= Representation ring of a semisimple Lie algebra
{title2=$R(\mathfrak g)\cong\mathbb Z[P]^W$}
The representation ring of finite-dimensional modules for a complex <semisimple Lie algebra> is the abelian group generated by module classes, with $[V\oplus U]=[V]+[U]$, and multiplication $[V][U]=[V\otimes U]$. <Weyl complete reducibility theorem> makes the classes of <irreducible representations> a free integral basis. The <formal character> map identifies this ring with the <Weyl group> invariant part of the <group ring of a weight lattice>. Injectivity follows from distinct <highest weights>; surjectivity follows by subtracting the character with each maximal dominant weight, whose highest-weight coefficient is one, from a Weyl-invariant finite sum.
// Target: semisimple-lie-algebra.bigb
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