The Category O

## The category $𝒪$

Let $ℱ = ⨁ i ∈ ℤ ℱ i be a ℤ -graded algebra$ such that

1. $\mathrm{dim}\left({ℱ}_{i}\right)\le \infty$,
2. ${ℱ}_{0}$ is reductive, and
3. $ℱ$ is semisimple as a ${ℱ}_{0}$-module (under adjoint action).

The category $𝒪$ for $ℱ=\underset{i\in ℤ}{⨁}{ℱ}_{i}$ is the category of $ℤ -graded ℱ -modules M = ⨁ i ∈ ℤ M i$ such that

1. $M$ is semisimple as a ${ℱ}_{0}$-module, and
2. if $m\in M$ then $\mathrm{dim}\left({U}^{\ge 0}m\right)<\infty$,
where ${U}^{\ge 0}=U\left({ℱ}_{\ge 0}\right)$ with ${ℱ}_{\ge 0}=\underset{i\in {ℤ}_{\ge 0}}{⨁}{ℱ}_{i}$.