Factorization in Loop Groups
نویسنده
چکیده
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 Weyl Groups 34 2.1. The Affine Weyl Group Waff 34 2.2. Infinite Reduced Words and Infinite Elements of Waff 40 2.3. Coweight Elements, and their Reduced Words in Type An−1 42 2.4. Permutation Realizations of the Classical Affine Weyl Groups 48 2.5. Reduced Words for the Coweight Elements of Other Classical Types 52 2.6. Lam & Pylyavskyy’s Method for Finding Reduced Words 72 3. Factorization and Coordinates for Unitary Loop Groups 77 3.1. Affine Lie Algebras 77 3.2. Loop Groups and Extensions 78 3.3. Parametrizations of Birkhoff Strata for Infinite Elements 81 3.4. Factorizations for Σ 1 83 References 85
منابع مشابه
Triple factorization of non-abelian groups by two maximal subgroups
The triple factorization of a group $G$ has been studied recently showing that $G=ABA$ for some proper subgroups $A$ and $B$ of $G$, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. In this paper we study two infinite classes of non-abelian finite groups $D_{2n}$ and $PSL(2,2^{n})$...
متن کامل$n$-factorization Property of Bilinear Mappings
In this paper, we define a new concept of factorization for a bounded bilinear mapping $f:Xtimes Yto Z$, depended on a natural number $n$ and a cardinal number $kappa$; which is called $n$-factorization property of level $kappa$. Then we study the relation between $n$-factorization property of level $kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity...
متن کاملTotal Positivity for Loop Groups Ii: Chevalley Generators
This is the second in a series of papers developing a theory of total positivity for loop groups. In this paper, we study infinite products of Chevalley generators. We show that the combinatorics of infinite reduced words underlies the theory, and develop the formalism of infinite sequences of braid moves, called a braid limit. We relate this to a partial order, called the limit weak order, on ...
متن کاملConstructions of Factorization Systems in Categories
In [2] we constructed homological localizations of spaces, groups, and 17"modules; here we generalize those constructions to give "factorization systems" and "homotopy factorization systems" for maps in categories. In Section 2 we recall the definition and basic properties of factorization systems, and in Section 3 we give our first existence theorem (3.1)for such systems. It can be viewed as a...
متن کاملFactorization in One-Loop Gauge Theory
Factorization properties of one-loop gauge theory amplitudes have been used as checks on explicitly computed amplitudes and in the construction of ansätze for higher-point ones. In massless theories, such as QCD at high energies, infrared divergences complicate factorization. Here we prove that factorization in such theories is described by a set of universal functions. In particular, a proof o...
متن کامل