Algebra objects in direct limit completions of compact Lie group duals and the classification of $c=1$ vertex operator algebras
Let $G$ be a complex reductive affine algebraic group with its symmetric tensor category $\mathcal{C}_G$ of finite-dimensional rational representations. We prove that every simple commutative algebra object with at most countable dimension in the direct limit completion $\operatorname{Ind}(\mathcal{C}_G)$ of $\mathcal{C}_G$ is isomorphic to the algebra $\mathcal{O}(G/H)$ of regular functions on the homogeneous space $G/H$, for some reductive algebraic subgroup $H$ of $G$. Then we apply this result to the theory of vertex operator algebra extensions. In particular, assuming the strong rationality of the $ \operatorname{A}_5$-orbifold of the vertex operator algebra $V_{\mathcal{L}_2}$ associated with the rank-one root lattice $\mathcal{L}_2:= \sqrt{2}\mathbb{Z}$, we classify all the not necessarily rational simple CFT type preunitary vertex operator algebra extensions of the simple unitary Virasoro vertex operator algebra $L(1,0)$ with central charge $c=1$ satisfying a certain spectrum condition. This result is the vertex operator algebra analogue of a conformal net result by Feng Xu. Every strongly rational preunitary vertex operator algebra extension of $L(1,0)$ satisfies the above spectrum condition because of the congruence subgroup modularity property of its characters. As a consequence, we get a complete classification result for strongly rational $c=1$ vertex operator algebras, up to the strong rationality of $V_{\mathcal{L}_2}^{\operatorname{A}_5}$.