Shiquan Ren Februry, 2010 Abstract: I want to write a summary of my study of complex semisimple Lie algebras, as my thesis for a bachelor degree. The classification of complex semisimple Lie algebras is obtained through the classification of simple root systems. Parallel to it, the classification of compact real Lie algebras can be obtained in a similar way. Only after these two classifications...

When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits Cartan subalgebra, Renault's work on subalgebras implies existence another description c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such subalgebra arises from subgroupoid $\mathcal{S}$ $\mathcal{G}$. this study relationship between origina...

In a paper by Xu, some simple Lie algebras of generalized Cartan type were constructed, using the mixtures of grading operators and down-grading operators. Among them, are the simple Lie algebras of generalized Witt type, which are in general nongraded and have no torus. In this paper, some representations of these simple Lie algebras of generalized Witt type are presented. 1. INTRODUCTION The ...

For a smooth manifold equipped with a compact Lie group action, we construct an equivariant cohomology theory which takes values in a vertex algebra, and contains the classical equivariant cohomology as a subalgebra. The main idea is to synthesize the algebraic approach to the classical equivariant cohomology theory due to H. Cartan, with the chiral de Rham algebra of Malikov-Schechtman-Vaintro...

Let G be a semisimple noncompact Lie group with finite center and K a maximal compact subgroup ofG andX = G/K the corresponding Riemannian symmetric space of noncompact type. We have a Cartan decomposition g = k+ p and we choose a maximal abelian subalgebra a of p. In what follows, Σ corresponds to the root system of g and Σ+ to the positive roots. We have the root space decomposition g = g0 + ...

In this paper we introduce a general axiomatic framework for algebras with triangular decomposition, which allows for a systematic study of the Bernstein-Gelfand-Gelfand Category O. Our axiomatic framework can be stated via three relatively simple axioms, and it encompasses a very large class of algebras studied in the literature. We term the algebras satisfying our axioms as regular triangular...

Let A ⊂M be a MASA in a II1 factor M. We describe the von Neumann subalgebra of M generated by A and its normalizer N (A) as the set Nw q (A) consisting of those elements m ∈M for which the bimodule AmA is discrete. We prove that two MASAs A and B are conjugate by a unitary u ∈ Nw q (A) iff A is discrete over B and B is discrete over A in the sense defined by Feldman and Moore [5]. As a consequ...

The nonlinear scalar-field realisation of w 1+∞ symmetry in d = 2 dimensions is studied in analogy to the nonlinear realisation of d = 4 conformal symmetry SO(4, 2). The w 1+∞ realisation is derived from a coset-space construction in which the divisor group is generated by the non-negative modes of the Virasoro algebra, with subsequent application of an infinite set of co-variant constraints. T...

These are informal notes of my talk during the Atlas of Lie groups workshop at AIM in Palo Alto, July 2003. All the ideas below are due to others, the primary references are Mickelsson [M] and Zhelebenko [Z1], [Z2]. Zhelebenko’s papers contain statements without proofs, therefore one should probably verify the results independently. Assuming the results, one gets a fairly explicit algorithm for...

A basic problem in the representation theory of a compact Lie group is to calculate the restriction of an irreducible representation of K to a closed subgroup. Most classical results on this problem concern connected groups. We’ll recall some of those, and talk about some of the modifications needed to work with disconnected groups. 1. Weight multiplicities Suppose g is a complex semisimple Lie...