Character of a finite-dimensional representation of a semi-simple Lie algebra
The function that assigns to every weight of the representation the dimension of the corresponding weight subspace. If is a Cartan subalgebra of a semi-simple Lie algebra
over an algebraically closed field
of characteristic
,
is a linear representation and
is the weight subspace corresponding to
, then the character of the representation
(or of the
-module
) can be written in the form
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and can be regarded as an element of the group ring . If
and
, where
is an analytic linear representation of a Lie group
with Lie algebra
, then
can be regarded as the function on
suggested by the notation and
coincides with the function
(
), where
is the character of the representation
. Characters of a representation of a Lie algebra have the following properties:
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References
[1] | J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) |
[2] | J. Dixmier, "Algèbres enveloppantes" , Gauthier-Villars (1974) |
Character of a finite-dimensional representation of a semi-simple Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Character_of_a_finite-dimensional_representation_of_a_semi-simple_Lie_algebra&oldid=44944