Existence and structure of the simple affine VOA quotient

By [F2], $V^{\mathrm{\ell }}(\mathfrak{g})$ is spanned by PBW monomials $x_{-n_1}^1\cdots x_{-n_r}^r\mathbf{1}$ with every $n_i\ge 1$. Give such a monomial degree $n_1+\cdots +n_r$. The empty monomial is the unique degree-zero basis vector, and each fixed degree has finitely many monomials because $\mathfrak{g}$ is finite dimensional and there are finitely many ordered compositions of that degree. Thus $V^{\mathrm{\ell }}={⨁}_{n\ge 0}{V}_n$ with $V_0=\mathbb{C}\mathbf{1}$ and finite-dimensional $V_n$. To extend the induced module to the full affine algebra, use the derivation $D(x_n)=n{x}_n$, $D(c)=0$ from [F3]. For $n\ge 0$, $D(x_n)=n{x}_n\in \mathfrak{g}[t]$, so $D$ preserves $\mathfrak{p}=\mathfrak{g}[t]\oplus \mathbb{C}c$. Extend $D$ to the tensor algebra by the Leibniz rule; it preserves every relation $xy-yx-[x,y]$ because $D$ is a Lie derivation, so it descends to a derivation $D_U$ of $U(\overline{L\mathfrak{g}})$. By [F7], the vacuum character ${\chi }_{\mathrm{\ell }}$ is zero on $\mathfrak{g}[t]$ and has ${\chi }_{\mathrm{\ell }}(c)=\mathrm{\ell }$. Thus ${\chi }_{\mathrm{\ell }}(D_U(a))=0$ for each generator of $U(\mathfrak{p})$, and the derivation and character rules give it for every $a\in U(\mathfrak{p})$. The formula $d(u\otimes 1):=D_U(u)\otimes 1$ respects $ua\otimes 1=u\otimes {\chi }_{\mathrm{\ell }}(a)$, since the difference is $u\otimes {\chi }_{\mathrm{\ell }}(D_U(a))=0$; hence it is well-defined. It has $d\mathbf{1}=0$ and $[d,x_n]=n{x}_n$. On the PBW monomial above, $d$ acts by $-(n_1+\cdots +n_r)$, so the full affine Cartan acts on $\mathbf{1}$ by $c\mapsto \mathrm{\ell }$, $d\mapsto 0$, and $\mathfrak{h}\mapsto 0$. The action of $\mathfrak{g}[t]$ kills $\mathbf{1}$ by [F7], so under the loop/GCM identification in [F5] it is an affine highest vector of weight $\mathrm{\ell }{\mathrm{\Lambda }}_0$ by [F3].

Put $q=\mathrm{\ell }+h^{\vee }$, which is nonzero because $\mathrm{\ell }>0$ and the normalized dual Coxeter number is positive. For currents $A(z)=J^{x_a}(z)$ and $B(z)=J^{x^a}(z)$, the state-field product identity [F9] and the residue definition in [F9] give $Y(x_a(-1)x^a(-1)\mathbf{1},z)={\mathrm{Res}}_x\text{\hspace{0.17em}⁣}({\iota }_{x,z}(x-z{)}^{-1}A(x)B(z)-{\iota }_{z,x}(x-z{)}^{-1}B(z)A(x))$. Expanding the first binomial as ${\sum }_{j\ge 0}{z}^j{x}^{-j-1}$ selects precisely the negative modes of $A$, while the second expansion $-{\sum }_{j\ge 0}{x}^j{z}^{-j-1}$, with its preceding minus sign, selects the nonnegative modes of $A$ to the right of $B$. Thus this field is $:J^{x_a}(z)J^{x^a}(z):$ in the stated normal-order convention. The Sugawara state $\omega =\frac{1}{2q}{\sum }_a{x}_a(-1)x^a(-1)\mathbf{1}$ therefore has exactly the field whose coefficients are the $L_m$ of [F4]. Restrictedness follows from [F8], so [F4] gives the Virasoro relations. At mode $m=-1$, [F4] says $[L_{-1},x_n]=-n{x}_{n-1}$, the same commutator as $T$ in [F3]. Both operators kill $\mathbf{1}$, and the negative-current PBW monomials span; induction on their length gives $L_{-1}=T$. Similarly $L_0\mathbf{1}=0$ from normal ordering and $[L_0,x_{-n}]=n{x}_{-n}$, so $L_0$ acts by the PBW degree from step 1.1. This is a CFT-type conformal structure in the sense of [F6].

Every vertex-algebra ideal is $L_0$-invariant and therefore graded: a vector has only finitely many homogeneous components, and polynomial interpolation in its distinct integer $L_0$-eigenvalues extracts each component while staying in the ideal. A proper ideal has zero intersection with $V_0$: if it contains a nonzero scalar multiple of $\mathbf{1}$, it contains $\mathbf{1}$, and then ${\mathbf{1}}_{(-1)}v=v$ for every $v\in V$ by the vacuum-field axiom, forcing the ideal to be all of $V$. The sum of all proper ideals, interpreted as finite sums of their elements, is again an ideal and still has zero degree-zero component, hence is proper. Call this sum $J_{\mathrm{\ell }}$. It contains every proper ideal, so it is the unique maximal proper ideal. The quotient has no proper nonzero ideal and is simple; its inherited grading has nonnegative degrees, finite-dimensional pieces, and degree-zero space $\mathbb{C}\mathbf{1}$.

The ideal $J_{\mathrm{\ell }}$ is stable under every current mode because currents are vertex states. Conversely, let $U\subseteq V^{\mathrm{\ell }}$ be stable under every current mode. For states $a,b$, the module Jacobi residue formula [F10] gives $((a_{(r)}b{)}^M{)}_{(m)}v={\sum }_{j\ge 0}(-1{)}^j\left(\genfrac{}{}{0px}{}{r}{j}\right)a_{(r-j)}^M{b}_{(m+j)}^{M}v-(-1{)}^r{\sum }_{j\ge 0}(-1{)}^j\left(\genfrac{}{}{0px}{}{r}{j}\right)b_{(m+r-j)}^M{a}_{(j)}^{M}v$. For fixed $v$, both sums are finite by lower truncation of the rightmost field in each term. Each PBW state is built recursively as $a_{(-n)}b$ with $a$ a current state and $n\ge 1$; induction on PBW length therefore shows every mode of every state preserves $U$, so $U$ is a vertex ideal. Hence the simple quotient in step 3.1 is simple for the full affine current action and, with the $d$-action from step 1.1, for the full affine algebra. Its vacuum is a highest vector of weight $\mathrm{\ell }{\mathrm{\Lambda }}_0$ by step 1.1. By [F5], the unique simple highest-weight module of that weight is the quotient. This proves the affine-module identification in the Statement.