The Torsion Free Pieri Formula

1998 ◽  
Vol 50 (2) ◽  
pp. 266-289 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Tensor Product ◽  
Arbitrary Degree ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional ◽  
Pieri Formula ◽  
Free Modules ◽  
Central Characters ◽  
Torsion Free

AbstractCentral to the study of simple infinite dimensional g𝓵(n, C)-modules having finite dimensional weight spaces are the torsion free modules. All degree 1 torsion free modules are known. Torsion free modules of arbitrary degree can be constructed by tensoring torsion free modules of degree 1 with finite dimensional simple modules. In this paper, the central characters of such a tensor product module are shown to be given by a Pieri-like formula, complete reducibility is established when these central characters are distinct and an example is presented illustrating the existence of a nonsimple indecomposable submodule when these characters are not distinct.

Tensor Product Realizations of Simple Torsion Free Modules

2001 ◽  
Vol 53 (2) ◽  
pp. 225-243 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Complex Numbers ◽  
Infinite Dimensional ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Elementary Construction ◽  
Free Modules ◽  
Torsion Free

AbstractLet be a finite dimensional simple Lie algebra over the complex numbers C. Fernando reduced the classification of infinite dimensional simple -modules with a finite dimensional weight space to determining the simple torsion free -modules for of type A or C. Thesemodules were determined by Mathieu and using his work we provide a more elementary construction realizing each one as a submodule of an easily constructed tensor product module.

Graded torsion-free 𝔰𝔩2(ℂ)-modules of rank 2

2021 ◽  
pp. 2250229
Author(s):  
Yuri Bahturin ◽  
Abdallah Shihadeh
Keyword(s):  
Lie Algebras ◽  
Simple Lie Algebras ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Graded Modules ◽  
Finite Dimensional ◽  
Rank 2 ◽  
Torsion Free

In this paper, we explore the possibility of endowing simple infinite-dimensional [Formula: see text]-modules by the structure of graded modules. The gradings on the finite-dimensional simple modules over simple Lie algebras have been studied in 7, 8.

scholarly journals On Some Infinite Dimensional Representations of Semi-Simple Lie Algebras

1965 ◽  
Vol 25 ◽  
pp. 211-220 ◽  
Author(s):  
Hiroshi Kimura
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional

Let g be a semi-simple Lie algebra over an algebraically closed field K of characteristic 0. For finite dimensional representations of g, the following important results are known; 1) H1(g, V) = 0 for any finite dimensional g space V. This is equivalent to the complete reducibility of all the finite dimensional representations,2) Determination of all irreducible representations in connection with their highest weights.3) Weyl’s formula for the character of irreducible representations [9].4) Kostant’s formula for the multiplicity of weights of irreducible representations [6],5) The law of the decomposition of the tensor product of two irreducible representations [1].

Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras

1996 ◽  
Vol 39 (1) ◽  
pp. 111-114
Author(s):  
F. Okoh
Keyword(s):  
Rational Functions ◽  
Indecomposable Module ◽  
Torsion Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dynkin Diagrams ◽  
The Status ◽  
Hereditary Algebras ◽  
Divisible Modules ◽  
Torsion Free

AbstractIf R is a Dedekind domain, then div splits i.e.; the maximal divisible submodule of every R-module M is a direct summand of M. We investigate the status of this result for some finite-dimensional hereditary algebras. We use a torsion theory which permits the existence of torsion-free divisible modules for such algebras. Using this torsion theory we prove that the algebras obtained from extended Coxeter- Dynkin diagrams are the only such hereditary algebras for which div splits. The field of rational functions plays an essential role. The paper concludes with a new type of infinite-dimensional indecomposable module over a finite-dimensional wild hereditary algebra.

scholarly journals Cyclotomic Nazarov-Wenzl Algebras

2006 ◽  
Vol 182 ◽  
pp. 47-134 ◽  
Author(s):  
Susumu Ariki ◽  
Andrew Mathas ◽  
Hebing Rui
Keyword(s):  
Irreducible Representations ◽  
Arbitrary Field ◽  
Dimensional Algebra ◽  
Simple Modules ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Brauer Algebras

AbstractNazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain “cyclotomic quotients” of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank rn(2n−1)!! (when Ω is u-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.

