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We consider the compact quantum group U q (2) for q ∈ C \ { 0 } with | q | ≠ 1 , and decompose the tensor product of two irreducible representations into irreducible components. The decomposition is realized in terms of a basis of homogeneous polynomials in four variables involving the matrix elements of the irreducible representations of U q (2) . Then, we compute the Clebsch–Gordan coefficients in terms of the q-hypergeometric series 3 ϕ 2 . When q is real, the Clebsch–Gordan coefficients are real and its expression can be written in terms of the q-Hahn polynomial Q n (x ; a , b , N | q 2) . [ABSTRACT FROM AUTHOR]

The non-commutative unitary C ∗ -algebra U n nc is the universal unital C ∗ -algebra generated by elements u ij , 1 ≤ i , j ≤ n satisfying the relations which make [ u ij ] a unitary matrix. In this paper, we study several structural properties of this C ∗ -algebra, showing in particular that it possesses the Lifting Property and is primitive. We also study the quantum semigroup structure of this C ∗ -algebra and the resulting compact semigroup structure of its state space, characterize all of its invertible elements. [ABSTRACT FROM AUTHOR]

We prove a number of results having to do with equipping type-I C*-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the C*-algebra in question is an extension of a non-zero finite direct sum of elementary C*-algebras by a commutative unital C*-algebra then it must be finitedimensional. [ABSTRACT FROM AUTHOR]

In this paper, we show that the quantum disk, i.e. the quantum space corresponding to the Toeplitz C * -algebra does not admit any compact quantum group structure. We prove that if such a structure existed the resulting compact quantum group would simultaneously be of Kac type and not of Kac type. The main tools used in the solution come from the theory of operators on Hilbert spaces. [ABSTRACT FROM AUTHOR]

For each λ ∈ ] 0 , 1 ] we exhibit an uncountable family of compact quantum groups G such that the von Neumann algebra L ∞ (G) is the injective factor of type III λ with separable predual. We also show that uncountably many injective factors of type III 0 arise as L ∞ (G) for some compact quantum group G. We introduce natural invariants of quantum groups related to the scaling group, modeled on the Connes invariant T for von Neumann algebras. We compute the values of these invariants in our examples, which allows us to distinguish between them. We also investigate their connection to the Connes invariants T (L ∞ (G)) , S (L ∞ (G)). In the final section we show that for a compact quantum group G the von Neumann algebra L ∞ (G) cannot have a direct summand of the form B (H) with dim ⁡ H = ∞. In particular, factors of type I (of any dimension) cannot be obtained as L ∞ (G) for a non-trivial compact quantum group G. [ABSTRACT FROM AUTHOR]

We show that a natural notion of irreducibility implies connectedness in the Compact Quantum Group setting. We also investigate the converse implication and show it is related to Kaplansky's conjectures on group algebras.

In this paper, we obtain a complete list of inequivalent irreducible representations of the compact quantum group U q (2) for nonzero complex deformation parameters q , which are not roots of unity. The matrix coefficients of these representations are described in terms of the little q -Jacobi polynomials. The Haar state is shown to be faithful and an orthonormal basis of L 2 (U q (2)) is obtained. Thus, we have an explicit description of the Peter–Weyl decomposition of U q (2). As an application, we discuss the Fourier transform and establish the Plancherel formula. We also describe the decomposition of the tensor product of two irreducible representations into irreducible components. Finally, we classify the compact quantum group U q (2). [ABSTRACT FROM AUTHOR]

In this paper we study the states of Poisson type and infinitely divisible states on compact quantum groups. Each state of Poisson type is infinitely divisible, i.e., it admits n-th root for all n ≥ 1 . The main result is that on finite quantum groups infinitely divisible states must be of Poisson type. This generalizes Böge's theorem concerning infinitely divisible measures (commutative case) and Parthasarathy's result on infinitely divisible positive definite functions (cocommutative case). Two proofs are given. [ABSTRACT FROM AUTHOR]

