= Highest-weight classification of finite-dimensional semisimple Lie algebra modules
{title2=$\{\text{irreducibles}\}\longleftrightarrow P_+$}
= Highest-weight classification
{synonym}
For a complex <semisimple Lie algebra> and a choice of <positive roots>, every finite-dimensional <Irreducible Lie algebra representation> has a unique <highest weight>, which is a <dominant integral weight>. Conversely each dominant integral weight labels exactly one such <irreducible module> up to <isomorphism>. Its <highest-weight vector> spans the highest-weight space and generates the <module>. This classification and <Weyl complete reducibility theorem> reduce decomposing any finite-dimensional <module> to its irreducible highest-weight summands.
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