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Kac-Moody algebras

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The authors study two kinds of actions of a discrete amenable Kac algebra. The first one is an action whose modular part is normal. They construct a new invariant which generalizes a characteristic invariant for a discrete group action, and we will present a complete classification. The second is a centrally free action. By constructing a Rohlin tower in an asymptotic centralizer, the authors show that the Connes-Takesaki module is a complete invariant.

Kac-Moody algebras. --- Lie algebras. --- Injective modules (Algebra)

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Ordered algebraic structures --- Kac-Moody algebras. --- Kac-Moody, Algèbres de. --- Representations of algebras. --- Représentations d'algèbres. --- Kac-Moody algebras --- Representations of algebras --- Algebra --- Algebras, Kac-Moody --- Lie algebras

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"The book starts with an outline of the classical Lie theory, used to set the scene. Part II provides a self-contained introduction to Kac-Moody algebras. The heart of the book is Part III, which develops an intuitive approach to the construction and fundamental properties of Kac-Moody groups. It is complemented by two appendices, respectively offering introductions to affine group schemes and to the theory of buildings. "- publisher

Buildings (Group theory). --- Kac-Moody algebras. --- Lie groups. --- Linear algebraic groups.

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"The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac-Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the (spin) rigid tableaux, and prove that they are equinumerous to the weight multiplicities of the highest weight modules under our consideration. These new classes of Young tableaux arise from crystal basis elements for dominant maximal weights of the integrable highest weight modules over affine Kac-Moody algebras. By applying combinatorics of tableaux such as the Robinson-Schensted algorithm and new insertion schemes, and using integrals over orthogonal groups, we reveal hidden structures in the sets of weight multiplicities and obtain explicit closed formulas for the weight multiplicities. In particular we show that some special families of weight multiplicities form the Pascal, Catalan, Motzkin, Riordan and Bessel triangles"--

Affine algebraic groups. --- Combinatorial analysis. --- Representations of algebras. --- Young tableaux. --- Kac-Moody algebras. --- Quantum groups.