Graded contractions of Lie algebras, representations and tensor products

1992 ◽  
Author(s):  
J. Patera
Keyword(s):  
Lie Algebras ◽  
Tensor Products

scholarly journals A generalization of the alcove model and its applications

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Cristian Lenart ◽  
Arthur Lubovsky
Keyword(s):  
Lie Algebras ◽  
Energy Function ◽  
Tensor Products ◽  
Present Evidence ◽  
Semisimple Lie Algebras ◽  
Highest Weight ◽  
International Audience ◽  
Alcove Model

International audience The alcove model of the first author and Postnikov describes highest weight crystals of semisimple Lie algebras. We present a generalization, called the quantum alcove model, and conjecture that it uniformly describes tensor products of column shape Kirillov-Reshetikhin crystals, for all untwisted affine types. We prove the conjecture in types $A$ and $C$. We also present evidence for the fact that a related statistic computes the energy function. Le modèle des alcôves du premier auteur et Postnikov décrit les cristaux de plus haut poids des algèbres de Lie semi-simples. Nous présentons une généralisation, appelée le modèle des alcôves quantique, et nous conjecturons qu’il décrit dans une manière uniforme les produits tensoriels des cristaux de Kirillov-Reshetikhin de type colonne, pour toutes les types affines symétriques. Nous prouvons la conjecture dans les types $A$ et $C$. Nous fournissons aussi des preuves qu’une statistique associée donne la fonction d’énergie.

NEW EXCEPTIONAL $(A_1^{(1)})^{\oplus r} $ INVARIANTS AND THE ASSOCIATED GEPNER MODELS

1992 ◽  
Vol 07 (10) ◽  
pp. 2245-2264 ◽  
Author(s):  
JÜRGEN FUCHS ◽  
ALBRECHT KLEMM ◽  
MICHAEL G. SCHMIDT ◽  
DIRK VERSTEGEN
Keyword(s):  
Lie Algebras ◽  
Infinite Series ◽  
Tensor Products ◽  
Heterotic String ◽  
Partition Functions ◽  
Affine Lie Algebras ◽  
Affine Algebra ◽  
Modular Invariant ◽  
Fusion Rules ◽  
Modular Invariants

New modular invariant partition functions for tensor products of [Formula: see text] affine Lie algebras are presented. These exceptional modular invariants can be understood in terms of automorphisms of the fusion rules of the affine algebra or of its extensions. There are three isolated cases, as well as an infinite series of new invariants. As an application, the new modular invariants are employed to produce new Gepner type compactifications of the heterotic string.

scholarly journals Demazure Crystals, Kirillov-Reshetikhin Crystals, and the Energy Function

10.37236/2184 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Anne Schilling ◽  
Peter Tingley
Keyword(s):  
Lie Algebras ◽  
Energy Function ◽  
Tensor Products ◽  
Macdonald Polynomials ◽  
Close Connection ◽  
Whittaker Functions ◽  
Demazure Character ◽  
Demazure Crystals

It has previously been shown that, at least for non-exceptional Kac-Moody Lie algebras, there is a close connection between Demazure crystals and tensor products of Kirillov-Reshetikhin crystals. In particular, certain Demazure crystals are isomorphic as classical crystals to tensor products of  Kirillov-Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural affine grading on Demazure crystals with a combinatorially defined energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy function, which has applications to Macdonald polynomials and $q$-deformed Whittaker functions.

scholarly journals A non-abelian tensor product of Lie algebras

1991 ◽  
Vol 33 (1) ◽  
pp. 101-120 ◽  
Author(s):  
Graham J. Ellis
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Algebraic Theory ◽  
Tensor Products ◽  
Algebraic Structures ◽  
Commutative Algebras ◽  
Group Case ◽  
Do So ◽  
Product Of Groups

A generalized tensor product of groups was introduced by R. Brown and J.-L. Loday [6], and has led to a substantial algebraic theory contained essentially in the following papers: [6, 7, 1, 5, 11, 12, 13, 14, 18, 19, 20, 23, 24] ([9, 27, 28] also contain results related to the theory, but are independent of Brown and Loday's work). It is clear that one should be able to develop an analogous theory of tensor products for other algebraic structures such as Lie algebras or commutative algebras. However to do so, many non-obvious algebraic identities need to be verified, and various topological proofs (which exist only in the group case) have to be replaced by purely algebraic ones. The work involved is sufficiently non-trivial to make it interesting.

scholarly journals Categorification via blocks of modular representations for

2020 ◽  
pp. 1-29
Author(s):  
Vinoth Nandakumar ◽  
Gufang Zhao
Keyword(s):  
Lie Algebras ◽  
Positive Characteristic ◽  
Tensor Products ◽  
Derived Categories ◽  
Modular Representations ◽  
Combinatorial Approach ◽  
Standard Representation ◽  
Coherent Sheaves ◽  
Cotangent Bundles ◽  
Representations Of Lie Algebras

Abstract Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak {sl}_2$ , where they use singular blocks of category $\mathcal {O}$ for $\mathfrak {sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak {s}\mathfrak {l}_n$ over a field $\mathbf {k}$ of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig’s conjectures for representations of Lie algebras in positive characteristic.

scholarly journals Tensor products of positive definite quadratic forms, VII

1984 ◽  
Vol 96 ◽  
pp. 133-137 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Form ◽  
Lie Algebras ◽  
Quadratic Forms ◽  
Tensor Products ◽  
Positive Definite ◽  
Positive Definite Quadratic Form ◽  
Constant Multiple ◽  
The Third ◽  
Cartan Matrices

In this paper we generalize results of the third paper of this series. As a corollary we can show the following: Let Li (1 ≤ i ≤ n) be a positive definite quadratic form which is equivalent to one of Cartan matrices of Lie algebras of type An (n ≥ 2), Dn (n ≥ 4), E6, E7, E8 and assume that is positive definite quadratic forms and satisfies that rk Mt ≥ 2 and implies rk K or rk L = 1. Then we have n = m and Lt is equivalent to a constant multiple of Ms(i) for some permutation s. Therefore we get the uniqueness of decompositions with respect to tensor products in this case.

Tensor products and fusion rules

1994 ◽  
Vol 72 (7-8) ◽  
pp. 527-536 ◽  
Author(s):  
M. A. Walton
Keyword(s):  
Lie Algebras ◽  
Tensor Products ◽  
Fusion Rules ◽  
Semisimple Lie Algebras

Methods of decomposing tensor products of integrable representations of semisimple Lie algebras are described. They include a formula due to Zelobenko, the descendant of a conjecture made by Parthasarathy et al., and identities found by Feingold. The methods are adapted to the calculation of fusion rules in Wess–Zumino–Novikov–Witten models.

WZW fusion rules for the classical Lie algebras

1994 ◽  
Vol 72 (7-8) ◽  
pp. 342-344 ◽  
Author(s):  
C. J. Cummins ◽  
R. C. King
Keyword(s):  
Lie Algebras ◽  
Young Diagram ◽  
Tensor Products ◽  
Fusion Rules

Fusion rules for WZW models based on the Lie algebras su(n), sp(2n) and so(n) are considered. Previous work has shown how, in the cases of su(n) and sp(2n), fusion rules may be computed using Young diagram methods and applying fusion modification rules similar to the rank modification rules required for tensor products. In this note we extend these results to include the so(n) case.

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