Related papers: On quantum Freidel-Maillet algebra for non-ultralo…

The integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic Poisson algebra, whose the general form is investigated. Each classical superintegrable…

We propose a generalization of quantization as a categorical way. For a fixed Poisson algebra quantization categories are defined as subcategories of R-module category with the structure of classical limits. We construct the generalized…

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always…

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL_n. We construct the action of the quantum loop algebra U_v(Lsl_n) in the equivariant K-theory of Laumon spaces…

We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite dimensional algebra, whose representation theory is analogous to blocks of Bernstein--Gelfand--Gelfand category $\mathcal O$.…

The problem of construction of irreducible representations of quantum $A^q_n$ algebras is solved at the level of explicit integration of the linear (inhomogeneous) system in finite differences in the n-dimensional space. The general…

Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical…

Let U be the quantum group associated to a symmetrizable generalized Cartan matrix. We give a realization of U from the category of the representations of certain product valued quiver.

To a tree of semi-simple algebras we associate a qurve (or formally smooth algebra) S. We introduce a Zariski- and etale quiver describing the finite dimensional representations of S. In particular, we show that all quotient varieties of…

There is a standard way to define two symplectic (hamiltonian) structures, the first and second Gelfand-Dikii brackets, on the space of ordinary linear differential operators of order $m$, $L = -d^m + U_1 d^{m-1} + U_2 d^{m-2} + \ldots +…

The minimal irreducible representations of $U_q[gl(m|n)]$, i.e. those irreducible representations that are also irreducible under $U_q[osp(m|n)]$ are investigated and shown to be affinizable to give irreducible representations of the…

The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal…

For a quantum affine algebra of type A, we describe the composition series of the tensor product of a general minimal affinization with a Kirillov-Resehtikhin module associated to an extreme node of the Dynkin diagram of the underlying…

We clarify the algebraic structure of continuous and discrete quasi-exactly solvable spectral problems by embedding them into the framework of the quantum inverse scattering method. The quasi-exactly solvable hamiltonians in one dimension…

We develop the theory of $q$-characters for quantum affine superalgebras of type $A$ in connection with deformed Cartan matrices. To achieve this, we establish a Khoroshkin-Tolstoy-type multiplicative formula of the universal $R$-matrix of…

We study special class of matrices with noncommutative entries and demonstrate their various applications in integrable systems theory. They appeared in Yu. Manin's works in 87-92 as linear homomorphisms between polynomial rings; more…

In this paper we discuss some properties of Baxter's TQ equation for the eight-vertex elliptic Sklyanin algebra it its compact representation based on the elliptic Gamma-functions. As the main result, we establish the structure of the…

Let q be a finite-dimensional Lie algebra and $\theta$ an automorphism of q of order m. We extend $\theta$ to an automorphism of the loop algebra of q and consider the fixed-point subalgebra $q[t,t^{-1}]^{\theta}$. Using a splitting of…

This paper provides the foundations of quantum Clifford analysis in $q$-commutative variables with symmetric difference operators. We consider a $q$-Dirac operator on the quantum Euclidean space that factorizes the $U_q(\frak{o})$-invariant…

It is known that there is a correspondence between representations of superalgebras and ordinary (non-graded) algebras. Keeping in mind this type of correspondence between the twisted quantum affine superalgebra $U_{q}(gl(2r|1)^{(2)})$ and…