Actions Of Quantum Groups Research Articles

The noncommutative space of stochastic diffusion systems

In the matrix product ground states approach to n-species diffusion processes the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra determined by the dynamics of the process. We show that the quadratic algebra defines a noncommutative space with a GLq(n) quantum group action as its symmetry. Boundary processes account for the appearance of parameter-dependent linear terms in the algebraic relations. We argue that for systems with boundary conditions the diffusion algebras are also obtained either by a shift of basis in the n-dimensional quantum plane or by an appropriate change of basis in a lower dimensional one which leads to a reduction of the GLq(n) symmetry.

The Unitary Implementation of a Locally Compact Quantum Group Action

In this paper we study actions of locally compact quantum groups on von Neumann algebras and prove that every action has a canonical unitary implementation, paralleling Haagerup's classical result on the unitary implementation of a locally compact group action. This result is an important tool in the study of quantum groups in action. We will use it in this paper to study subfactors and inclusions of von Neumann algebras. When α is an action of the locally compact quantum group (M, Δ) on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion Nα⊂N↪Mα⋉N is a basic construction. Here Nα denotes the fixed point algebra and Mα⋉N is the crossed product. When α is an outer and integrable action on a factor N we prove that the inclusion Nα⊂N is irreducible, of depth 2 and regular, giving a converse to the results of M. Enock and R. Nest (1996, J. Funct. Anal.137, 466–543; 1998, J. Funct. Anal.154, 67–109). Finally we prove the equivalence of minimal and outer actions and we generalize the main theorem of Yamanouchi (1999, Math. Scand.84, 297–319): every integrable outer action with infinite fixed point algebra is a dual action.

Generalized noiseless quantum codes utilizing quantum enveloping algebras

A generalization of the results of Rasetti and Zanardi concerning avoiding errors in quantum computers by using states preserved by evolution is presented. The concept of the dynamical symmetry is generalized from the level of classical Lie algebras and groups, to the level of a dynamical symmetry based on quantum Lie algebras and quantum groups (in the sense of Woronowicz). An intrinsic dependence of the concept of dynamical symmetry on the differential calculus (which holds also in the classical case) is stressed. A natural connection between quantum states invariant under a quantum group action, and quantum states preserved by the dynamical evolution is discussed.

Quantum Group Actions on the Cuntz Algebra

The Cuntz Algebra carries in a natural way the structure of a module algebra over the quantized universal enveloping algebra U q (g), and the structure of a co-module algebra over the quantum group G q associated with U q (g). We determine the fixed and co-fixed point algebras in the case of the classical quantum algebras and groups. We also show that the Hopf algebraic structure of the classical quantum groups can be recovered from a knowledge of the co-fixed point algebras.

The R-matrix action of untwisted affine quantum groups at roots of 1

Let gˆ be an untwisted affine Kac–Moody algebra. The quantum group Uq(gˆ) is known to be a quasitriangular Hopf algebra (to be precise, a braided Hopf algebra). Here we prove that its unrestricted specializations at odd roots of 1 are braided too: in particular, specializing q at 1 we have that the function algebra F[Hˆ] of the Poisson proalgebraic group Ĥ dual of Ĝ (a Kac–Moody group with Lie algebra gˆ) is braided. This in turn implies also that the action of the universal R-matrix on the tensor products of pairs of Verma modules can be specialized at odd roots of 1.

Hopf stars, twisted Hopf stars and scalar products on quantum spaces

The properties of Hopf star operations and twisted Hopf star operations on quantum groups are discussed in relation with the theory of representations (star representations). Invariant Hermitian sesquilinear forms (scalar products) on modules or module-algebras are then defined and analyzed. Particular attention is paid to scalar products that can be associated with the Killing form (when it exists) or with the left (or right) invariant integrals on the quantum group. Our results are systematically illustrated in the case of a family of non-semisimple and finite dimensional quantum groups that are obtained as Hopf quotients of the quantum enveloping algebra U q(sl(2, C)) , q being an Nth root of unity. Many explicit results concerning the case N=3 are given. We also mention several physical motivations for the present work: conformal field theory, spin chains, integrable models, generalized Yang–Mills theory with quantum group action and the search for finite quantum groups symmetries in particle physics.

Ergodic Actions of Universal Quantum Groups on Operator Algebras

We construct ergodic actions of compact quantum groups on C^*-algebras and von Neumann algebras, and exhibit phenomena of such actions that are of a different nature from ergodic actions of compact Lie groups. In particular, we construct: (1). ergodic actions of the compact quantum groups $A_u(Q)$ on the Powers factors; (2). ergodic actions of the compact quantum groups $A_u(n)$ on the hyperfinite II_1 factor; (3). ergodic actions of the compact quantum groups $A_u(Q)$ on the Cuntz algebras; (4). ergodic actions of general compact quantum groups on their homogeneous spaces and an example of a non-homogeneous classical space that nevertheless admits an ergodic action of a compact quantum group.

