Overview of Root System of Type E7

Lattice Roots Root count: Short Hand Notation and Rules where where , Basis High Root Poset of Abelian Radical Geometrically, Abelian Radical is the face of the Root polyhedron which contains (at long end of Dynkin graph) and is parallel to the hyperplane spanned by the remaining roots. Algebraically, it consists of roots which -coefficient… Read More »

Motivation for Root Systems

Classification of Lattices Irreducibility and Root System A lattice is a configuration of evenly-spaced points in an ordinary Euclidean space such as that made by atoms (or molecules) distributed in a homogeneous solid. More precisely: A discrete collection of points is a lattice if a translation determined by any pair keeps the collection unchanged. The… Read More »

E6 Abelian Radical Root Tree

Lattice Roots Root count: Basis Restriction from Extension from High Root Abelian Radical Geometrically, Abelian Radical is the face of the Root polyhedron which contains a long-leg extreme root in the Dynkin graph and is parallel to the hyperplane spanned by the remaining roots. Here we choose that end to be . Algebraically, it consists… Read More »

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 »

Dynkin Diagrams

Graphic Representation of Root Systems Just like any finite set of Euclidean vectors, a Basis of a Root System is fully described by listing the length of each vector and the angle between each pair. We can draw a graph which nodes represent elements of and edges which width represent the fractional angles. In view… Read More »

Parabolic and Abelian Sets

Parabolic Sets Let be a Root System. A set is called Closed if any root which is a sum of 2 elements from is also in . Note that is a decomposition of into closed sets. More generally, we call Parabolic if is closed and . If is parabolic then its complement is also closed.… Read More »