Universal affine vacuum module

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra, let $B$ be the normalized invariant form in Untwisted affine central extension, and let $k\in \mathbb{C}$. Write $\widehat{\mathfrak{g}}=\overline{L\mathfrak{g}}$ and let $K$ denote its central generator (called $c$ in that definition). Put $\mathfrak{p}=\mathfrak{g}[t]\oplus \mathbb{C}K$ and let ${\mathbb{C}}_k$ be the one-dimensional $\mathfrak{p}$-module from The affine vacuum character exists at every complex level. The universal affine vacuum module at level $k$ is $V^k(\mathfrak{g}):=U(\widehat{\mathfrak{g}}){\otimes }_{U(\mathfrak{p})}{\mathbb{C}}_k,$ with vacuum $\mathbf{1}=1\otimes 1$. We write $x_n$ for the action of $x\otimes t^n$ and $J^x(z)={\sum }_{n\in \mathbb{Z}}{x}_n{z}^{-n-1}$ for its current distribution when it is a field. The central element acts as $k\mathrm{id}$.

This is an algebraic induced module. The affine Lie algebra here has no degree derivation, and no completion, unitarity, integrability, simple quotient, or conformal vector is part of this definition. The nonzero vacuum and ordered negative-mode basis are established by PBW basis of the universal affine vacuum module ↗.