Grothendieck Groups of a Class of Quantum Doubles

2008 ◽  
Vol 15 (03) ◽  
pp. 431-448 ◽  
Author(s):  
Yun Zhang ◽  
Feng Wu ◽  
Ling Liu ◽  
Hui-Xiang Chen
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Grothendieck Group ◽  
Ring Structure ◽  
Quantum Double ◽  
Simple Modules ◽  
Loewy Length ◽  
Finite Dimensional ◽  
Grothendieck Groups

Let k be a field and An(ω) be the Taft n2-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(An(ω)) of An(ω) is a ribbon Hopf algebra. We have constructed an n4-dimensional Hopf algebra Hn(p,q) which is isomorphic to D(An(ω)) if p ≠ 0 and q = ω-1, and studied the irreducible and finite-dimensional representations of Hn(1,q). In this paper, we continue our study of Hn(1,q), examine the Grothendieck group G0(Hn(1,q)) ≅ G0(D(An(ω)), and describe its ring structure. We also give the Loewy length of the tensor product of two simple modules over Hn(1,q).

scholarly journals Representations of infinite soluble groups

1983 ◽  
Vol 24 (1) ◽  
pp. 43-52 ◽  
Author(s):  
Ian M. Musson
Keyword(s):  
Group Algebra ◽  
Soluble Group ◽  
Polycyclic Group ◽  
Soluble Groups ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Torsion Free ◽  
Free Soluble Group

The purpose of this paper is to study the following two questions.(1) When does the group algebra of a soluble group have infinite dimensional irreducible modules?(2) When is the group algebra of a torsion free soluble group primitive?In relation to the first question, Roseblade [13] has proved that if G is a polycyclic group and k an absolute field then all irreducible kG-modules are finite dimensional. Here we prove a converse.

DEGENERATE TORSION-FREE G3-CONNECTIONS

1998 ◽  
Vol 09 (08) ◽  
pp. 945-955 ◽  
Author(s):  
QUO-SHIN ChI
Keyword(s):  
Moduli Space ◽  
Technical Condition ◽  
Analytic Torsion ◽  
Differential Systems ◽  
Exterior Differential Systems ◽  
Infinite Dimensional ◽  
Exterior Differential ◽  
Finite Dimensional ◽  
Torsion Free

Using exterior differential systems (EDS), Bryant proved that the moduli space of nondegenerate analytic torsion-free G3-connections depends on four functions in three variables. Here nondegeneracy is a technical condition which imposes the nonvanishing everywhere of a certain determinant pertinent to a torsion-free G3-connection for EDS to carry through. The finite-dimensional moduli of homogeneous torsion-free G3-connections are degenerate examples, where the determinant vanishes identically. We establish in fact that the moduli space of analytic inhomogeneous degenerate torsion-free G3-connections is infinite-dimensional.

scholarly journals Geometric Quantities of Manifolds with grassmann structure

2005 ◽  
Vol 180 ◽  
pp. 45-76 ◽  
Author(s):  
N. Bokan ◽  
P. Matzeu ◽  
Z. Rakić
Keyword(s):  
Tensor Product ◽  
Point Of View ◽  
Vector Spaces ◽  
Grassmann Manifolds ◽  
Simple Modules ◽  
Complete Decomposition ◽  
Curvature Tensors ◽  
Torsion Free ◽  
Product Vector

AbstractWe study geometry of manifolds endowed with a Grassmann structure which depends on symmetries of their curvature. Due to this reason a complete decomposition of the space of curvature tensors over tensor product vector spaces into simple modules under the action of the group G = GL(p, ℝ) ⊗ GL(q, ℝ) is given. The dimensions of the simple submodules, the highest weights and some projections are determined. New torsion-free connections on Grassmann manifolds apart from previously known examples are given. We use algebraic results to reveal obstructions to the existence of corresponding connections compatible with some type of normalizations and to enlighten previously known results from another point of view.

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