Category Archives: Math

Affine Geometry

Formal Definition Affine theory deals with 2 sets: One which elements are called locations and another, which elements are called directions. The directions form an additive group. Every additive group carries a derived composition between its elements and scalars. It is defined recursively as (composition symbol implied). Greek letters will be reserved for scalar names… Read More »

Numbers

Formal Definition A Field is a set which elements are called Scalars. 2 scalars can be combined in 2 ways, One is called their sum and is denoted by . Another is called their product and is denoted  by (with composition symbol implied). Both compositions are associative and commutative and both have neutral elements called… Read More »

Language of Mathematics

Introduction Mathematics is a language. Like any other language it is written by combining characters into words and words into grammatical sentences. In this post we use plain English to describes this grammar and situations where the language can be useful. Mathematics make statements about elements of a Set. It does not matter what Sets… Read More »

Exterior Algebra

Interior Antiderivations If is a vector space then any form extends to a unique anti-derivation of the exterior algebra , i.e., for homogenous If is a basis of V, the dual basis and if denotes we have the following action on monomials (The sign is negative if the number of terms preceding the matching one… Read More »

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

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 »

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 »