Dual Quantum Group Research Articles

On ideals of L1-algebras of compact quantum groups

We develop a notion of a non-commutative hull for a left ideal of the [Formula: see text]-algebra of a compact quantum group [Formula: see text]. A notion of non-commutative spectral synthesis for compact quantum groups is proposed as well. It is shown that a certain Ditkin’s property at infinity (which includes those [Formula: see text] where the dual quantum group [Formula: see text] has the approximation property) is equivalent to every hull having synthesis. We use this work to extend recent work of White that characterizes the weak[Formula: see text] closed ideals of a measure algebra of a compact group to those of the measure algebra of a coamenable compact quantum group. In the sequel, we use this work to study bounded right approximate identities of certain left ideals of [Formula: see text] in relation to coamenability of [Formula: see text].

Noncommutative Poisson boundaries and Furstenberg–Hamana boundaries of Drinfeld doubles

We clarify the relation between noncommutative Poisson boundaries and Furstenberg–Hamana boundaries of quantum groups. Specifically, given a compact quantum group G, we show that in many cases where the Poisson boundary of the dual discrete quantum group Gˆ has been computed, the underlying topological boundary either coincides with the Furstenberg–Hamana boundary of the Drinfeld double D(G) of G or is a quotient of it. This includes the q-deformations of compact Lie groups, free orthogonal and free unitary quantum groups, quantum automorphism groups of finite dimensional C⁎-algebras. In particular, the boundary of D(Gq) for the q-deformation of a compact connected semisimple Lie group G is Gq/T (for q≠1), in agreement with the classical results of Furstenberg and Moore on the Furstenberg boundary of GC.We show also that the construction of the Furstenberg–Hamana boundary of D(G) respects monoidal equivalence and, in fact, can be carried out entirely at the level of the representation category of G. This leads to a notion of the Furstenberg–Hamana boundary of a rigid C⁎-tensor category.

On the Sobolev embedding properties for compact matrix quantum groups of Kac type

We study the optimal order of natural analogues of Sobolev embedding properties within the framework of compact matrix quantum groups of Kac type. One of the main results of this paper is that the optimal order is given by the polynomial growth order of dual discrete quantum groups in a broad class, which covers all connected compact Lie groups, duals of polynomially growing discrete groups, $ O_2^+ $ and $ S_4^+ $. Outside the realm of co-amenable compact quantum groups, we prove that the optimal order is $ 3 $ for duals of free groups and free quantum groups $ O_N^+ $ and $ S_N^+ $, and that Sobolev embedding properties can be generalized for all compact matrix quantum groups of Kac type whose duals have the rapid decay property. In addition, we generalize sharpened Hausdorff-Young inequalities, compute degrees of the rapid decay property for duals of $ O_N^+,S_N^+ $ and prove sharpness of Hardy-Littlewood inequalities on duals of free groups.

Inner amenability and approximation properties of locally compact quantum groups

We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction. We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau--Paterson, resolve a recent conjecture of Ng--Viselter, and prove that, for inner amenable quantum groups $\mathbb{G}$, approximation properties of the dual operator algebras can be averaged to approximation properties $\mathbb{G}$. Similar homological techniques are used to prove that $\ell^1(\mathbb{G})$ is not relatively operator biflat for any non-Kac discrete quantum group $\mathbb{G}$; a discrete Kac algebra $\mathbb{G}$ with Kirchberg's factorization property is weakly amenable if and only if $L^1_{cb}(\widehat{\mathbb{G}})$ is operator amenable, and amenability of a locally compact quantum group $\mathbb{G}$ implies $C_u(\widehat{\mathbb{G}})=L^1(\widehat{\mathbb{G}})\widehat{\otimes}_{L^1(\widehat{\mathbb{G}})}C_0(\widehat{\mathbb{G}})$ completely isometrically. The latter result allows us to partially answer a conjecture of Voiculescu when $\mathbb{G}$ has the approximation property.

Mixed quantum skew Howe duality and link invariants of type A

We define a ribbon category Sp(β), depending on a parameter β, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for β=m−n the monoidal category of representations of Uq(glm|n) generated by exterior powers of the vector representation and their duals. We identify this category Sp(β) with a direct limit of quotients of a dual idempotented quantum group U˙q(glr+s), proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category Sp(β) gives a unified natural setting for defining the colored glm|n link invariant (for β=m−n) and the colored HOMFLY-PT polynomial (for β generic).

Group-like projections for locally compact quantum groups

Let $\mathbb{G}$ be a locally compact quantum group. We give a 1-1 correspondence between group-like projections in $L^\infty(\mathbb{G})$ preserved by the scaling group and idempotent states on the dual quantum group $\widehat{\mathbb{G}}$. As a byproduct we give a simple proof that normal integrable coideals in $L^\infty(\mathbb{G})$ which are preserved by the scaling group are in 1-1 correspondence with compact quantum subgroups of $\mathbb{G}$.

