= Category O
{c}
{title2=$\mathcal O$}
For a finite-dimensional complex <semisimple Lie algebra> with a fixed <triangular decomposition of a Lie algebra>, category $\mathcal O$ consists of finitely generated <universal enveloping algebra> modules that are direct sums of Cartan weight spaces and are locally finite under the positive nilpotent subalgebra. Their <weight spaces> are finite-dimensional and their weights lie in a finite union of downward positive-root cones. <Verma modules>, finite-dimensional modules, submodules and quotients belong to this category. Closure under submodules uses the noetherian property of the enveloping algebra, obtained from the <Poincaré-Birkhoff-Witt theorem>. The category condition does not require the module to be finite-dimensional.
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