Related papers: Universal integrability objects
Definition of Dirac operators on the quantum group $SU_{q}(2)$ and the quantum sphere $S^{2}_{q \mu}$ are discussed. In both cases similar $SU_{q}(2)$-invariant form is obtained. It is connected with corresponding Laplace operators.
Representations of Quantum Groups U_q (g_n), g_n any semi simple Lie algebra of rank n, are constructed from arbitrary representations of rank n-1 quantum groups for q a root of unity. Representations which have the maximal dimension and…
We discuss the dynamical quantum systems which turn out to be bi-unitary with respect to the same alternative Hermitian structures in a infinite-dimensional complex Hilbert space. We give a necessary and sufficient condition so that the…
The concept of quantum representation of finite groups (QRFG) has been a fundamental aspect of quantum computing for quite some time, playing a role in every corner, from elementary quantum logic gates to the famous Shor's and Grover's…
A problem of universality in simulation of evolution of quantum system and in theory of quantum computations is related with the possibility of expression or approximation of arbitrary unitary transformation by composition of specific…
Vertex operators associated with level two $U_q(\widehat{sl}_2)$ modules are constructed explicitly using bosons and fermions. An integral formula is derived for the trace of products of vertex operators. These results are applied to give…
This is a short, self-contained expository survey, focused on algebraic and analytic aspects of quantum groups. Topics covered include the definition of ``quantum group,'' the Yang-Baxter equation, quantized universal enveloping algebras,…
We show that in quantum computation almost every gate that operates on two or more bits is a universal gate. We discuss various physical considerations bearing on the proper definition of universality for computational components such as…
The two-parametric quantum superalgebra $U_{p,q}[gl(2/1)]$ is consistently defined. A construction procedure for induced representations of $U_{p,q}[gl(2/1)]$ is described and allows us to construct explicitly all (typical and nontypical)…
When the $q$-deformed creation and annihilation operators are used in a second quantization procedure, the algebra satisfied by basis vectors (orthogonal complete set) should be also deformed such as a field operator remains invariant under…
The integrable structure of the two dimensional superconformal field theory is considered. The classical counterpart of our constructions is based on the $\hat{osp}(1|2)$ super-KdV hierarchy. The quantum version of the monodromy matrix…
Metal--insulator transitions and transitions between different quantum Hall liquids are used to describe the physical ideas forming the basis of quantum phase transitions and the methods of application of theoretical results in processing…
This paper is meant to be an informal introduction to Quantum Groups, starting from its origins and motivations until the recent developments. We call in particular the attention on the newly descovered relationship among quantum groups,…
The universal quantum computer is a device capable of simulating any physical system and represents a major goal for the field of quantum information science. Algorithms performed on such a device are predicted to offer significant gains…
We study the Universality and Membership Problems for gate sets consisting of a finite number of quantum gates. Our approach relies on the techniques from compact Lie groups theory. We also introduce an auxiliary problem called Subgroup…
Let ${\mathbf U}(n)$ be the quantum enveloping algebra of ${\frak {gl}}_n$ over $\mathbb Q(v)$, where $v$ is an indeterminate. We will use $q$-Schur algebras to realize the integral form of ${\mathbf U}(n)$. Furthermore we will use this…
An approach to study a generalization of the classical-quantum transition for general systems is proposed. In order to develop the idea, a deformation of the ladder operators algebra is proposed that contains a realization of the quantum…
The representation theory of the quantum group su$_q(2)$ is used to introduce $q$-analogues of the Wigner rotation matrices, spherical functions, and Legendre polynomials. The method amounts to an extension of variable separation from…
We introduce the concept of quantum supermap, describing the most general transformation that maps an input quantum operation into an output quantum operation. Since quantum operations include as special cases quantum states, effects, and…
The two-parametric quantum superalgebra $U_{pq}[gl(2/2)]$ and its representations are considered. All finite-dimensional irreducible representations of this quantum superalgebra can be constructed and classified into typical and nontypical…