On Representations of Quantum Conjugacy Classes of GL(n)

Let O be a closed Poisson conjugacy class of the complex algebraic Poisson group GL(n) relative to the Drinfeld-Jimbo factorizable classical r-matrix. Denote by T the maximal torus of diagonal matrices in GL(n). With every ${a \in O \cap T}$ we associate a highest weight module M _a over the quantum group ${U_q \bigl(\mathfrak{g} \mathfrak{l}(n)\bigr)}$ and an equivariant quantization ${\mathbb{C}_{\hbar,a}[O]}$ of the polynomial ring ${\mathbb{C}[O]}$ realized by operators on M _a . All quantizations ${\mathbb{C}_{\hbar,a}[O]}$ are isomorphic and can be regarded as different exact representations of the same algebra, ${\mathbb{C}_{\hbar}[O]}$ . Similar results are obtained for semisimple adjoint orbits in ${\mathfrak{g} \mathfrak{l}(n)}$ equipped with the canonical GL(n)-invariant Poisson structure.     

Differential Calculus on Quantum Matrix Lie Groups

Following the discussions on quantum groups by Reshetikhin, Takhadshyan and Faddeev, we thoroughly studied the differential calculus on quantum groups, and obtained the explicit expreeeione in the case of SLq(2). We also compared our results with those of Woronowics and Podles under the restriction of SLq(2) to SUq(2).     

Non-Levi Closed Conjugacy Classes of SP q (2n)

We construct an explicit quantization of semi-simple conjugacy classes of the complex symplectic group SP(2n) with non-Levi isotropy subgroups through an operator realization on highest weight modules over the quantum group ${U_q\bigl(\mathfrak{sp}(2n)\bigr)}$ .     

量子态的等价类与量子逻辑门

在量子力学中,态的演化是一个幺正演化过程,态的演化过程可以用演化算子对态的作用来表示,幺正演化过程是时间可逆的.基于这一基本事实,Gerard't Hoofl引进了量子态的等价类概念,并用两组等价类之间的变换来描述量子态的幺正演化.本文利用等价类的概念及其变换来探究构建量子信息论中常用的通用量子门,给出通用量子门的推广形式.最后说明这些通用量子门可以基于双qubit体系内在的相互作用Hamilton量得以实现.     

Quantum Matrix Pairs

The notion of quantum matrix pairs is defined. These are pairs of matrices with noncommuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of powers of the matrices obey the same pattern of internal relations as the original pair. Such matrices appear in an approach by the authors to quantizing gravity in 2 space and 1 time dimensions with negative cosmological constant on the torus. Explicit examples and transformations which generate new pairs from a given pair are presented.     

Exchange Dynamical Quantum Groups

For any simple Lie algebra ? and any complex number q which is not zero or a nontrivial root of unity, %but may be equal to 1 we construct a dynamical quantum group (Hopf algebroid), whose representation theory is essentially the same as the representation theory of the quantum group U _q (?). This dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of U _q (?), and is an algebraic structure standing behind these relations. Received: 24 March 1998 / Accepted: 14 February 1999     

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik–Zamolodchikov–Bernard equation, to throw some light on the quasi-Hopf structure of conformal field theory and (perhaps) the Calogero–Moser models.     

Representations of Multiparameter Quantum Groups

We construct representations of the quantum algebras U_q,q(gl(n)) and U_q,q(sl(n)) which are in duality with the multiparameter quantum groups GL_qq(n), SL_qq(n), respectively. These objects depend on n(n − 1)/2+ 1 deformation parameters q, q_ij (1 ≤ i< j ≤ n) which is the maximal possible number in the case of GL(n). The representations are labelled by n − 1 complex numbers r_i and are acting in the space of formal power series of n(n − 1)/2 non-commuting variables. These variables generate quantum flag manifolds of GL_qq(n), SL_qq(n). The case n = 3 is treated in more detail.

     

Semiinfinite Cohomology of Quantum Groups

We define a new cohomology theory of associative algebras called semiinfinite cohomology in the derived categories' setting. We investigate the case of a small quantum group u, calculate semiinfinite cohomology spaces of the trivial u-module and express them in terms of local cohomology of the nilpotent cone for the corresponding semisimple Lie algebra. We discuss the connection between the semiinfinite homology of u and the conformal blocks' spaces. Received: 14 October 1996 / Accepted: 25 February 1997     

Quantum Mechanics on Finite Groups

Although a few new results are presented, this is mainly a review article on the relationship between finite-dimensional quantum mechanics and finite groups. The main motivation for this discussion is the hidden subgroup problem of quantum computation theory. A unifying role is played by a mathematical structure that we call a Hilbert ^*-algebra. After reviewing material on unitary representations of finite groups we discuss a generalized quantum Fourier transform. We close with a presentation concerning position-momentum measurements in this framework.     

Quantum Systems on Linear Groups

Discussed are quantized dynamical systems on orthogonal and affine groups. The special stress is laid on geodetic systems with affinely-invariant kinetic energy operators. The resulting formulas show that such models may be useful in nuclear and hadronic dynamics. They differ from traditional Bohr–Mottelson models where SL(n,ℝ) is used as a so-called non-invariance group. There is an interesting relationship between classical and quantized integrable lattices. PACS: 11.30.Ly, 02.20.-a, 21.60.Ev.     

Quantum Groups and Fuss--Catalan Algebras

The categories of representations of compact quantum groups of automorphisms of certain inclusions of finite dimensional ℂ^*-algebras are shown to be isomorphic to the categories of Fuss–Catalan diagrams. Received: 9 March 2001 / Accepted: 12 November 2001     

Duality for Coloured Quantum Groups

Duality between the coloured quantum group and the coloured quantum algebra corresponding to GL(2) is established. The coloured L ^± functionals are constructed and the dual algebra is derived explicitly. These functionals are then employed to give a coloured generalisation of the differential calculus on quantum GL(2) within the framework of the R-matrix approach.     

Quantum Random Walk Approximation on Locally Compact Quantum Groups

A natural scheme is established for the approximation of quantum Lévy processes on locally compact quantum groups by quantum random walks. We work in the somewhat broader context of discrete approximations of completely positive quantum stochastic convolution cocycles on C*-bialgebras.     

Bicross-Product Structure of Affine Quantum Groups

We show that the affine quantum group is isomorphic to a bicross-product central extension of the quantum loop group by a quantum cocycle in R-matrix form.     

Quantum Symmetry Groups of Noncommutative Spheres

We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for certain compact quantum groups. We give a general construction of homogeneous spaces which support noncommutative spin geometries. Received: 28 February 2001 / Accepted: 12 March 2001     

Simple Current Actions of Cyclic Groups

Permutation actions of simple currents on the primaries of a Rational Conformal Field Theory are considered in the framework of admissible weighted permutation actions. The solution of admissibility conditions is presented for cyclic quadratic groups: an irreducible WPA corresponds to each subgroup of the quadratic group. As a consequence, the primaries of a RCFT with an order n integral or half-integral spin simple current may be arranged into multiplets of length k² (where k is a divisor of n) or 3k² if the spin of the simple current is half-integral and k is odd.Mathematical Subject Classifications (2000). 81T40.Work supported by grant OTKA TS044839.     

Regular Nilpotent Elements and Quantum Groups

We suggest new realizations of quantum groups U _q (?) corresponding to complex simple Lie algebras, and of affine quantum groups. These new realizations are labeled by Coxeter elements of the corresponding Weyl group and have the following key feature: The natural counterparts of the subalgebras U(?), where ?⊂? is a maximal nilpotent subalgebra, possess non-singular characters. Received: 29 May 1998 / Accepted: 12 January 1999