We show that for every N ≥ 3 the free unitary group U N + is topologically generated by its classical counterpart U N and the lower-rank U N − 1 + . This allows for a uniform inductive proof that a number of finiteness properties, known to hold for all N ≠ 3 , also hold at N = 3. Specifically, all discrete quantum duals U N + ̂ and O N + ̂ are residually finite, and hence also have the Kirchberg factorization property and are hyperlinear. As another consequence, U N + are topologically generated by U N and their maximal tori ℤ ∗ N ̂ (dual to the free groups on N generators) and similarly, O N + are topologically generated by O N and their tori ℤ 2 ∗ N ̂. [ABSTRACT FROM AUTHOR]

When a compact quantum group H coacts freely on unital C ⁎ -algebras A and B , the existence of equivariant maps A → B may often be ruled out due to the incompatibility of some invariant. We examine the limitations of using invariants, both concretely and abstractly, to resolve the noncommutative Borsuk–Ulam conjectures of Baum–Dąbrowski–Hajac. Among our results, we find that for certain finite-dimensional H , there can be no well-behaved invariant which solves the Type 1 conjecture for all free coactions of H. This claim is in stark contrast to the case when H is finite-dimensional and abelian. In the same vein, it is possible for all iterated joins of H to be cleft as comodules over the Hopf algebra associated to H. Finally, two commonly used invariants, the local-triviality dimension and the spectral count, may both change in a θ -deformation procedure. [ABSTRACT FROM AUTHOR]

We generalize the Fejér-Riesz operator systems defined for the circle group by Connes and van Suijlekom to the setting of compact matrix quantum groups and their ergodic actions on C*-algebras. These truncations form filtrations of the containing C*-algebra. We show that when they and the containing C*-algebra are equipped with suitable quantum metrics, then under suitable conditions they converge to the containing C*-algebra for quantum Gromov-Hausdorff distance. Among other examples, our results are applicable to the quantum groups S U q (2) and their homogeneous spaces S q 2. [ABSTRACT FROM AUTHOR]

Abstract Kazhdan's property (T) has been studied for several discrete group-like structures, including standard invariants of Jones' subfactors and discrete quantum groups. We prove a Żuk-type spectral gap criterion for property (T) in this setting. This allows us to construct a family of property (T) discrete quantum groups, as well as subfactor standard invariants, given by generators and a triangle presentation. These are the first examples of property (T) discrete quantum groups that are not directly related to a discrete group with property (T). [ABSTRACT FROM AUTHOR]

In this paper, we review some recent developments of compact quantum groups that arise as θ-deformations of compact Lie groups of rank at least two. A θ-deformation is merely a 2-cocycle deformation using an action of a torus of dimension higher than 2. Using the formula (Lemma 5.3) developed in [11], we derive the noncommutative 7-sphere in the sense of Connes and Landi [3] as the fixed-point subalgebra. [ABSTRACT FROM AUTHOR]

Abstract For an algebraic compact quantum group H we establish a bijection between the set of right coideal ⁎-subalgebras A → H and that of left module quotient ⁎-coalgebras H → C. It turns out that the inclusion A → H always splits as a map of right A -modules and right H -comodules, and the resulting expectation E : H → A is positive (and lifts to a positive map on the full C ⁎ completion on H) if and only if A is invariant under the squared antipode of H. The proof proceeds by Tannaka-reconstructing the coalgebra C corresponding to A → H by means of a fiber functor from H -equivariant A -modules to Hilbert spaces, while the characterization of those A → H which admit positive expectations makes use of a Fourier transform turning elements of H into functions on C. [ABSTRACT FROM AUTHOR]

Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group O n + , the so-called easy quantum groups, introduced by Banica and Speicher in 2009. This correspondence was generalized to two-colored set partitions, which, in addition, assign a black or white color to each point of a set. Globally colorized categories of partitions are those categories that are invariant with respect to arbitrary permutations of colors. This paper presents a classification of globally colorized categories. In addition, we show that the corresponding unitary quantum groups can be constructed from the orthogonal ones using tensor complexification. [ABSTRACT FROM AUTHOR]

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations. [ABSTRACT FROM AUTHOR]

