Affine action
Encyclopedia
Let 
be the Weyl group
of a semisimple Lie algebra
(associate to fixed choice of a Cartan subalgebra 
). Assume that a set of simple root
s in
is chosen.
The affine action (also called the dot action) of the Weyl group on the space
is
where
is the sum of all fundamental weights, or, equivalently, the half of the sum of all positive roots.
be the Weyl groupWeyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
of a semisimple Lie algebra
(associate to fixed choice of a Cartan subalgebra ). Assume that a set of simple rootSimple root
in mathematics the term simple root can refer to one of two unrelated notions:*A simple root of a polynomial is a root of multiplicity one*A simple root in a root system is a member of a subset determined by a choice of positive roots...
s in
is chosen.The affine action (also called the dot action) of the Weyl group on the space
iswhere
is the sum of all fundamental weights, or, equivalently, the half of the sum of all positive roots.