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 $\text{O}(n)$ and the symplectic group $\text{Sp}(2n)$, respectively. Based on Sergeev's theorems on the invariant theory of $\text{Osp}(1,2n)$ 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.

Final version

References in corpus (4)

• Orbifolds of symplectic fermion algebras
• A Hilbert theorem for vertex algebras
• Invariant theory and the W_{1+\infty} algebra with negative integral central charge
• W-Symmetry of the Adelic Grassmannian

Cited by in corpus (14)

• Schur-Weyl Duality for Heisenberg Cosets
• Cosets of affine vertex algebras inside larger structures
• Simple current extensions beyond semi-simplicity
• Trialities of $\mathcal{W}$-algebras
• Orbifolds and cosets of minimal $\mathcal{W}$-algebras
• $\mathcal{W}$-algebra Modules, Free Fields, and Gukov-Witten Defects
• Universal two-parameter ${\mathcal{W}}_{\infty }$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots ,N)$
• Universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra
• Cosets of the ${\mathcal{W}}^k({\mathfrak{sl}}_4,f_{\text{subreg}})$-algebra
• $N=4$ superconformal algebras and diagonal cosets
• Generalized parafermions of orthogonal type
• Twisted logarithmic modules of free field algebras
• On the structure of W-algebras in type A
• Vertex Algebras and Commutative Algebras