Open AccessBook
Lie Algebras of Finite and Affine Type
Reads0
Chats0
TLDR
In this article, the authors introduce the notion of universal enveloping algebras and derive universal constructions for simple Lie algeses and Kac-Moody algebraes.Abstract:
1. Basic concepts 2. Representations of soluble and nilpotent Lie algebras 3. Cartan subalgebras 4. The Cartan decomposition 5. The root systems and the Weyl group 6. The Cartan matrix and the Dynkin diagram 7. The existence and uniqueness theorems 8. The simple Lie algebras 9. Some universal constructions 10. Irreducible modules for semisimple Lie algebras 11. Further properties of the universal enveloping algebra 12. Character and dimension formulae 13. Fundamental modules for simple Lie algebras 14. Generalized Cartan matrices and Kac-Moody algebras 15. The classification of generalised Cartan matrices 16 The invariant form, root system and Weyl group 17. Kac-Moody algebras of affine type 18. Realisations of affine Kac-Moody algebras 19. Some representations of symmetrisable Kac-Moody algebras 20. Representations of affine Kac-Moody algebras 21. Borcherds Lie algebras Appendix.read more
Citations
More filters
Positive Representations with Zero Casimirs
Ivan Chi-Ho Ip,Ryuichi Man +1 more
TL;DR: In this paper , a new family of generalization of the positive representations of split-real quantum groups based on the degeneration of the Casimir operators acting as zero on some Hilbert spaces was constructed.
Dissertation
Root systems of Levi type for Lie algebras of affine type
TL;DR: In this article, the Levi type root system is defined for the case of affine types and the normalizers of parabolic subgroups of finite and affine Weyl groups of classical types are defined.
Journal ArticleDOI
Distributions Defined by $q$-Supernomials, Fusion Products, and Demazure Modules
TL;DR: This result is a central limit theorem for a serious class of graded tensors and serves as an indication towards universal behavior: the central string functions and the basic specialization of fusion and, in particular, Demazure modules behave asymptotically normal, as the number of fusions scale linearly in an asymPTotic parameter.
Dissertation
Weyl’s Construction of the Irreducible Regular Representations of the Complex Classical Groups
TL;DR: The main objective of as mentioned in this paper is to construct all the regular representations of the complex classical groups by using Weyl's method, which provides an explicit and concrete realization of each of the desired modules.
Straightening Identities in the Onsager Algebra
TL;DR: In this paper, straightening identities in the Onsager algebra have been studied and proved for linear combinations of products which are in a different order, allowing one to rewrite specific products of basis elements as linear combinations.