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The microscopic theories of quantum gravity related to integrable lattice models can be constructed as special deformations of pure gravity. Each such deformation is defined by a second order differential operator acting on the coupling…
We perform generalizations of Witt and Virasoro algebras, and derive the corresponding Korteweg-de Vries equations from known R(p,q)-deformed quantum algebras previously introduced in J. Math. Phys. 51, 063518, (2010). Related relevant…
The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized root systems suggested by V. Serganova.…
This paper is a continuation of "Quantization of Lie bialgebras I-IV". The goal of this paper is to define and study the notion of a quantum vertex operator algebra in the setting of the formal deformation theory and give interesting…
The present article is devoted to one class of generalizations of the Salem functions. To construct such functions by systems of functional equations, the generalized shift operator is used.
Ideas from deformation quantization applied to algebras with one generator lead to methods to treat a nonlinear flat connection. It provides us elements of algebras to be parallel sections. The moduli space of the parallel sections is…
Surprising links between the deformation of 2D quantum field theories induced by the composite $\textrm{T} \bar{\textrm{T}}$ operator, effective string models and the $AdS/$CFT correspondence, have recently emerged. The purpose of this…
To each representation of the elliptic quantum group $E_{\tau,\eta}(sl_2)$ is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebraic Bethe ansatz method. Special cases of this…
We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to affine Kac--Moody algebras. Our experience with the Gaudin…
We consider a natural generalization of the Carlsson-Okounkov Ext operator on the K-theory groups of the moduli spaces of stable sheaves on a smooth projective surface. We compute the commutation relations between the Ext operator and the…
In this paper, we present an explicit realization of q-deformed Calogero-Vasiliev algebra whose generators are first-order q-difference operators related to the generalized discrete q-Hermite II polynomials recently introduced in [13].…
The q-deformed traces and orbits for the two parametric quantum groups $GL_{qp}(2)$ and $GL_{qp}(1|1)$ are defined. They are subsequently used in the construction of $q$-orbit invariants for these groups. General $qp$-(super)oscillator…
We elaborate the generalizations of the approach to gauge-invariant deformations of the gauge theories developed in our previous work [1]. In the given paper we construct the exact transformations defying the gauge-invariant deformed theory…
Starting with any R-matrix with spectral parameter, obeying the Yang-Baxter equation and a unitarity condition, we construct the corresponding infinite dimensional quantum group U_{R} in term of a deformed oscillators algebra A_R. The…
Generalized quantum cluster algebras introduced in [1] are quantum deformation of generalized cluster algebras of geometric types. In this paper, we prove that the Laurent phenomenon holds in these generalized quantum cluster algebras. We…
We obtain the Baxter Q-operators in the $U_q(\hat{sl}_2)$ invariant integrable models as a special limits of the quantum transfer matrices corresponding to different spins in the auxiliary space both from the functional relations and from…
We formulate equations of motion and conservation laws for a quantum many-body system in a co-moving Lagrangian reference frame. It is shown that generalized inertia forces in the co-moving frame are described by Green's deformation tensor…
We propose a conjectural correspondence between the spectra of the Bethe algebra for the quantum toroidal $\mathfrak{gl}_2$ algebra on relaxed Verma modules, and $q$-hypergeometric opers with apparent singularities. We introduce alongside…
For a real or complex semisimple Lie group $G$ and two nested parabolic subgroups $Q\subset P\subset G$, we study parabolic geometries of type $(G,Q)$. Associated to the group $P$, we introduce a class of relative natural bundles and…
We discuss the deformation quantization approach for the teaching of quantum mechanics. This approach has certain conceptual advantages which make its consideration worthwhile. In particular, it sheds new light on the relation between…