Simple Cn modules with multiplicities 1 and applications

1994 ◽  
Vol 72 (7-8) ◽  
pp. 326-335 ◽  
Author(s):  
D. J. Britten ◽  
J. Hooper ◽  
F. W. Lemire
Keyword(s):  
Weyl Group ◽  
Tensor Products ◽  
Weight Space ◽  
One Dimensional ◽  
Infinite Dimensional ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Recursion Formulas ◽  
Completely Reducible ◽  
Index 2

In this paper we show that there exist exactly two nonequivalent simple infinite dimensional highest weight Cn modules having the property that every weight space is one dimensional. The tensor products of these modules with any finite-dimensional simple Cn module are proven to be completely reducible and we provide an explicit decomposition for such tensor products. As an application of these decompositions, we obtain two recursion formulas for computing the multiplicities of simple finite dimensional Cn modules. These formulas involve a sum over subgroups of index 2 in the Weyl group of Cn.

Note on Weight Spaces of Irreducible Linear Representations

1968 ◽  
Vol 11 (3) ◽  
pp. 399-403 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Irreducible Representation ◽  
Weight Function ◽  
Wide Class ◽  
Irreducible Representations ◽  
Weight Space ◽  
Closed Field ◽  
One Dimensional ◽  
Highest Weight ◽  
Linear Representations ◽  
Finite Dimensional

Let L denote a finite dimensional, simple Lie algebra over an algebraically closed field F of characteristic zero. It is well known that every weight space of an irreducible representation (ρ, V) admitting a highest weight function is finite dimensional. In a previous paper [2], we have established the existence of a wide class of irreducible representations which admit a one-dimensional weight space but no highest weight function. In this paper we show that the weight spaces of all such representations are finite dimensional.

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 SOLVABILITY OF THE F4 INTEGRABLE SYSTEM

2001 ◽  
Vol 16 (29) ◽  
pp. 4769-4801 ◽  
Author(s):  
KONSTANTIN G. BORESKOV ◽  
JUAN CARLOS LOPEZ VIEYRA ◽  
ALEXANDER V. TURBINER
Keyword(s):  
Weyl Group ◽  
Differential Operators ◽  
Coupling Constants ◽  
Dimensional Representation ◽  
Algebraic Form ◽  
Finite Dimensional Representation ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Exactly Solvable ◽  
New Variables

It is shown that the F4 rational and trigonometric integrable systems are exactly-solvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions are by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are invariant with respect to the Weyl group of F4 root system and can be obtained by averaging over an orbit of the Weyl group. An alternative way of finding these variables exploiting a property of duality of the F4 model is presented. It is demonstrated that in these variables the Hamiltonian of each model can be expressed as a quadratic polynomial in the generators of some infinite-dimensional Lie algebra of differential operators in a finite-dimensional representation. Both Hamiltonians preserve the same flag of spaces of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasi-exactly-solvable generalization of the rational F4 model depending on two continuous and one discrete parameters is found.

A New Family of Irreducible Representations of An

1975 ◽  
Vol 18 (4) ◽  
pp. 543-546 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Lie Algebra ◽  
Irreducible Representations ◽  
Weight Space ◽  
Complex Numbers ◽  
One Dimensional ◽  
Simple Lie Algebra ◽  
New Family ◽  
Highest Weight ◽  
The Family

For a simple Lie algebra L over the complex numbers ℂ all irreducible representations admitting a highest weight have been constructed and characterized for example in [3, 6]. In [1] Bouwer considered the family of all irreducible representations of L admitting at least one one-dimensional weight space (this includes, of course, all those having a highest weight space) and showed, by construction, that this is a strictly larger class of representations.

scholarly journals KOSTANT'S PROBLEM AND PARABOLIC SUBGROUPS

2009 ◽  
Vol 52 (1) ◽  
pp. 19-32 ◽  
Author(s):  
JOHAN KÅHRSTRÖM
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Verma Module ◽  
Parabolic Subgroups ◽  
Linear Transformations ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Simple Quotient ◽  
Finite Linear

AbstractLet be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let I be the corresponding semi-simple subalgebra of . Denote by WI the Weyl group of I and let w○ and wI○ be the longest elements of W and WI, respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight I-module LI(x) of highest weight x ⋅ 0, x ∈ WI, as the answer for the simple highest weight -module L(xwI○w○) of highest weight xwI○w○ ⋅ 0. We also give a new description of the unique quasi-simple quotient of the Verma module Δ(e) with the same annihilator as L(y), y ∈ W.