Finite dimensional quantum group covariant differential calculus on a complex matrix algebra

Using the fact that the algebra M 3( C) of 3×3 complex matrices can be taken as a reduced quantum plane, we build a differential calculus Ω(S) on the quantum space S defined by the algebra C ∞(M)⊗M 3( C) , where M is a space-time manifold. This calculus is covariant under the action and coaction of finite dimensional dual quantum groups. We study the star structures on these quantum groups and the compatible one in M 3( C) . This leads to an invariant scalar product on the later space. We analyse the differential algebra Ω(M 3( C)) in terms of quantum group representations, and consider in particular the space of 1-forms on S since its elements can be considered as generalized gauge fields.

Actions of compact quantum groups on 𝐶*-algebras

In this paper we investigate a structure of the fixed point algebra under an action of compact matrix quantum group on a C ∗ C^* -algebra B \mathcal {B} . We also show that the categories of C \mathcal C -comodules in B \mathcal B and inner endomorphisms restricted to the fixed point algebra coincide when the relative commutant of the fixed point algebra is trivial. Next we show a version of the Tannaka duality theorem for twisted unitary groups.

Solutions of Klein-Gordon and Dirac equations on quantum Minkowski spaces

Covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified. In the case of quantum Minkowski spaces they have the same dimensions as in the classical case. Formal solutions of the corresponding Klein--Gordon and Dirac equations are found. The Fock space construction is sketched.

Quantum spin chains with quantum group symmetry

We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasi-local) tails of the action, we find that there are no actions of a properly quantum group commuting with lattice translations. The non-locality arises from the ordering of factors in the quantum groupC *-algebra, and can be made one-sided, thus allowing semi-local actions on a half chain. Under such actions, localized quantum group invariant elements remain localized. Hence the notion of interactions invariant under the quantum group and also under translations, recently studied by many authors, makes sense even though there is no global action of the quantum group. We consider a class of such quantum group invariant interactions with the property that there is a unique translation invariant ground state. Under weak locality assumptions, its GNS representation carries no unitary representation of the quantum group.

Canonical bases for the Brauer centralizer algebra

In this paper we construct canonical bases for the Birman-Wenzl algebra BWn, the q-analogue of the Brauer centralizer algebra, and so define left, right and two-sided cells. We describe these objects combinatorially (generalizing the Robinson-Schensted algorithm for the symmetric group) and show that each left cell carries an irreducible representation of BWn. In particular, we obtain canonical bases for each representation, defined over Z. The same technique generalizes to an arbitrary tangle algebra and Rmatrix [R]; in particular to centralizers of the quantum group action on V ⊗r, for V a finite dimensional representation of a quantum group. BWn occurs for particular values of the parameters (q, r, x) as the centralizers of the action of Uqsp2k or Uqok on the n-th tensor power of its standard representation V . One may presumably transfer the bases of the BWn modules to give a basis of representations occurring in V ⊗n (as in [GL]), and it is natural to conjecture that the basis so obtained coincides with that of [L,§27]. Of the Weyl groups, only in the symmetric group are the cell representations irreducible. In this respect BWn is similar to Sn. One would expect this because of the relation with quantum groups, which also behave like Hecke algebras of type A [L]. Moreover, our main new insight into the structure of BWn is precisely of this form—we show that every representation is induced from a representation of a symmetric group in a precise way (see §6.5). This paper is essentially self-contained, except for an appeal to the solution of the corresponding problem for Sn in [KL,1.4]. In particular, we make no further mention of quantum groups and use no previous work on the structure of BWn (e.g. [BW,HR,W]) except for its description as a

Rings of skew polynomials and Gel'fand-Kirillov conjecture for quantum groups

We introduce and study action of quantum groups on skew polynomial rings and related rings of quotients. This leads to a “q-deformation” of the Gel'fand-Kirillov conjecture which we partially prove. We propose a construction of automorphisms of certain non-commutative rings of quotients coming from complex powers of quantum group generators; this is applied to explicit calculation of singular vectors in Verma modules over\(U_q (\mathfrak{s}\mathfrak{l}_{n + 1} )\). We finally give a definition of aq-connection with coefficients in a ring of skew polynomials and study the structure of quantum group modules twisted by aq-connection.

Quantum groups and WZNW models

An explanation of the appearance of quantum groups in chiral WZNW models is given. Invariance of the theory under quantum group action is discussed.