Related papers: Fock space representations for the quantum affine …
By using the ideas of Feigin and Stoyanovsky and Calinescu, Lepowsky and Milas we introduce and study the principal subspaces associated with the Etingof-Kazhdan quantum affine vertex algebra of integer level $k\geqslant 1$ and type…
We prove that the Hall algebra U^-_n of the cyclic quiver of type A^(1)_{n-1} decomposes as a direct product of the quantum negative nilpotent subalgebra U_q^-(\hat{sl}_{n)) and C[q,q^{-1},z_1,z_2...]. We use this to prove a conjecture of…
Some problems related to an algebraic approach to quantum statistics are discussed. Generalized quantum statistics is described as a result of interactions. The Fock space representation is discussed. The problem of existence of…
We study canonical basis elements in higher-level Fock spaces associated with the quantum group $U_q(\mathfrak{gl}_\infty)$, which are conjecturally related to Calogero-Moser theory for complex reflection groups. We generalize the…
Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_q(\hat{\mathfrak{g}})$ the corresponding quantum affine algebra. We prove that every irreducible finite-dimensional $U_q(\hat{\mathfrak{g}})$-module gives rise to a family of…
We construct and study in detail various dual pairs acting on some Fock representations between a finite dimensional Lie group and a completed infinite rank affine algebra associated to an infinite affine Cartan matrix. We give explicit…
The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. A representation for the quantum system in terms of non-commutative functions on the (dual) Lie algebra, and a…
We construct Fock and MacMahon modules for the quantum toroidal superalgebra $\mathcal{E}_\mathbf{s}$ associated with the Lie superalgebra $\mathfrak{gl}_{m|n}$ and parity $\mathbf{s}$. The bases of the Fock and MacMahon modules are labeled…
We discuss quantum deformation of the affine transformation group and its Lie algebra. It is shown that the quantum algebra has a non-cocommutative Hopf algebra structure, simple realizations and quantum tensor operators. The deformation of…
We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W_{1+infty} introduced by Miki. Our construction involves trivalent intertwining…
We investigate the characters of some finite-dimensional representations of the quantum affine algebras $U_q(\hat{g})$ using the action of the copy of $U_q(g)$ embedded in it. First, we present an efficient algorithm for computing the…
In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional fermion algebra, and we investigate the properties of this category. The categorical…
For generic $q$ we give expressions for the transformations of all essentially typical finite-dimensional modules of the Hopf superalgebra $U_q[gl(3/2)]$. The latter is a deformation of the universal enveloping algebra of the Lie…
The recent discoveries of new forms of quantum statistics require a close look at the under-lying Fock space structure. This exercise becomes all the more important in order to provide a general classification scheme for various forms of…
Given any representation of an arbitrary Lie algebra L over a field k of characteristic 0, we construct representations of L on bosonic and fermionic Fock space. The method gives an explicit formula for a (sometimes trivial) 2-cocycle in…
Fractional supersymmetric quantum mechanics of order $\lambda$ is realized in terms of the generators of a generalized deformed oscillator algebra and a Z$_{\lambda}$-grading structure is imposed on the Fock space of the latter. This…
Attention is focused on quantum spaces of particular importance in physics, i.e. two-dimensional quantum plane, q-deformed Euclidean space in three or four dimensions, and q-deformed Minkowski space. Each of these quantum spaces can be…
In this study, we focus on the positive part $U_q^{+}$ of the quantum affine superalgebra $U_q(C(2)^{(2)})$. This algebra admits a presentation with two two generators $e_{\alpha}$ and $e_{\delta-\alpha}$, which satisfy the cubic $q$-Serre…
We give a realization of the quantum affine Lie algebras $\uqa$ and $\uqc$ in terms of anyons defined on a one-dimensional chain (or on a two-dimensional lattice), the deformation parameter $q$ being related to the statistical parameter…
A representation of the quantum affine algebra $U_{q}(\hat{sl_{2}})$ of an arbitrary level $k$ is realized in terms of three boson fields, whose $q \rightarrow 1$ limit becomes the Wakimoto representation. An analogue of the screening…