scholarly journals On q-Galois Extensions of Simple Rings

1966 ◽  
Vol 27 (2) ◽  
pp. 485-507 ◽  
Author(s):  
Hisao Tominaga
Keyword(s):  
Galois Theory ◽  
Galois Extensions ◽  
Late Professor ◽  
Simple Ring ◽  
Infinite Dimensional ◽  
Intermediate Ring ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
Ring Extensions

In 1952, the late Professor T. Nakayama succeeded in constructing the Galois theory for finite dimensional simple ring extensions [7]. And, we believe, the theory was essentially due to the following proposition: If a simple ring A is Galois and finite over a simple subring B then A is B′-A-completely reducible for any simple intermediate ring B′ of A/B [7, Lemmas 1.1 and 1.2]. Moreover, as was established in [5], Nakayama’s idea was still efficient in considering the infinite dimensional Galois theory of simple rings.

Completely Reducible Operator Algebras and Spectral Synthesis

1982 ◽  
Vol 34 (5) ◽  
pp. 1025-1035 ◽  
Author(s):  
Shlomo Rosenoer
Keyword(s):  
Operator Algebras ◽  
Spectral Synthesis ◽  
Invariant Subspaces ◽  
Bounded Operators ◽  
Von Neumann ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
Weakly Closed ◽  
Reductive Algebra

An algebra of bounded operators on a Hilbert space H is said to be reductive if it is unital, weakly closed and has the property that if M ⊂ H is a (closed) subspace invariant for every operator in , then so is M⊥. Loginov and Šul'man [6] and Rosenthal [9] proved that if is an abelian reductive algebra which commutes with a compact operator K having a dense range, then is a von Neumann algebra. Note that in this case every invariant subspace of is spanned by one-dimensional invariant subspaces. Indeed, the operator KK* commutes with . Hence its eigenspaces are invariant for , so that H is an orthogonal sum of the finite-dimensional invariant subspaces of From this our claim easily follows.

scholarly journals DUALITY IN INFINITE DIMENSIONAL FOCK REPRESENTATIONS

1999 ◽  
Vol 01 (02) ◽  
pp. 155-199 ◽  
Author(s):  
WEIQIANG WANG
Keyword(s):  
Dual Pair ◽  
Affine Algebra ◽  
Dual Pairs ◽  
Fock Representation ◽  
Infinite Dimensional ◽  
Infinite Rank ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Fock Representations ◽  
Infinite Dimensional Lie Algebra

We construct and study in detail various dual pairs acting on some infinite dimensional Fock representations between a finite dimensional classical Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix. We give explicit decompositions of a Fock representation into a direct sum of irreducible isotypic subspaces with respect to the action of a dual pair, present explicit formulas for the highest weight vectors and calculate the corresponding highest weights. We further outline applications of these dual pairs to the study of tensor products of modules of such an infinite dimensional Lie algebra.

scholarly journals An FDA-Based Approach for Clustering Elicited Expert Knowledge

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 184-204
Author(s):  
Carlos Barrera-Causil ◽  
Juan Carlos Correa ◽  
Andrew Zamecnik ◽  
Francisco Torres-Avilés ◽  
Fernando Marmolejo-Ramos
Keyword(s):  
Functional Data ◽  
Expert Knowledge ◽  
Probability Distributions ◽  
Knowledge Elicitation ◽  
Parallel Analysis ◽  
Data Sets ◽  
Dimensional Problem ◽  
Prior Distributions ◽  
Infinite Dimensional ◽  
Finite Dimensional

Expert knowledge elicitation (EKE) aims at obtaining individual representations of experts’ beliefs and render them in the form of probability distributions or functions. In many cases the elicited distributions differ and the challenge in Bayesian inference is then to find ways to reconcile discrepant elicited prior distributions. This paper proposes the parallel analysis of clusters of prior distributions through a hierarchical method for clustering distributions and that can be readily extended to functional data. The proposed method consists of (i) transforming the infinite-dimensional problem into a finite-dimensional one, (ii) using the Hellinger distance to compute the distances between curves and thus (iii) obtaining a hierarchical clustering structure. In a simulation study the proposed method was compared to k-means and agglomerative nesting algorithms and the results showed that the proposed method outperformed those algorithms. Finally, the proposed method is illustrated through an EKE experiment and other functional data sets.

scholarly journals Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Carlos A. M. André ◽  
João Dias
Keyword(s):  
Local Field ◽  
Unit Group ◽  
Basic Algebra ◽  
Local Fields ◽  
Dimensional Representation ◽  
Smooth Representation ◽  
One Dimensional ◽  
Finite Dimensional

Abstract We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

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