Piotr M. Sołtan

Quantum families of maps and quantum semigroups on finite quantum spaces

Abstract Quantum families of maps between quantum spaces are defined and studied. We prove that quantum semigroup (and sometimes quantum group) structures arise naturally on such objects out of more fundamental properties. As particular cases we study quantum semigroups of maps preserving a fixed state and quantum commutants of given quantum families of maps.

Quantum SO(3) groups and quantum group actions on M2

Answering a question of Shuzhou Wang we give a description of quantum SO.3/ groups of Podleś as universal compact quantum groups acting on the C*-algebra M2 and preserving the Powers state. We use this result to give a complete classification of all continuous compact quantum group actions on M2. Mathematics Subject Classification (2010). 46L89, 58B34, 58B32; 17B37, 16W30.

A Remark on Manageable Multiplicative Unitaries

We propose a weaker condition for multiplicative unitary operators related to quantum groups, than the condition of manageability introduced by S. L. Woronowicz. We prove that all the main results of the theory of manageable multiplicative unitaries remain true under this weaker condition. We also show that multiplicative unitaries arising naturally in the construction of some recent examples of noncompact quantum groups satisfy our condition, but fail to be manageable.

EMBEDDABLE QUANTUM HOMOGENEOUS SPACES

Abstract We discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups. On the von Neumann algebra level we recall an interesting duality for such objects studied earlier by M. Izumi, R. Longo, S. Popa for compact Kac algebras and by M. Enock in the general case of locally compact quantum groups. A definition of a quantum homogeneous space is proposed along the lines of the pioneering work of Vaes on induction and imprimitivity for locally compact quantum groups. The concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces. Among various examples we single out the quantum analog of the quotient of the Cartesian product of a quantum group with itself by the diagonal subgroup, analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles. The former turns out to be an interesting application of the duality mentioned above.

Noncommutative homogeneous spaces: The matrix case

Abstract Given a quantum subgroup G ⊂ U n and a number k ≤ n we can form the homogeneous space X = G / ( G ∩ U k ) , and it follows from the Stone–Weierstrass theorem that C ( X ) is the algebra generated by the last n − k rows of coordinates on G . In the quantum group case the analogue of this basic result does not necessarily hold, and we discuss here its validity, notably with a complete answer in the group dual case. We focus then on the “easy quantum group” case, with the construction and study of several algebras associated to the noncommutative spaces of type X = G / ( G ∩ U k + ) .

Quantum Families of Invertible Maps and Related Problems

The notion of families of quantum invertible maps (

Projective limits of quantum symmetry groups and the doubling construction for Hopf algebras

C^*

QUANTUM AUTOMORPHISM GROUPS OF FINITE QUANTUM GROUPS ARE CLASSICAL

-algebra homomorphisms satisfying Podle\s condition) is employed to strengthen and reinterpret several results concerning universal quantum groups acting on finite quantum spaces. In particular Wangs quantum automorphism groups are shown to be universal with respect to quantum families of invertible maps. Further the construction of the Hopf image of Banica and Bichon is phrased in the purely analytic language and employed to define the quantum subgroup generated by a family of quantum subgroups or more generally a family of quantum invertible maps.

Compactifications of Discrete Quantum Groups

The quantum symmetry group of the inductive limit of C*-algebras equipped with orthogonal filtrations is shown to be the projective limit of the quantum symmetry groups of the C*-algebras appearing in the sequence. Some explicit examples of such projective limits are studied, including the case of quantum symmetry groups of the duals of finite symmetric groups, which do not fit directly into the framework of the main theorem and require further specific study. The investigations reveal a deep connection between quantum symmetry groups of discrete group duals and the doubling construction for Hopf algebras.

Self-Adjoint Extensions of Symmetric Operators

Abstract In a recent paper of Bhowmick, Skalski and Soltan the notion of a quantum group of automorphisms of a finite quantum group was introduced and, for a given finite quantum group G , existence of the universal quantum group acting on G by automorphisms was proved. We show that this universal quantum group is in fact a classical group. The key ingredient of the proof is the use of multiplicative unitary operators, and we include a thorough discussion of this notion in the context of finite quantum groups.