Generalized finite and affine $W$-algebras in type $A$

We construct a new family of affine $W$-algebras $W^k(λ,μ)$ parameterized by partitions $λ$ and $μ$ associated with the centralizers of nilpotent elements in $\mathfrak{gl}_N$. The new family unifies a few known classes of $W$-algebras. In particular, for the column-partition $λ$ we recover the affine $W$-algebras $W^k(\mathfrak{gl}_N,f)$ of Kac, Roan and Wakimoto, associated with nilpotent elements $f\in\mathfrak{gl}_N$ of type $μ$. Our construction is based on a version of the BRST complex of the quantum Drinfeld-Sokolov reduction. We show that the application of the Zhu functor to the vertex algebras $W^k(λ,μ)$ yields a family of generalized finite $W$-algebras $U(λ,μ)$ which we also describe independently as associative algebras.

💡 Research Summary

The paper “Generalized finite and affine W‑algebras in type A” by Dong‑Jun Choi, Alexander Molev and Uhi Rinn Suh introduces a broad new family of vertex algebras and associative algebras that simultaneously generalize several previously known constructions of W‑algebras in the setting of the general linear Lie algebra glₙ. The authors start by fixing a nilpotent element e∈glₙ whose Jordan blocks are described by a partition λ of N (the size of the matrix). The centralizer a=glₙᵉ is then described explicitly using a left‑justified pyramid associated with λ; its basis consists of elements E^{(r)}_{ij} indexed by triples (i,j,r) belonging to a combinatorial set Sₑ.

A second partition µ of the number of rows n of λ is introduced. The µ‑pyramid determines a ℤ‑grading on a by assigning degree deg_µ(E^{(r)}{ij}) = col_µ(j) – col_µ(i). The positively graded part a(>0) is denoted a{>0}, and the subspace n_{λ,µ}=⊕{i>0}a(i) plays the role of the “nilpotent” subalgebra in the Drinfeld–Sokolov reduction. A linear functional χ on n{λ,µ} is defined by χ(E^{(λ_i+1−1)}_{i,i+1})=1 whenever the rows of µ satisfy the adjacency condition; all other basis elements are sent to zero. This choice mirrors the standard character used in quantum Hamiltonian reduction.

With these data the authors construct a BRST complex C_k(λ,µ) = V_k(a) ⊗ F(n_{λ,µ}), where V_k(a) is the affine Kac–Moody vertex algebra of a at level k (built from the invariant bilinear form (·|·) defined in (3.1)), and F(n_{λ,µ}) is the free fermion vertex algebra generated by odd fields ϕ_{(i,j,r)} and their duals. The BRST differential Q is defined in the usual way, mixing the current fields of V_k(a) with the fermions according to the character χ. The authors verify Q²=0, and then prove (Theorem 3.6) that the zeroth cohomology H⁰(C_k(λ,µ),Q) inherits a vertex algebra structure; they denote this algebra by W^k(λ,µ).

A substantial part of the paper is devoted to describing explicit generating sets for W^k(λ,µ). Theorem 3.12 and Corollary 3.13 give minimal collections of currents and fermionic composites that generate the whole algebra, depending on the combinatorics of λ and µ. Several specializations recover known objects:

• If λ is the column partition (1ⁿ) and µ is arbitrary, then W^k(λ,µ) coincides with the affine W‑algebra W^k(glₙ,f) of Kac‑Roan‑Wakimoto, where f∈glₙ has Jordan type µ.
• If µ is the row partition (N) and λ is arbitrary, W^k(λ,µ) reduces to the affine vertex algebra V_k(a) introduced in earlier work of the authors.
• If µ is the column partition (1ⁿ), the algebra W^k(λ,µ) becomes the “universal affine vertex algebra” V_k(a) associated with the centralizer a itself.

Thus the family W^k(λ,µ) interpolates between principal, trivial, and intermediate W‑algebras. The authors also note that in general W^k(λ,µ) need not be conformal; Example 5.2 exhibits a non‑conformal case.

The second major construction is the generalized finite W‑algebra U(λ,µ). Starting from the universal enveloping algebra U(a), they form the left ideal I_{λ,µ}=U(a)⟨n+χ(n)⟩ generated by the shifted nilpotent subalgebra, and then take the n_{λ,µ}‑invariants in the quotient:

 U(λ,µ) = (U(a)/I_{λ,µ})^{n_{λ,µ}}.

Definition 2.1 shows that this is a well‑defined associative algebra. The authors prove (Theorem 4.6) that the Zhu algebra of the vertex algebra W^k(λ,µ) is precisely U(λ,µ), extending the result of De Sole–Kac for ordinary affine W‑algebras to this broader setting.

Several concrete families of U(λ,µ) are identified:

• When λ is the column partition (1ⁿ) and µ arbitrary, U(λ,µ) is the usual finite W‑algebra U(g,e_µ) associated with glₙ and a nilpotent of type µ.
• When µ is the column partition (1ⁿ) and λ arbitrary, U(λ,µ) coincides with the universal enveloping algebra U(a) of the centralizer a.
• When λ has equal parts λ_i=p (so a≅gl_n