Characters of Lie Groups and Lie Algebras
Let G be a Lie group with associated Lie algebra, and let H and be the Cartan subgroup/subalgebra.
Let V be a representation of G If we write the weight spaces of V as, then, we can define the formal character of the Lie group and Lie algebra as
where we sum over all weights of the weight lattice. In the above expression, is a formal object satisfying . This formal character is related to the regular one for other groups. If, where H is the Cartan subgroup of G (that is, ), then
The above discussion for the decomposition of tensor products and other representations continue to hold true for the formal character. In the case of a compact Lie group, the Weyl character formula can be used to calculate the formal character.
Read more about this topic: Character Theory
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