TOPICS
Search

Weyl Group


Let L be a finite-dimensional split semisimple Lie algebra over a field of field characteristic 0, H a splitting Cartan subalgebra, and Lambda a weight of H in a representation of L. Then

 Lambda^'=LambdaS_alpha=lambda-(2(Lambda,alpha))/((alpha,alpha))(alpha)

is also a weight. Furthermore, the reflections S_alpha with alpha a root, generate a group of linear transformations in H_0^* called the Weyl group W of L relative to H, where H^* is the algebraic conjugate space of H and H_0^* is the Q-space spanned by the roots (Jacobson 1979, pp. 112, 117, and 119).

Weyl group 1
Weyl group 2
WeylGroup3

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 g to h) 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 A_3 (cf. Dynkin diagram), which is the special linear Lie algebra, sl_4.


See also

Cartan Matrix, Dynkin Diagram, Lie Algebra, Lie Algebra Root, Lie Group, Macdonald's Constant-Term Conjecture, Root System, Root Lattice, Semisimple Lie Algebra

Portions of this entry contributed by Todd Rowland

Explore with Wolfram|Alpha

References

Jacobson, N. Lie Algebras. New York: Dover, 1979.

Referenced on Wolfram|Alpha

Weyl Group

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Weyl Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeylGroup.html

Subject classifications