Let be a finite-dimensional split semisimple Lie algebra over a field of field characteristic 0, a splitting Cartan subalgebra, and a weight of in a representation of . Then
is also a weight. Furthermore, the reflections with a root, generate a group of linear transformations in called the Weyl group of relative to , where is the algebraic conjugate space of and is the Q-space spanned by the roots (Jacobson 1979, pp. 112, 117, and 119).
|
|
|
The Weyl group acts on the roots of a semisimple Lie algebra, and it is a finite group. The animations above illustrate this action for Weyl Group acting on the roots of a homotopy from one Weyl matrix to the next one (i.e., it slides the arrows from to ) in the first two figures, while the third figure shows the Weyl Group acting on the roots of the Cartan matrix of the infinite family of semisimple lie algebras (cf. Dynkin diagram), which is the special linear Lie algebra, .