# Category Archives: Lie Overview

References Intro to Lie Algebras by V. Kac

## Cartan Subalgebra Decomposition

is called Cartan subalgebra if it is nilpotent and is its own normalizer (i.e., implies ). Cartan algebras exist and are unique up to an automorphism of . For let denote eigenspace of . If then is called a root of (relative to ). Let be the set of roots. We have a decomposition of… Read More »

## Hypergeometric Lie Algebras

References GKZ Hypergeometric Structures Definition Let be a finite dimensional vector space, be its dual and be a finite set of nonzero vectors such that . Let where is a family of one dimensional vector spaces. In the following, will be a selection of . We make a Lie algebra using is Abelian is Abelian… Read More »

## Simple Lie Algebra of a Root System

Overview It turns out that the shape of a Root System contains information needed to construct a well defined Lie Algebra structure on a vector space extending the ambient vector space of the system and such that the automorphisms of the System extend to Lie Authomorphisms of . The scalar product on is, up to… Read More »