𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Basic Representations for Classical Affine Lie Algebras

✍ Scribed by Mirko Primc

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
342 KB
Volume
228
Category
Article
ISSN
0021-8693

No coin nor oath required. For personal study only.

✦ Synopsis

Presented here is a construction of certain bases of basic representations for classical affine Lie algebras. The starting point is a ‫-ޚ‬grading ᒄ s ᒄ q ᒄ q ᒄ y1 0 1 of a classical Lie algebra ᒄ and the corresponding decomposition ᒄ s ᒄ q ᒄ q ˜˜ỹ 1 0 ᒄ of the affine Lie algebra ᒄ. By using a generalization of the Frenkel᎐Kac vertex ˜1 operator formula for A Ž1. one can construct a spanning set of the basic ᒄ-module 1 in terms of monomials in basis elements of ᒄ and certain group element e. These ˜1 monomials satisfy certain combinatorial Rogers᎐Ramanujan type difference conditions arising from the vertex operator formula, and the main result is that these differences coincide with the energy function of a perfect crystal corresponding to the ᒄ -module ᒄ . The linear independence of the constructed spanning set of the 0 1 basic ᒄ-module is proved by using a crystal base character formula for standard modules due to S.

📜 SIMILAR VOLUMES

Crystals for Demazure Modules of Classic
Crystals for Demazure Modules of Classical Affine Lie Algebras
✍ Atsuo Kuniba; Kailash C. Misra; Masato Okado; Taichiro Takagi; Jun Uchiyama 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 336 KB

We study, in the path realization, crystals for Demazure modules of affine Lie algebras of types A Ž1. , B Ž1. , C Ž1. , D Ž1. , A Ž2. , A Ž2. , and D Ž2. . We find a special sequence of affine Weyl group elements for the selected perfect crystal, and show that if the highest weight is l⌳ , the Dem

Irreducible Representations for Toroidal
Irreducible Representations for Toroidal Lie Algebras
✍ Stephen Berman; Yuly Billig 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 284 KB

In this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal Lie algebras. To accomplish this we develop a general theory of ‫ޚ‬ n -graded Lie algebras with polynomial multiplication. We construct modules by the standard inducing

Real Characters for Demazure Modules of
Real Characters for Demazure Modules of Rank Two Affine Lie Algebras
✍ Yasmine B. Sanderson 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 211 KB

Using Littelmann's path model for highest weight representations of Kac᎐Moody algebras, we obtain explicit combinatorial expressions for certain specialized characters of all Demazure modules of A Ž1. and A Ž2. .