Monthly Archives: October 2017

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 »

Geometry

Overview Term ‘Geometry’ is used here broadly to indicate applying a particular ‘axiomatic’ approach to describe some part of Physical World. We are interested in one particular approach, originated in late 19th century, based on the notion of an (abstract) ‘Group’. This is a departure from an earlier approach where axioms were statements about coincidence… Read More »