Difference between revisions of "Character of a finite-dimensional representation of a semi-simple Lie algebra"
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− | + | The function that assigns to every weight of the representation the dimension of the corresponding weight subspace. If $ \mathfrak h $ | |
+ | is a Cartan subalgebra of a semi-simple Lie algebra $ \mathfrak g $ | ||
+ | over an algebraically closed field $ k $ | ||
+ | of characteristic $ 0 $, | ||
+ | $ \phi : \mathfrak g \rightarrow \mathfrak g \mathfrak l (V) $ | ||
+ | is a linear representation and $ V _ \lambda $ | ||
+ | is the weight subspace corresponding to $ \lambda \in \mathfrak h ^ {*} $, | ||
+ | then the character of the representation $ \phi $( | ||
+ | or of the $ \mathfrak g $- | ||
+ | module $ V $) | ||
+ | can be written in the form | ||
− | + | $$ | |
+ | \mathop{\rm ch} V = \ | ||
+ | \sum _ {\lambda \in \mathfrak h ^ {*} } | ||
+ | ( \mathop{\rm dim} V _ \lambda ) e ^ \lambda | ||
+ | $$ | ||
− | + | and can be regarded as an element of the group ring $ \mathbf Z [ \mathfrak h ^ {*} ] $. | |
+ | If $ k = \mathbf C $ | ||
+ | and $ \phi = d \Phi $, | ||
+ | where $ \Phi : G \rightarrow \mathop{\rm GL} (V) $ | ||
+ | is an analytic linear representation of a Lie group $ G $ | ||
+ | with Lie algebra $ \mathfrak g $, | ||
+ | then $ e ^ \lambda $ | ||
+ | can be regarded as the function on $ \mathfrak h $ | ||
+ | suggested by the notation and $ \mathop{\rm ch} \phi $ | ||
+ | coincides with the function $ x \mapsto \chi _ \Phi ( e ^ {x} ) $( | ||
+ | $ x \in \mathfrak h $), | ||
+ | where $ \chi _ \Phi $ | ||
+ | is the character of the representation $ \Phi $. | ||
+ | Characters of a representation of a Lie algebra have the following properties: | ||
+ | |||
+ | $$ | ||
+ | \mathop{\rm ch} (V _ {1} \oplus V _ {2} ) = \ | ||
+ | \mathop{\rm ch} V _ {1} + \mathop{\rm ch} V _ {2} , | ||
+ | $$ | ||
+ | |||
+ | $$ | ||
+ | \mathop{\rm ch} (V _ {1} \otimes V _ {2} ) = \mathop{\rm ch} V _ {1} \cdot \mathop{\rm ch} V _ {2} . | ||
+ | $$ | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Dixmier, "Algèbres enveloppantes" , Gauthier-Villars (1974)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Dixmier, "Algèbres enveloppantes" , Gauthier-Villars (1974)</TD></TR></table> |
Revision as of 16:25, 31 March 2020
The function that assigns to every weight of the representation the dimension of the corresponding weight subspace. If $ \mathfrak h $
is a Cartan subalgebra of a semi-simple Lie algebra $ \mathfrak g $
over an algebraically closed field $ k $
of characteristic $ 0 $,
$ \phi : \mathfrak g \rightarrow \mathfrak g \mathfrak l (V) $
is a linear representation and $ V _ \lambda $
is the weight subspace corresponding to $ \lambda \in \mathfrak h ^ {*} $,
then the character of the representation $ \phi $(
or of the $ \mathfrak g $-
module $ V $)
can be written in the form
$$ \mathop{\rm ch} V = \ \sum _ {\lambda \in \mathfrak h ^ {*} } ( \mathop{\rm dim} V _ \lambda ) e ^ \lambda $$
and can be regarded as an element of the group ring $ \mathbf Z [ \mathfrak h ^ {*} ] $. If $ k = \mathbf C $ and $ \phi = d \Phi $, where $ \Phi : G \rightarrow \mathop{\rm GL} (V) $ is an analytic linear representation of a Lie group $ G $ with Lie algebra $ \mathfrak g $, then $ e ^ \lambda $ can be regarded as the function on $ \mathfrak h $ suggested by the notation and $ \mathop{\rm ch} \phi $ coincides with the function $ x \mapsto \chi _ \Phi ( e ^ {x} ) $( $ x \in \mathfrak h $), where $ \chi _ \Phi $ is the character of the representation $ \Phi $. Characters of a representation of a Lie algebra have the following properties:
$$ \mathop{\rm ch} (V _ {1} \oplus V _ {2} ) = \ \mathop{\rm ch} V _ {1} + \mathop{\rm ch} V _ {2} , $$
$$ \mathop{\rm ch} (V _ {1} \otimes V _ {2} ) = \mathop{\rm ch} V _ {1} \cdot \mathop{\rm ch} V _ {2} . $$
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=17019