Simple positive-integral-level affine vertex operator algebra

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with invariant form normalized so that every long root has squared length $2$. For a positive integer $\mathrm{\ell }$, the simple affine vertex operator algebra $L_{\mathrm{\ell }}(\mathfrak{g})$ is the quotient of the universal affine vertex algebra $V^{\mathrm{\ell }}(\mathfrak{g})$ by its unique maximal proper vertex-algebra ideal. It carries the noncritical Sugawara conformal vector; its underlying untwisted affine Kac–Moody module is the irreducible highest-weight module of highest weight $\mathrm{\ell }{\mathrm{\Lambda }}_0$. Existence, uniqueness, the conformal grading, and the affine-module identification are established by the local quotient lemma. This is an algebraic vertex operator algebra of CFT type; no Hilbert-space inner product, unitarity, or full two-dimensional conformal field theory is implicit.

Loop algebra of a simple Lie algebra

Untwisted affine central extension

Universal affine vacuum module

Universal affine vertex algebras exist at every complex level

Conformal vertex algebra of CFT type

Sugawara modes and their normal ordering

Affine node-zero fundamental weight

Existence and structure of the simple affine VOA quotient