Related papers: Quantum groups and functional relations for arbitr…
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary --- or, in general, projective unitary --- representations implement the action of an abstract symmetry group on physical states…
A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given $R$-matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known…
We formalize the correspondence between quantum states and quantum operations isometrically, and harness its consequences. This correspondence was already implicit in the various proofs of the operator sum representation of Completely…
This is a self-contained review on the theory of quantum group and its applications to modern physics. A brief introduction is given to the Yang-Baxter equation in integrable quantum field theory and lattice statistical physics. The quantum…
By resorting to the Fock--Bargmann representation, we incorporate the quantum Weyl--Heisenberg algebra, $q$-WH, into the theory of entire analytic functions. The $q$--WH algebra operators are realized in terms of finite difference operators…
These notes are intended as an introduction to a study of applications of noncommutative calculus to quantum statistical Physics. Centered on noncommutative calculus we describe the physical concepts and mathematical structures appearing in…
The connection between q-analogs of special functions and representations of quantum algebras has been developed recently. It has led to advances in the theory of q-special functions that we here review.
The so called quantized algebras of functions on affine Hecke algebras of type A and the corresponding q-Schur algebras are defined and their irreducible unitarizable representations are classified.
$Q$-systems and $T$-systems are systems of integrable difference equations that have recently attracted much attention, and have wide applications in representation theory and statistical mechanics. We show that certain $\tau$-functions,…
We introduce a category $\mathcal O$ of representations of the elliptic quantum group associated with $\mathfrak{sl}_2$ with well-behaved $q$-character theory. We derive separation of variables relations for asymptotic representations in…
We study finite dimensional algebras that appear as fibers of quantum orders over a given point of variety of center. We present the formula for the number of irreducible representations and check it for it for the algebra of twisted…
In quantum mechanics, one can express the evolution operator and other quantities in terms of functional integrals. The main goal of this paper is to prove corresponding results in the geometric approach to quantum theory. We apply these…
Lie groups and quantum algebras are connected through their common universal enveloping algebra. The adjoint action of Lie group on its algebra is naturally extended to related q-algebra and q-coalgebra. In such a way, quantum structure can…
The elements of the wide class of quantum universal enveloping algebras are prooved to be Hopf algebras $H$ with spectrum $Q(H)$ in the category of groups. Such quantum algebras are quantum groups for simply connected solvable Lie groups…
We explore a wider theoretical framework that has quantum field theory built-in, taking the fact that quantum mechanics is reconstructed from quantum field theory as a hint. We formulate a quantum theory with an embedded structure by…
Baxter's Q-operator for the quantum transfer matrix of the XXZ spin-chain is constructed employing the representation theory of quantum groups. The spectrum of this Q-operator is discussed and novel functional relations which describe the…
We give closed formulae for the q-characters of the fundamental representations of the quantum loop algebra of a classical Lie algebra in terms of a family of partitions satisfying some simple properties. We also give the multiplicities of…
A class of two-dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one-dimensional factorized systems. By using standard methods to find the shape-invariant…
The quantum mechanics of one degree of freedom exhibiting the exact conformal SL(2,R) symmetry is presented. The starting point is the classification of the unitary irreducible representations of the SL(2,R) group (or, to some extent, its…
This is a review of the results related to generalizations of the notion of $\tau$-function and integrable hierarchies and to their interpretation within the group theory framework that admits an immediate quantization procedure. Different…