The Structure of the Kac–Wang–Yan Algebra

Abstract

The Lie algebra $${\mathcal{D}}$$ of regular differential operators on the circle has a universal central extension $${\hat{\mathcal{D}}}$$ . The invariant subalgebra $${\hat{\mathcal{D}}^+}$$ under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum $${\hat{\mathcal{D}}^+}$$ -module with central charge $${c \in \mathbb{C}}$$ , and its irreducible quotient $${\mathcal{V}_c}$$ , possess vertex algebra structures, and $${\mathcal{V}_c}$$ has a nontrivial structure if and only if $${c \in \frac{1}{2}\mathbb{Z}}$$ . We show that for each integer $${n > 0}$$ , $${\mathcal{V}_{n/2}}$$ and $${\mathcal{V}_{-n}}$$ are $${\mathcal{W}}$$ -algebras of types $${\mathcal{W}(2, 4,\dots,2n)}$$ and $${\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}$$ , respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group $${{\rm O}(n)}$$ and the symplectic group $${{\rm Sp}(2n)}$$ , respectively. Based on Sergeev’s theorems on the invariant theory of $${{\rm Osp}(1, 2n)}$$ we conjecture that $${\mathcal{V}_{-n+1/2}}$$ is of type $${\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}$$ , and we prove this for $${n = 1}$$ . As an application, we show that invariant subalgebras of $${\beta\gamma}$$ -systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.