We study the topological structure of the automorphism groups of compact quantum groups showing that, in parallel to a classical result due to Iwasawa, the connected component of identity of the automorphism group and of the “inner” automorphism group coincide. For compact matrix quantum groups, which can be thought of as quantum analogues of compact Lie groups, we prove that the inner automorphism group is a compact Lie group and the outer automorphism group is discrete. Applications of this to the study of group actions on compact quantum groups are highlighted. We end by providing examples of compact matrix quantum groups with infinitely-generated fusion rings, in stark contrast with the classical situation. Along the way we study the invariant theory of finite group actions on free Laurent rings and show that the rings of invariants are, in general, not finitely generated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the co-amenability of compact quantum groups. Combining with some properties of regular C*-norms on algebraic compact quantum groups, we show that the quantum double of co-amenable compact quantum groups is unique. Based on this, this paper proves that co-amenability is preserved under formulation of the quantum double construction of compact quantum groups, which exhibits a type of nice symmetry between the co-amenability of quantum groups and the amenability of groups.

We define a notion of quantum automorphism groups of graph C ∗ -algebras for finite, connected graphs. Under the assumption that the underlying graph does not have any multiple edge or loop, the quantum automorphism group of the underlying directed graph in the sense of Banica [Quantum automorphism groups of homogeneous graphs, J. Funct. Anal.224 (2005) 243–280] (which is also the symmetry object in the sense of [S. Schmidt and M. Weber, Quantum symmetry of graph C ∗ -algebras, arXiv:1706.08833 ] is shown to be a quantum subgroup of quantum automorphism group in our sense. Quantum symmetries for some concrete graph C ∗ -algebras have been computed. [ABSTRACT FROM AUTHOR]

We consider the Type 1 and Type 2 noncommutative BorsukUlam conjectures of Baum, Da browski, and Hajac: there are no equivariant morphisms A → A ⊛δ H or H → A ⊛δ H, respectively, when H is a nontrivial compact quantum group acting freely on a unital C∗-algebra A. Here A ⊛δ H denotes the equivariant noncommutative join of A and H; this join procedure is a modification of the topological join that allows a free action of H on A to produce a free action of H on A ⊛δ H. For the classical case H = C(G), G a compact group, we present a reduction of the Type 1 conjecture and counterexamples to the Type 2 conjecture. We also present some examples and conditions under which the Type 2 conjecture does hold. [ABSTRACT FROM AUTHOR]

Suppose that a compact quantum group Q acts faithfully on a smooth, compact, connected manifold M, i.e. has a C* (co)-action α on C(M), such that the action α is isometric in the sense of [GOS09] for some Riemannian structure on M. We prove that Q must be commutative as a C* algebra i.e. Q≅C(G) for some compact group G acting smoothly on M. In particular, the quantum isometry group of M (in the sense of [GOS09]) coincides with C(ISO(M)). [ABSTRACT FROM AUTHOR]

We study quantum isometry groups, denoted by , of spectral triples on for a finitely generated discrete group coming from the word-length metric with respect to a symmetric generating set . We first prove a few general results about including: • For a group with polynomial growth property, the dual of has polynomial growth property provided the action of on has full spectrum. • for any discrete abelian group , where is a suitable metric on the dual compact abelian group . We then carry out explicit computations of for several classes of examples including free and direct product of cyclic groups, Baumslag-Solitar group, Coxeter groups etc. In particular, we have computed quantum isometry groups of all finitely generated abelian groups which do not have factors of the form or for some in the direct product decomposition into cyclic subgroups. [ABSTRACT FROM AUTHOR]

Let q = | q | e i π θ be a nonzero complex number such that | q | ≠ 1 and consider the compact quantum group U q (2). For θ ∉ Q ∖ { 0 , 1 } , we obtain the K -theory of the underlying C ⁎ -algebra C (U q (2)). We construct a spectral triple on U q (2) which is equivariant under its own comultiplication action. The spectral triple obtained here is even, 4 + -summable, non-degenerate, and the Dirac operator acts on two copies of the L 2 -space of U q (2). The K -homology class of the associated Fredholm module is shown to be nontrivial. [ABSTRACT FROM AUTHOR]