Related papers: Notes on affine W-algebras
We define a quantum version of Hamiltonian reduction by stages, producing a construction in type A for a quantum Hamiltonian reduction from the W-algebra $U(\mathfrak{g},e_1)$ to an algebra conjecturally isomorphic to $U(\mathfrak{g},e_2)$,…
The quantum Hamiltonian reduction on the OSp(1,2) super Kac-Moody algebra is described in the BRST formalism. Using a free field representation of the KM currents, the super Kac-Moody algebra is shown to be reduced to a superconformal one…
We give explicit formulae for certain elements occurring in the Bernstein presentation of an affine Hecke algebra, in terms of the usual Iwahori- Matsumoto generators. We utilize certain minimal expressions for said elements and we give a…
In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations…
Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the…
Recently a remarkable map between 4-dimensional superconformal field theories and vertex algebras has been constructed \cite{BLLPRV15}. This has lead to new insights in the theory of characters of vertex algebras. In particular it was…
Let $C$ be a symmetrizable generalized Cartan Matrix, and $q$ an indeterminate. ${\fg}(C)$ is the Kac-Moody Lie algebra and $U=U_q({\fg}(C))$ the associated quantum enveloping algebra over $ k={\Bbb Q}(q)$. The quantum function algebra…
The partition functions of a 2D conformal system - the modular invariant one or the generalized ones, coming from the introduction of defect lines - are expressed in terms of a set of coefficients that have the particularity to form nimreps…
We give a geometric proof of inverse Hamiltonian reduction for all finite W-algebras in type $A$, a certain embedding of the finite W-algebra corresponding to an arbitrary nilpotent in $\mathfrak{gl}_N$ into that corresponding to a larger…
We study the structure and representations of a family of vertex algebras obtained from affine superalgebras by quantum reduction. As an application, we obtain in a unified way free field realizations and determinant formulas for all…
For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, we describe a new family of relaxed highest weight representations of $V_k(\mathfrak{g})$. They are simple quotients of representations of the affine Kac-Moody algebra…
We derive sufficient conditions under which the ``second'' Hamiltonian structure of a class of generalized KdV-hierarchies defines one of the classical $\cal W$-algebras obtained through Drinfel'd-Sokolov Hamiltonian reduction. These…
The goal of this paper is to present some results and (more importantly) state a number of conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products categorifies certain structures in the…
We develop a quantum harmonic analysis framework for the affine group. This encapsulates several examples in the literature such as affine localization operators, covariant integral quantizations, and affine quadratic time-frequency…
We further develop the abstract representation theory of affine Hecke algebras with arbitrary positive parameters. We establish analogues of several results that are known for reductive p-adic groups. These include: the relation between…
We introduce a twisted quantum affine algebra associated to each simply laced finite dimensional simple Lie algebra. This new algebra is a Hopf algebra with a Drinfeld-type comultiplication. We obtain this algebra by considering its vertex…
In \cite{Kawasetsu:2018irs}, Kawasetsu proved that the simple W-algebra associated with a minimal nilpotent element $W_{k}(\mathfrak{g},f_\theta)$ is rational and $C_2$-cofinite for $\mathfrak{g}=D_4,E_6,E_7,E_8$ with non-admissible level…
In the previous three papers in this series, [WKO1]-[WKO3] (arXiv:1405.1956, arXiv:1405.1955, and to appear), Z. Dancso and I studied a certain theory of "homomorphic expansions" of "w-knotted objects", a certain class of knotted objects in…
This is a survey paper about affine Hecke algebras. We start from scratch and discuss some algebraic aspects of their representation theory, referring to the literature for proofs. We aim in particular at the classification of irreducible…
We develop a relative version of Kostant's harmonic theory and use this to prove a relative version of Kostant's theorem on Lie algebra (co)homology. These are associated to two nested parabolic subalgebras in a semisimple Lie algebra. We…