Related papers: The Baxter's Q-operator for the W-algebra $W_N$

A $q$-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the $2$-sphere, is obtained as the commutant of the $\mathfrak{o}_{q^{1/2}}(2) \oplus \mathfrak{o}_{q^{1/2}}(2)$ subalgebra of…

We define and study the quantum equivariant $K$-theory of cotangent bundles over Grassmannians. For every tautological bundle in the $K$-theory we define its one-parametric deformation, referred to as quantum tautological bundle. We prove…

We find an explicit formula for the conformal vector of any quantum affine $W$-algebra in its free field realization.

In this paper, we shall describe all the Rota-Baxter operators with any weight on split semi-quaternion algebra. Firstly, we give the matrix characterization of the Rota-Baxter operator on split semi-quaternion algebra. Then we give the…

For the case of quantum loop algebras $\mathrm U_q(\mathcal L(\mathfrak{sl}_{l + 1}))$ with $l = 1, 2$ we find the $\ell$-weights and the corresponding $\ell$-weight vectors for the representations obtained via Jimbo's homomorphism, known…

Level-one representations of the quantum affine superalgebra $U_q[\hat{gl(N|N)}]$ associated to the appropriate non-standard system of simple roots and $q$-vertex operators (intertwining operators) associated with the level-one modules are…

For quantum integrable models with elliptic R-matrix, we construct the Baxter Q-operator in infinite-dimensional representations of the algebra of observables.

By factorization of the Hamiltonian describing the quantum mechanics of the continuous q-Hermite polynomial, the creation and annihilation operators of the q-oscillator are obtained. They satisfy a q-oscillator algebra as a consequence of…

We construct a realization of the elliptic quantum algebra $U_{q,p}(\hat{sl_N})$ for any given level $k$ in terms of free boson fields and their twisted partners. It can be considered as the elliptic deformation of the Wakimoto realization…

An algebraic interpretation of the $q$-Meixner polynomials is obtained. It is based on representations of $\mathcal{U}_q(\mathfrak{su}(1,1))$ on $q$-oscillator states with the polynomials appearing as matrix elements of unitary…

We construct Q-operators for the open spin-1/2 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators…

Applying the Pasquier-Gaudin procedure we construct the Baxter's Q-operator for the homogeneous XXX model as integral operator in standard representation of SL(2). The connection between Q-operator and local Hamiltonians is discussed. It is…

We establish the $Q \widetilde{Q}$-systems for the twisted quantum affine algebras that were conjectured in arXiv:1606.05301. We develop the representation theory of Borel subalgebra of twisted quantum affine algebras and we construct their…

The properties of the quantum Minkowski space algebra are discussed. Its irreducible representations with highest weight vectors are constructed and relations to other quantum algebras: $su_{q}(2)$, $q$-oscillator, $q$-sphere are pointed…

We re-examine the level-one irreducible highest weight representations of the quantum affine superalgebra $U_q(\hat{sl}(2|1))$ and derive the characters and supercharacters associated with these representations. We calculate the exchange…

We define categories $\mathcal{O}^w$ of representations of Borel subalgebras $\mathcal{U}_q\mathfrak{b}$ of quantum affine algebras $\mathcal{U}_q\hat{\mathfrak{g}}$, which come from the category $\mathcal{O}$ twisted by Weyl group elements…

Baxter's Q-operator is generally believed to be the most powerful tool for the exact diagonalization of integrable models. Curiously, it has hitherto not yet been properly constructed in the simplest such system, the compact spin-1/2…

We construct Baxter operators as generalized transfer matrices being traces of products of generic $R$ matrices. The latter are shown to factorize into simpler operators allowing for explicit expressions in terms of functions of a Weyl pair…

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…

Given a real number $q$ such that $0<q<1$, the natural setting for the mathematics of a $q$-oscillator is an infinite-dimensional, separable Hilbert space that is said to provide an interpolation between the Bargmann-Segal space of…