The Weyl group of a semisimple Lie algebra is the group generated by reflections
in its roots. 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 finite-dimensional 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 
-vector space of linear
combinations of the roots (Jacobson 1979,
pp. 112, 117, and 119).
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The Weyl group acts on the roots of a semisimple Lie algebra, and it is a finite group. The first
two animations illustrate this action on the roots,
interpolating from one Weyl matrix to another. The third
figure shows the Weyl group acting on the roots
of the rank-3 semisimple Lie algebra of
type 
, a member of the infinite family 
(cf. Dynkin
diagram), which is the special linear
Lie algebra, 
.
