S. L. Woronowicz

Compact matrix pseudogroups

The compact matrix pseudogroup is a non-commutative compact space endowed with a group structure. The precise definition is given and a number of examples is presented. Among them we have compact group of matrices, duals of discrete groups and twisted (deformed)SU(N) groups. The representation theory is developed. It turns out that the tensor product of representations depends essentially on their order. The existence and the uniqueness of the Haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived. The form of these relations differs from that in the group case. This is due to the fact that the Haar measure on pseudogroups is not central in general. The corresponding modular properties are discussed. The Haar measures on the twistedSU(2) group and on the finite matrix pseudogroup are found.

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.

Quantum deformation of Lorentz group

A one parameter quantum deformationSμL(2,ℂ) ofSL(2,ℂ) is introduced and investigated. An analog of the Iwasawa decomposition is proved. The compact part of this decomposition coincides withSμU(2), whereas the solvable part is identified as a Pontryagin dual ofSμU(2). It shows thatSμL(2,ℂ) is the result of the dual version of Drinfelds double group construction applied toSμU(2). The same construction applied to any compact quantum groupGc is discussed in detail. In particular the explicit formulae for the Haar measures on the Pontryagin dualGd ofGc and on the double groupG are given. We show that there exists remarkable 1-1 correspondence between representations ofG and bicovariant bimodules (“tensor bundles”) overGc. The theory of smooth representations ofSμL(2,ℂ) is the same as that ofSL(2,ℂ) (Clebsh-Gordon coefficients are however modified). The corresponding “tame” bicovariant bimodules onSμU(2) are classified. An application to 4D+ differential calculus is presented. The nonsmooth case is also discussed.

Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups

SummaryThe notion of concrete monoidalW*-category is introduced and investigated. A generalization of the Tannaka-Krein duality theorem is proved. It leads to new examples of compact matrix pseudogroups. Among them we have twistedSU(N) groups denoted bySμU(N). It is shown that the representation theory forSμU(N) is similar to that ofSU(N): irreducible representations are labeled by Young diagrams and formulae for dimensions and multiplicity are the same as in the classical case.

Positive maps of low dimensional matrix algebras

Abstract It is shown that any positive map from M 2 into M 3 is a sum of completely positive and completely copositive maps. This result does not hold for maps into M 4 . A generalization of the Kadison inequality is suggested and proved for positive maps defined on M 2 or M 3 .

Unbounded elements affiliated withC*-algebras and non-compact quantum groups

The affiliation relation that allows to include unbounded elements (operators) into theC*-algebra framework is introduced, investigated and applied to the quantum group theory. The quantum deformation of (the two-fold covering of) the group of motions of Euclidean plane is constructed. A remarkable radius quantization is discovered. It is also shown that the quantumSU(1, 1) group does not exist on theC*-algebra level for real value of the deformation parameter.

Functional calculus for sesquilinear forms and the purification map

The paper gives a proposition of a functional calculus for positive sesquilinear forms. A definition of any homogeneous function of two positive sesquilinear forms is given. The purification map for states on C∗-algebras is described in terms of the geometrical mean of two positive forms related to states in a natural way. Properties of the geometrical mean are investigated.

Passive states and KMS states for general quantum systems

We characterize equilibrium states of quantum systems by a condition of passivity suggested by the second principle of thermodynamics. Ground states and β-KMS states for all inverse temperatures β≧0 are completely passive. We prove that these states are the only completely passive ones. For the special case of states describing pure phases, assuming the passivity we reproduce the results of Haag et al.

FROM MULTIPLICATIVE UNITARIES TO QUANTUM GROUPS

An alternative version of the theory of multiplicative unitaries is presented. Instead of the original regularity condition of Baaj and Skandalis we formulate another condition selecting manageable multiplicative unitaries. The manageability is the property of multiplicative unitaries coming from the quantum group theory. For manageable multiplicative unitaries we reproduce all the essential results of the original paper of Baaj and Skandalis and much more. In particular the existence of the antipode and its polar decomposition is shown.

A C*-ALGEBRAIC FRAMEWORK FOR QUANTUM GROUPS

We develop a general framework to deal with the unitary representations of quantum groups using the language of C*-algebras. Using this framework, we prove that the duality holds in a general context. This extends the framework of the duality theorem using the language of von Neumann algebras previously developed by Masuda and Nakagami.