Restricted affine-current module

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with the invariant form used in Untwisted affine central extension. A representation $\rho$ of the loop central extension $\overline{L\mathfrak{g}}$ is a restricted affine-current module of level $k\in \mathbb{C}$ if the central generator acts as $k\mathrm{id}$ and, for every vector $v$ and current $x\in \mathfrak{g}$, there is an integer $N(v,x)$ such that $\rho (x\otimes t^n)v=0$ for all $n\ge N(v,x)$.

The algebra in this definition is the central extension only; it does not contain the degree derivation. Restrictedness is a mode-truncation condition and does not itself supply an energy grading. A compatible positive-energy grading is extra structure on this current algebra.