Category Archives: System of Type G

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 »

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 »

Equations

Let be an orthonormal basis. Let denote the group of cyclic permutations of . Then is the standard representation of a Root System of type with basis The set of simple roots orthogonal to the Highest Root is and the corresponding Parabolic set is . Its complement is a 5-dimensional Heisenberg Set The 4 maximal… Read More »

Useful Plugins & Links

Themes Iconic One Plugins WP Date Remover Hides creation date in post header Category Order and Taxonomy Terms Order Allows to order categories WP QuickLaTeX Enables LaTeX Tasks Order Posts Edit Timestamp Menu Navigate to Appearance > Menus. Select a name. Build menu items from categories by dragging between left and right panes. Build hierarchy… Read More »