Weyl Chamber

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In mathematics , the Weyl chamber (named after Hermann Weyl ) is a term from the theory of Lie groups . Weyl chambers are needed in defining positive and simple roots , and they also play a central role in the theory of buildings .

definition

Let be a finite dimensional semi-simple Lie algebra , a Cartan subalgebra and the associated root system .

For a root denote

the associated hyperplane in .

Then the connected components are called of

the Weyl chambers of the root system.

Effect of the Weyl Group

The Weyl group of acts on and permutes the set of Weyl chambers, i.e. H. the effect of the Weyl group on the set of Weyl chambers is simply transitive and the number of Weyl chambers is the cardinality of the Weyl group.

The conclusion of a Weyl chamber is a fundamental area for the effect of the Weyl group on .

Weyl chambers in symmetrical spaces

Let it be a symmetrical space of a non-compact type . Then all containing flax are of the shape

for an Abelian sub-algebra . (Here is the exponential mapping into and the Cartan decomposition .)

In particular, the term Weyl chambers can be transferred to flax in symmetrical spaces: Weyl chambers in are (by definition) the images of the Weyl chambers in under the exponential map.

example

Root system A 2

Be it

and

.

The associated root system consists of the 6 roots

corresponding

.

The 's are three straight lines in two-dimensional vector space , they break down into six Weyl chambers.

In this case the Weyl group is the symmetrical group , it permutes the six Weyl chambers.

literature

  • Armand Borel : Linear algebraic groups. WA Benjamin, New York / Amsterdam 1969
  • Alexander Kirillov Jr .: An introduction to Lie groups and Lie algebras . In: Cambridge Studies in Advanced Mathematics , 113. Cambridge University Press, Cambridge 2008, ISBN 978-0-521-88969-8
  • Ira Gessel, Dorn Zeilberger: Random walk in a Weyl chamber . JSTOR 2159560

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