Book•
Lie Algebras of Finite and Affine Type
01 Jan 2005-
TL;DR: 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.
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01 Jan 2015
TL;DR: The lecture notes for the 5M reading course "Lie groups, Lie algebras, and their representations" at the University of Glasgow, autumn 2015 as mentioned in this paper were also available.
Abstract: These are the lecture notes for the 5M reading course ”Lie groups, Lie algebras, and their representations” at the University of Glasgow, autumn 2015.
157 citations
Cites background from "Lie Algebras of Finite and Affine T..."
...Other sources that treat the material in these notes are [1], [2], [4], [9] and [7]....
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TL;DR: The universal Askey-Wilson algebra AW=AW(3) as discussed by the authors is a central extension of AW, obtained from AW by reinterpreting certain parameters as central elements in the algebra.
Abstract: In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension $\Delta$ of AW, obtained from AW by reinterpreting certain parameters as central elements in the algebra. We call $\Delta$ the {\it universal Askey-Wilson algebra}. We give a faithful action of the modular group ${\rm {PSL}}_2({\mathbb Z})$ on $\Delta$ as a group of automorphisms. We give a linear basis for $\Delta$. We describe the center of $\Delta$ and the 2-sided ideal $\Delta[\Delta,\Delta]\Delta$. We discuss how $\Delta$ is related to the $q$-Onsager algebra.
106 citations
Book•
01 Jan 2012TL;DR: In this paper, the point particle, the quantum superstring, the classical bosonic string, the light-cone approach, and the spinors of the classical superstring are discussed.
Abstract: Preface 1. The point particle 2. The classical bosonic string 3. The quantum bosonic string 4. The light-cone approach 5. Clifford algebras and spinors 6. The classical superstring 7. The quantum superstring 8. Conformal symmetry and two-dimensional field theory 9. Conformal symmetry and string theory 10. String compactification and the heterotic string 11. The physical states and the no ghost theorem 12. Gauge covariant string theory 13. Supergravity theories in 4, 10 and 11 dimensions 14. Brane dynamics 15. D-branes 16. String theory and Lie algebras 17. Symmetries of string theory 18. String interactions Appendices Index.
104 citations
TL;DR: Using the formula for the universal R-matrix proposed by Khoroshkin and Tolstoy, this article gave a detailed derivation of L-operators for the quantum groups associated with the generalized Cartan matrices A(1)1 and A( 1)2.
Abstract: Using the formula for the universal R-matrix proposed by Khoroshkin and Tolstoy, we give a detailed derivation of L-operators for the quantum groups associated with the generalized Cartan matrices A(1)1 and A(1)2.
79 citations
TL;DR: In this article, the authors classified the Heisenberg action on the Fock space via the category O of cyclotomic rational double affine Hecke algebras.
Abstract: In this paper we categorify the Heisenberg action on the Fock space via the category O of cyclotomic rational double affine Hecke algebras. This permits us to relate the filtration by the support on the Grothendieck group of O to a representation theoretic grading defined using the Heisenberg action. This implies a recent conjecture of Etingof.
71 citations