Given a pair of nilpotent orbits in a simple Lie algebra, one can associate a pair of vertex algebras called affine W-algebras. Under some compatibility conditions on these orbits, we prove that one of these W-algebras can be obtained as the quantum Hamiltonian reduction of the other. This property is called reduction by stages. We provide several examples in classical and exceptional types.
To prove reduction by stages for affine W-algebras, we use our previous work on reduction by stages for the Slodowy slices associated with these nilpotent orbits, these slices being the associated varieties of the W-algebras. We also prove and use the fact that each W-algebra can be defined using several equivalent BRST cohomology constructions: choosing the right BRST complexes allows us to connect the two W-algebras in a natural way.
Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper, we prove that neighbouring generalized affine Grassmannian slices are related by Hamiltonian reduction by the action of the additive group. We also prove a weaker version of the same result for their quantizations, algebras known as truncated shifted Yangians.
Fix an affine Lie algebra $\widehat{\mathfrak{g}}_κ$ with associated principal affine W-algebra $\mathcal{W}_κ$. A basic conjecture of Frenkel--Kac--Wakimoto asserts that Drinfeld--Sokolov reduction sends admissible $\widehat{\mathfrak{g}}_κ$-modules to zero or cohomological shifts of minimal series $\mathcal{W}_κ$-modules. We prove this conjecture and a natural generalization to the spectrally flowed Drinfeld--Sokolov reduction functors. This extends the previous results of Arakawa for the minus reduction.
The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel'd-Sokolov (D-S) reduction associated to a Lie superalgebra $g$ and its even nilpotent element $f$, and we find a new definition of the classical affine W-superalgebra $W(g,f,k)$ via the D-S reduction. This new construction allows us to find free generators of $W(g,f,k)$, as a differential superalgebra, and two independent Lie brackets on $W(g,f,k)/\partial W(g,f,k).$ Moreover, we describe super-Hamiltonian systems with the P…
In this paper we propose a combinatorial framework to study a diagrammatic representation of the affine Temperley-Lieb algebra of type C introduced by Ernst. In doing this, we define two procedures, a decoration algorithm on diagrams and a reduction algorithm on heaps of independent interest. Using this approach, an explicit algorithmic description of Ernst representation map is provided from which its faithfulness can be deduced. We also give a construction of the inverse map.
Hamiltonian reductions as affine closures of cotangent bundles
State Key Laboratory of Mathematical Sciences, Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室、华罗庚数学中心）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）
We introduce a notion of $ξ$-stability on the affine grassmannian $\xx$ for the classical groups, this is the local version of the $ξ$-stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient $\xx^ξ/T$ of the stable part $\xx^ξ$ by the maximal torus $T$ exists as an ind-$k$-scheme, and we introduce a reduction process analogous to the Harder-Narasimhan reduction for vector bundles. For the group $\mathrm{SL}_{d}$, we calculate the Poincaré series of the quotient $\xx^ξ/T$.
We show that one can construct a classical affine W-algebra via a classical BRST complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional W-algebra has two compatible $λ$-brackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional W-algebra is associated to a minimal nilpotent, we d…
Drinfeld Zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra $\hat{sl}_n$. We introduce an affine, reduced, irreducible, normal quiver variety $Z$ which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on $Z$ in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian reduction of the corresponding quotient of its universal enveloping algebra…
We study modularity of the characters of a vertex (super)algebra equipped with a family of conformal structures. Along the way we introduce the notions of rationality and cofiniteness relative to such a family. We apply the results to determine modular transformations of trace functions on admissible modules over affine Kac-Moody algebras and, via BRST reduction, trace functions on minimal series representations of principal affine W-algebras.
We explore the relationship between de Rham and Lie algebra cohomologies in the finite dimensional and affine settings. As an application, we describe the BRST reduction of the chiral Hecke algebra as a vertex super algebra.
We prove an explicit condition on the level $k$ for the irreducibility of a vacuum module $V^{k}$ over a (non-twisted) affine Lie superalgebra, which was conjectured by M. Gorelik and V.G. Kac. An immediate consequence of this work is the simplicity conditions for the corresponding minimal W-algebras obtained via quantum reduction, in all cases except when the level $k$ is a non-negative integer.
We define deformations of W-algebras associated to complex semi-simple Lie algebras by means of quantum Drinfeld-Sokolov reduction procedure for affine quantum groups. We also introduce Wakimoto modules for arbitrary affine quantum groups and construct free field resolutions and screening operators for the deformed W-algebras. We compare our results with earlier definitions of q-W-algebras and of the deformed screening operators due to Awata, Kubo, Odake, Shiraishi (q-alg/9507034, q-alg/9508011, q-alg/9612001), Feigin, E. Frenkel (q-alg/9508009) and E. Frenkel, Reshetikhin (q-alg/9708006). The…
Hiraga, Ichino and Ikeda have conjectured an explicit expression for the Plancherel density of the group of points of a reductive group defined over a local field $F$, in terms of local Langlands parameters. In these lectures we shall present a proof of these conjectures for Lusztig's class of representations of unipotent reduction if $F$ is $p$-adic and $G$ is of adjoint type and splits over an unramified extension of $F$. This is based on the author's paper [Spectral transfer morphisms for unipotent affine Hecke algebras, Selecta Math. (N.S.) 22 (2016), no. 4, 2143--2207]. More generally for…
Quantum hamiltonian reduction of affine superalgebras is studied in the twisted case. The Ramond sector of "minimal" superconformal W-algebras is described in detail, the determinant formula is obtained. Extensive list of examples includes all the simple Lie superalgebras of rank up to 2. The paper generalizes the results of Kac and Wakimoto (math-ph/0304011) to the twisted case.
A cuspidal system for an affine Khovanov-Lauda-Rouquier algerba $R_\al$ yields a theory of standard modules. This allows us to classify the irreducible modules over $R_\al$ up to the so-called imaginary modules. We make a conjecture on reductions modulo $p$ of irreducible $R_\al$-modules, which generalizes James Conjecture. We also describe minuscule imaginary modules, laying the groundwork for future study of imaginary Schur-Weyl duality. We introduce colored imaginary tensor spaces and reduce a classification of imaginary modules to one color. We study the characters of cuspidal modules. We …
