On Virasoro-type reductions and inverse Hamiltonian reductions for $W$-algebras and $W_\infty$-algebras

TL;DR

The paper establishes a Virasoro-type reduction for height-two W-algebras in classical types and their inverse Hamiltonian reductions, linking $\mathrm{H}^{0}_{\,\ydiagram{2}\,}(\mathcal{W}^\mathsf{k}(\mathfrak{g},\mathbb{O}))$ to $\mathcal{W}^\mathsf{k}(\mathfrak{g},\widehat{\mathbb{O}})$ and providing embeddings into free-field algebras. It systematically constructs Wakimoto realizations to prove these isomorphisms across types $A,B,C,D$, including even/odd parity cases, and extends the framework to modules in the Kazhdan–Lusztig category. The results are lifted to universal objects via the 2-parameter $\mathcal{W}_\infty^{\mathfrak{sp}}(c,\mathsf{k})$, showing that several $\mathcal{W}$-algebras appear as 1-parameter quotients and that the Virasoro-type reduction interacts with two commuting Virasoro structures after suitable base changes. Overall, the work clarifies the hierarchical structure of W-algebras under height-two reductions, their universal counterparts, and their module categories, offering new tools for representation-theoretic and free-field analyses.

Abstract

In this article, the Virasoro-type reduction and the corresponding inverse reductions are established for W-algebras associated with classical Lie type and nilpotent orbits of height two. Moreover, these results are lifted to the universal objects by analyzing the Virasoro-type reduction of the vertex algebra $\mathcal{W}^{\mathfrak{sp}}_{\infty}$.

Figures (18)

• Figure 1: Generator $\tau$ for $\Gamma(D_n)$
• Figure 2: Generator $\sigma$ of $\Gamma(D_4)$
• Figure 3: Pyramid for $[N^2]$ in type $A$
• Figure 4: Pyramid for $[N^2,1]$, $N=2n$, in type $B$
• Figure 5: Pyramid for $[N^2,1]$, $N=2n+1$, in type $B$

Theorems & Definitions (37)

• Theorem A
• Theorem B
• Theorem C
• Conjecture A: CKL24
• Theorem 2.1
• proof
• Remark 2.2
• Corollary 2.3
• Conjecture 3.1: CFLN
• Conjecture 3.2: CKL24