Category Archives: Root Systems

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 »

Summary of Main Facts

Notational Conventions An Euclidean Vector Space will be denoted by and its scalar product by . will be called the Norm of H. In the following, will refer to the dimension of . For let be such that the map is proportional to and normalized by imposing . By definition, . In other words, is… Read More »

Links

http://www.ams.org/journals/tran/1988-306-02/S0002-9947-1988-0933314-6/S0002-9947-1988-0933314-6.pdf

Overview

Root Systems is a specific collection of finite sets in an Euclidean Space. This collection has a simple axiomatic definition and can also be listed explicitly. The significance of Root Systems lies in their relation to the Theory of Lie Groups.