= Group ring of a weight lattice
{title2=$\mathbb Z[P]$}
= Weight-lattice group ring
{synonym}
The <group ring> of a <weight lattice> $P$ consists of finite sums $\sum_\lambda a_\lambda e(\lambda)$ with <integer> coefficients, where $e(\lambda)e(\mu)=e(\lambda+\mu)$. A finite-dimensional <Lie algebra representation> has <formal character> $\sum_\lambda(\dim V_\lambda)e(\lambda)$ in this <ring>. The <Weyl group> acts by $e(\lambda)\mapsto e(w\lambda)$. The ambient <ring> need not consist of genuine <characters>; the <representation ring of a semisimple Lie algebra> of finite-dimensional <modules> for a complex <semisimple Lie algebra> identifies with the invariant subring $\mathbb Z[P]^W$.
// Target: semisimple-lie-algebra.bigb
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