Coherent states for quantum compact groups

Coherent states are introduced and their properties are discussed for simple quantum compact groupsA _l, B_l, C_l andD _l. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit. The coherent state is interpreted as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compactR-matrix formulation (generalizing this way theq-deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum group (the analogue of the Borel-Weil construction) is described using the concept of coherent state. The relation between representation theory and non-commutative differential geometry is suggested. Dedicated to Professor L.D. Faddeev on his 60th birthday     

A remark on compact matrix quantum groups

A simpler set of axioms of the theory of compact matrix quantum groups (pseudogroups) is found.Supported by Japan Society for Promoting Science. On leave from the Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoa 74, 00-682 Warsaw, Poland.     

Deformations of compact quantum groups via Rieffel's quantization

It is shown that compact quantum groups containing torus subgroups can be deformed into new compact quantum groups under Rieffel's quantization. This is applied to showing that the two classes of compact quantum groupsK _q ^u andK _q studied by Levendorkii and Soibelman are strict deformation quantization of each other, and that the quantum groupsA _u (m) have many deformations.     

On unitary representation theories of compact quantum groups

Categorical structure of unitary representation of compact quantum groups is studied with relation to a metrical structure encountered in the monoidal category of bimodules of finite Jones index.     

Faithful actions of locally compact quantum groups on classical spaces

We construct examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on connected classical (noncompact) smooth manifolds. However, none of these actions can be isometric in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009), leading to the conjecture that the result obtained by Goswami and Joardar (Rigidity of action of compact quantum groups on compact, connected manifolds, 2013. arXiv:1309.1294) about nonexistence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the noncompact case as well.     

Invariant star products and representations of compact semisimple Lie groups

Starting from the work by F. A. Berezin, and earlier paper by the author defined an invariant star product on every nonexceptional Kähler symmetric space. In this Letter a recursion formula is obtained to calculate the corresponding invariant Hochschild 2-cochains for spaces of types II and III. An invariant star product is defined on every integral symplectic (Kähler) homogeneous space of simply-connected compact Lie groups (on every integral orbit of the coadjoint representation). The invariant 2-cochains are obtained from the Bochner-Calabi function of the space. The leading term of the lth-2-cochain is determined by the l-power of the Laplace operator.     

Differential calculus on compact matrix pseudogroups (quantum groups)

The paper deals with non-commutative differential geometry. The general theory of differential calculus on quantum groups is developed. Bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied. Tensor algebra and external algebra constructions are described. It is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra, so the external derivative of higher order differential forms is well defined and obeys the usual properties. The proper form of the Cartan Maurer formula is found. The vector space dual to the space of left-invariant differential forms is endowed with a bilinear operation playing the role of the Lie bracket (commutator). Generalized antisymmetry relation and Jacobi identity are proved.     

Star products and quantum groups in quantum mechanics and field theory

Hopf algebras and quantum groups have recently been applied to the analysis of the combinatorics of Feynman graphs in relativistic quantum field theory. On the other hand, in accordance with the program of deformation quantization, the relation between star products and the perturbative expansion in field theory has also been the subject of intensive study. In the present work we clarify the relation between these two approaches. We show how these techniques can be applied in a unified way to quantum systems with a finite number of degrees of freedom and to quantum field theories. In particular, we find that the time-ordered product of quantum fields is the Weyl transform of a certain twisted product. We also show that one can pass from systems involving bosons to systems with fermions, essentially just by replacing the symmetric algebra of the relevant vector space by its exterior algebra.     

CQG algebras: A direct algebraic approach to compact quantum groups

The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar functional on any CQG algebra are established. It is shown that a CQG algebra can be naturally completed to aC ^*-algebra. The relations between our approach and several other approaches to compact quantum groups are discussed.     

On the Wigner-Eckart theorem for tensor operators connected with compact matrix quantum groups

In this Letter, the notion of a tensor operator connected with a unitary, smooth, finite-dimensional representation of a compact, matrix quantum group is introduced and investigated. It is proved that, for compact matrix, simply reducible quantum groups, there exists a theorem analogous to the famous Wigner-Eckart theorem.     

Ergodic endomorphisms of compact abelian groups

We show that for a surjective endomorphism of a compact abelian group ergodicity is equivalent to a condition which impliesr-mixing for allr1, and we characterize such maps algebraically. This is then used in proving the ergodicity of an extensive class of endomorphisms of the binary sequence space. As a simple corollary it is found that one-dimensional linear cellular automata and the accumulator automata arer-mixing for allr1.This work was supported in part by grants from NSERC     

Inhomogeneous quantum groups

Inhomogeneous quantum groups corresponding to the homogeneous quantum groupsU _q (N), SO _q (N) and theq-deformed Lorentz group acting on affine quantum spaces are constructed. Special representations of the translation part are investigated.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.     

Inhomogeneous quantum groups

Inhomogeneous quantum groups corresponding to the homogeneous quantum groupsU _q (N), SO _q (N) and theq-deformed Lorentz group acting on affine quantum spaces are constructed.     

Faithful compact quantum group actions on connected compact metrizable spaces

We construct faithful actions of quantum permutation groups on connected compact metrizable spaces. This disproves a conjecture of Goswami.     

Quantum dressing orbits on compact groups

The quantum double is shown to imply the dressing transformation on quantum compact groups and the quantum Iwasawa decompositon in the general case. Quantum dressing orbits are described explicitly as *-algebras. The dual coalgebras consisting of differential operators are related to the quantum Weyl elements. Besides, the differential geometry on a quantum leaf allows a remarkably simple construction of irreducible *-representations of the algebras of quantum functions. Representation spaces then consist of analytic functions on classical phase spaces. These representations are also interpreted in the framework of quantization in the spirit of Berezin applied to symplectic leaves on classical compact groups. Convenient coherent states are introduced and a correspondence between classical and quantum observables is given.