  
The Self ◽  
Gaudin Model ◽  
Bijective Correspondence ◽  
Dual Spaces ◽  
Finite Dimensional ◽  
Vector Representations ◽  
Evaluation Parameters

We derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G2. We compute explicit formulas for solutions of the Bethe ansatz equations associated to the tensor product of an arbitrary finite-dimensional irreducible module and the vector representation. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We show that the points of the spectrum of the Gaudin model in type G2 are in a bijective correspondence with self-self-dual spaces of polynomials. We study the set of all self-self-dual spaces — the self-self-dual Grassmannian. We establish a stratification of the self-self-dual Grassmannian with the strata labeled by unordered sets of dominant integral weights and unordered sets of nonnegative integers, satisfying certain explicit conditions. We describe closures of the strata in terms of representation theory.

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.

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