Related papers: Quantum W-algebras and Elliptic Algebras
W-algebras are constructed via quantum Hamiltonian reduction associated with a Lie algebra $\mathfrak{g}$ and an $\mathfrak{sl}(2)$-embedding into $\mathfrak{g}$. We derive correspondences among correlation functions of theories having…
We construct $q$-deformations of quantum $\mathcal{W}_N$ algebras with elliptic structure functions. Their spin $k+1$ generators are built from $2k$ products of the Lax matrix generators of ${\mathcal{A}_{q,p}(\widehat{gl}(N)_c)}$). The…
Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic…
It is shown that, classically, the W-algebras are directly related to the extrinsic geometry of the embedding of two-dimensional manifolds with chiral parametrisation (W-surfaces) into higher dimensional K\"ahler manifolds. We study the…
We suggest two explicit descriptions of the Poisson q-W algebras which are Poisson algebras of regular functions on certain algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g.…
We investigate the W-algebra resulting from Drinfel'd-Sokolov reduction of a $B_2$ WZW model with respect to the grading induced by a short root. The quantum algebra, which is generated by three fields of spin-2 and a field of spin-1, is…
The deformed $\mathcal W$ algebras of type $\textsf{A}$ have a uniform description in terms of the quantum toroidal $\mathfrak{gl}_1$ algebra $\mathcal E$. We introduce a comodule algebra $\mathcal K$ over $\mathcal E$ which gives a uniform…
Quantum algebras U_q(su_n) used as the algebras of flavour symmetry (usually described by SU(n)) to study static properties of hadrons lead to intriguing results. In this contribution we focus on the peculiar properties manifested by…
For any Hecke symmetry $R$ there is a natural quantization $A_n(R)$ of the Weyl algebra $A_n$. The aim of this paper is to study some general ring-theoretic aspects of $A_n(R)$ and its relations to formal deformations of $A_n$. We also…
We obtain a condensed reconstruction of algebraic quantum theory, emphasizing its foundational aspects and algebraic structure. We obtain the $W^*$-algebra structure from elementary assumptions about observers and how they can observe…
A Heisenberg-Clifford realization of a deformed $U(sl_{2})$ by two parameters $p$ and $q$ is discussed. The commutation relations for this deformed algebra have interesting connection with the theta functions.
It is shown that the notion of W_\infty-algebra originally carried out over a (compact) Riemann surface can be extended to n complex dimensional (compact) manifolds within a symplectic geometrical setup. The relationships with the…
Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic 0 and O be a nilpotent orbit in g. Then Orb is a symplectic algebraic variety and one can ask whether it is possible to quantize $\Orb$ (in an…
We address two problems regarding the structure and representation theory of finite W-algebras associated with the general linear Lie algebras. Finite W-algebras can be defined either via the Whittaker model of Kostant or, equivalently, by…
We consider the extended superconformal algebras of the Knizhnik-Bershadsky type with $W$-algebra like composite operators occurring in the commutation relations, but with generators of conformal dimension 1,$\frac{3}{2}$ and 2, only. These…
We report on generalizations of the KdV-type integrable hierarchies of Drinfel'd and Sokolov. These hierarchies lead to the existence of new classical $W$-algebras, which arise as the second Hamiltonian structure of the hierarchies. In…
This note for the Proceedings of the International Congress of Mathematical Physics gives an account of a construction of an ``elliptic quantum group'' associated with each simple classical Lie algebra. It is closely related to elliptic…
We generalize the framework of Higgsed networks of intertwiners to the quantum toroidal algebra associated to Lie algebra $\mathfrak{gl}_N$. Using our formalism we obtain a systems of screening operators corresponding to W-algebras…
Finite versions of W-algebras are introduced by considering (symplectic) reductions of finite dimensional simple Lie algebras. In particular a finite analogue of $W^{(2)}_3$ is introduced and studied in detail. Its unitary and non-unitary,…
We show explicitly a generalised Lie algebra embedded in the positive and negative parts of the Drinfeld-Jimbo quantum groups of type A_n. Such a generalised Lie algebra satisfy axioms closely related to the ones found by S.L. Woronowicz.…