Monthly Archives: September 2017

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 »

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 »

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 »

Equations

Standard representation of a Root System of type can be described in terms of an orthonormal basis as the set of non-zero differences For any such that we define a Parabolic Set by restricting the index of the first term to the interval and that of the second term to the complementing interval . Thus… Read More »

Differential Equations

It is the languge used to express laws of Physics. Ultimate goal of Physics is to provide a funtional relation between numeric observations. For example, the relation between altitude h and time t of an  object falling near the Earth surface is h =-1/2gt^2+v0t+h0 where g is a Physical constant (9.8) and v0,h0 are situation-dependent constants. A… Read More »