These are notes for a seminar talk at the MIT-Northeastern Spring 2017 Double Affine Hecke Algebras and Elliptic Hall Algebras (DAHAEHA) Seminar.

0. Introduction 1 0.1. The Finite-Dimensional Case SU(n) 2 0.2. The Infinite-Dimensional Case LSU(n) 4 0.3. Plan of this Dissertation 13 1. Factorization and Coordinates for Finite-Dimensional Lie Groups 16 1.1. Lie Algebras 16 1.2. Lie Groups 19 1.3. Parametrizations with Sequences of Simple Roots 22 1.4. Parametrizations with Inversion Sets 26 2. Reduced Words for Infinite Elements of Affine ...

Assume that $(N,L)$, is a pair of finite dimensional nilpotent Lie algebras, in which $L$ is non-abelian and $N$ is an ideal in $L$ and also $mathcal{M}(N,L)$ is the Schur multiplier of the pair $(N,L)$. Motivated by characterization of the pairs $(N,L)$ of finite dimensional nilpotent Lie algebras by their Schur multipliers (Arabyani, et al. 2014) we prove some properties of a pair of nilpoten...

A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their non-trivial second cohomology classes which give rise to their central extensions (the affine Kac–Moody groups and Lie algebras). Loop groups embed into the group GL∞ of continuous automorphisms of C((t)), and these classes come from a second cohomology class of GL∞. In a sim...

Explicit expressions are presented for general branching functions for cosets of affine Lie algebras ĝ with respect to subalgebras ĝ′ for the cases where the corresponding finite dimensional algebras g and g′ are such that g is simple and g′ is either simple or sums of u(1) terms. A special case of the latter yields the string functions. Our derivation is purely algebraical and has its origin i...

This paper is a part of the series [KL]; however, it can be read independently of the first two parts. In [D3], Drinfeld proved the existence of an equivalence between a tensor category of representations of a quantum group over C[[ ro]] and a tensor category of representations of an undeformed enveloping algebra over C[[ro]] , in which the associativity constraints are given by the Knizhnik-Za...

We use an argument of Romans showing that every Virasoro construction leads to realizations of W3, to construct W3 realizations on arbitrary affine Lie algebras. Solutions are presented for generic values of the level as well as for specific values of the level but with arbitrary parameters. We give a detailed discussion of the ŝu(2)l-case. Finally, we discuss possible applications of these rea...