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Cartan matrix
About: Cartan matrix is a research topic. Over the lifetime, 1148 publications have been published within this topic receiving 26400 citations.
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Book•
01 Jan 1983TL;DR: The invariant bilinear form and the generalized casimir operator are integral representations of Kac-Moody algebras and the weyl group as mentioned in this paper, as well as a classification of generalized cartan matrices.
Abstract: Introduction Notational conventions 1 Basic definitions 2 The invariant bilinear form and the generalized casimir operator 3 Integrable representations of Kac-Moody algebras and the weyl group 4 A classification of generalized cartan matrices 5 Real and imaginary roots 6 Affine algebras: the normalized cartan invariant form, the root system, and the weyl group 7 Affine algebras as central extensions of loop algebras 8 Twisted affine algebras and finite order automorphisms 9 Highest-weight modules over Kac-Moody algebras 10 Integrable highest-weight modules: the character formula 11 Integrable highest-weight modules: the weight system and the unitarizability 12 Integrable highest-weight modules over affine algebras 13 Affine algebras, theta functions, and modular forms 14 The principal and homogeneous vertex operator constructions of the basic representation Index of notations and definitions References Conference proceedings and collections of paper
4,653 citations
Journal Article•
TL;DR: In this article, the authors extend this connection with finite dimensional algebra representation theory into a central theme, which they call the tilting theory of finite-dimensional algebras.
Abstract: This paper continues the program begun by us in [8]), [9] (see also [15], [18]) in which the authors have begun to exploit in the modular representation theory of semisimple algebraic groups some of the powerful techniques of the theory of derived categories. As noted in the above references, the Inspiration for this work comes both from geometry, in the form of the classic algebraic work of Bernstein-Beilinson-Deligne [1] on singular spaces and perverse sheaves, and from the tilting theory of finite dimensional algebras [2], [3], [13], [14]. The present paper broadens and extends this connection with finite dimensional algebra representation theory into a central theme.
811 citations
Book•
01 Jan 1999
TL;DR: The Weil Model and the Cartan Model were proposed by Cartan as discussed by the authors, who considered the Weil model as an extension of Cartan's formula and showed that it can be used in the context of equivariant cohomology in topology.
Abstract: 1 Equivariant Cohomology in Topology.- 3 The Weil Algebra.- 4 The Weil Model and the Cartan Model.- 5 Cartan's Formula.- 6 Spectral Sequences.- 7 Fermionic Integration.- 8 Characteristic Classes.- 9 Equivariant Symplectic Forms.- 10 The Thom Class and Localization.- 11 The Abstract Localization Theorem.- Notions d'algebre differentielle application aux groupes de Lie et aux varietes ou opere un groupe de Lie: Henri Cartan.- La transgression dans un groupe de Lie et dans un espace fibre principal: Henri Cartan.
513 citations
TL;DR: In this article, the authors define a Hopf algebra u over Fp (the field with p elements, p = prime) as the restricted enveloping algebra of a simple Lie algebra over F p and fix an indecomposable positive definite symmetric Cartan matrix.
Abstract: 0.1. An important role in the theory of modular representations is played by certain finite dimensional Hopf algebras u over Fp (the field with p elements, p = prime). Originally, u was defined (Curtis [3]) as the restricted enveloping algebra of a "simple" Lie algebra over Fp For our purposes, it will be more convenient to define u as follows. Let us fix an indecomposable positive definite symmetric Cartan matrix
477 citations
Book•
01 Sep 1994
TL;DR: In this article, the authors present Axioms of a Hopf Algebra and Enveloping Algebras of algebraic groups, as well as the Hopf Dual and the Rosso Form.
Abstract: I. Hopf Algebras.- 1.1 Axioms of a Hopf Algebra.- 1.2 Group Algebras and Enveloping Algebras.- 1.3 Adjoint Action.- 1.4 The Hopf Dual.- 1.5 Comments and Complements.- 2. Excerpts from the Classical Theory.- 2.1 Lie Algebras.- 2.2 Algebraic Lie Algebras.- 2.3 Algebraic Groups.- 2.4 Lie Algebras of Algebraic Groups.- 2.5 Comments and Complements.- 3. Encoding the Cartan Matrix.- 3.1 Quantum Weyl Algebras.- 3.2 The Drinfeld Double.- 3.3 The Rosso Form and the Casimir Invariant.- 3.4 The Classical Limit and the Shapovalev Form.- 3.5 Comments and Complements.- 4. Highest Weight Modules.- 4.1 The Jantzen Filtration and Sum Formula.- 4.2 Kac-Moody Lie Algebras.- 4.3 Integrable Modules for Uq(gc).- 4.4 Demazure Modules and Product Formulae.- 4.5 Comments and Complements.- 5. The Crystal Basis.- 5.1 Operators in the Crystal Limit.- 5.2 Crystals.- 5.3 Ad-invariant Filtrations, Twisted Actions and the Crystal Basis for Uq(n-).- 5.4 The Grand Loop.- 5.5 Comments and Complements.- 6. The Global Bases.- 6.1 The ? Operation and the Embedding Theorem.- 6.2 Globalization.- 6.3 The Demazure Property.- 6.4 Littelmann's Path Crystals.- 6.5 Comments and Complements.- 7. Structure Theorems for Uq(g).- 7.1 Local Finiteness for the Adjoint Action.- 7.2 Positivity of the Rosso Form.- 7.3 The Separation Theorem.- 7.4 Noetherianity.- 7.5 Comments and Complements.- 8. The Primitive Spectrum of Uq(g).- 8.1 The Poincare Series of the Harmonic Space.- 8.2 Factorization of the Quantum PRV Determinants.- 8.3 Verma Module Annihilators.- 8.4 Equivalence of Categories.- 8.5 Comments and Complements.- 9. Structure Theorems for Rq[G].- 9.1 Commutativity Relations.- 9.2 Surjectivity and Injectivity Theorems.- 9.3 The Adjoint Action.- 9.4 The R-Matrix.- 9.5 Comments and Complements.- 10. The Prime Spectrum of Rq[G].- 10.1 Highest Weight Modules.- 10.2 The Quantum Weyl Group.- 10.3 Prime and Primitive Ideals of Rq[G].- 10.4 Hopf Algebra Automorphisms.- 10.5 Comments and Complements.- A.2 Excerpts from Ring Theory.- A.3 Combinatorial Identities and Dimension Theory.- A.4 Remarks on Constructions of Quantum Groups.- A.5 Comments and Complements.- Index of Notation.
445 citations