Category Archives: Lie Algebras

Orthogonal Overview

Overview Orthogonal Group is a symmetry group of a vector space equipped with a (non-degenerate) scalar product that is a group of linear transformations such that . If we present as a matrix with respect to some vector basis then is orthogonal when its coefficients satisfy certain algebraic equations which depend on how the basic… Read More »

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 »