Entropic uncertainty relations under localizations on discrete quantum groups

The uncertainty principle has been established within the framework of locally compact quantum groups in recent years. This paper demonstrates that entropic uncertainty relations can be strengthened under localizations on discrete quantum groups, which is the case if the dual compact quantum group G is the free orthogonal quantum group ON+ with N ≥ 3 or if G admits an infinite Λ(p) set with p > 2. On the other hand, this paper explains the reason why such phenomena do not appear when G is one of the connected semisimple compact Lie groups, O2+, and the quantum SU(2) groups. Also, we discuss the divergence of entropic uncertainty relations together with some explicit explanations.

Bekka-type amenabilities for unitary corepresentations of locally compact quantum groups

In this short note, we further Ng’s work by extending Bekka amenability and weak Bekka amenability to general locally compact quantum groups, and we generalize some of Ng’s results to the general case. In particular, we show that a locally compact quantum group G is coamenable if and only if the contra-corepresentation of its fundamental multiplicative unitary WG is Bekka-amenable, and that G is amenable if and only if its dual quantum group’s fundamental multiplicative unitary WGˆ is weakly Bekka-amenable.

Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological dimension, by connecting it with the Gelfand-Kirillov dimension of an algebra. Furthermore, we show that polynomial growth for a compact quantum group G of Kac type implies *-regularity of the Fourier algebra A(G), that is every closed ideal of C(G) has a dense intersection with A(G). In particular, A(G) has a unique C*-norm.

The spatial Rokhlin property for actions of compact quantum groups

We introduce the spatial Rokhlin property for actions of coexact compact quantum groups on C⁎-algebras, generalizing the Rokhlin property for both actions of classical compact groups and finite quantum groups. Two key ingredients in our approach are the concept of sequentially split ⁎-homomorphisms, and the use of braided tensor products instead of ordinary tensor products.We show that various structure results carry over from the classical theory to this more general setting. In particular, we show that a number of C⁎-algebraic properties relevant to the classification program pass from the underlying C⁎-algebra of a Rokhlin action to both the crossed product and the fixed point algebra. Towards establishing a classification theory, we show that Rokhlin actions exhibit a rigidity property with respect to approximate unitary equivalence. Regarding duality theory, we introduce the notion of spatial approximate representability for actions of discrete quantum groups. The spatial Rokhlin property for actions of a coexact compact quantum group is shown to be dual to spatial approximate representability for actions of its dual discrete quantum group, and vice versa.

Open quantum subgroups of locally compact quantum groups

The notion of an open quantum subgroup of a locally compact quantum group is introduced and given several equivalent characterizations in terms of group-like projections, inclusions of quantum group C⁎-algebras and properties of respective quantum homogeneous spaces. Open quantum subgroups are shown to be closed in the sense of Vaes and normal open quantum subgroups are proved to be in 1–1 correspondence with normal compact quantum subgroups of the dual quantum group.

Easy quantum groups and quantum subgroups of a semi-direct product quantum group

We consider homogeneous compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial symmetries with central support.We show that such quantum groups have a simple presentation as semi-direct product quantum groups of a group dual quantum group by an action of a permutation group. This general result allows us to completely classify easy quantum groups with the above property by certain reflection groups.We give four applications of our result. First, there are uncountably many easy quantum groups. Second, there are non-easy homogeneous hyperoctahedral quantum groups. Third, we study operator algebraic properties of the hyperoctahedral series. Finally, we prove a generalised de Finetti theorem for those easy quantum groups in the scope of this article.

Rieffel Deformation of Tensor Functor and Braided Quantum Groups

In this paper we apply Rieffel deformation to C*- tensor product viewed as a functor on the category of C*-algebras with an abelian group action. In the case of the Rieffel deformation of a quantum group with the action by automorphisms the deformed tensor product enables us to view the deformed object as a braided quantum group. We construct a bicharacter for a braided quantum group and the dual braided quantum group. We employ our method to get a braided quantum Minkowski space. Its description in terms of the deformed space-time coordinates is provided.

Completely positive definite functions and Bochnerʼs theorem for locally compact quantum groups

We prove two versions of Bochnerʼs theorem for locally compact quantum groups. First, every completely positive definite “function” on a locally compact quantum group G arises as a transform of a positive functional on the universal C⁎-algebra C0u(Gˆ) of the dual quantum group. Second, when G is coamenable, complete positive definiteness may be replaced with the weaker notion of positive definiteness, which models the classical notion. A counterexample is given to show that the latter result is not true in general. To prove these results, we show two auxiliary results of independent interest: products are linearly dense in L♯1(G), and when G is coamenable, the Banach ⁎-algebra L♯1(G) has a contractive bounded approximate identity.

Reduced operator algebras of trace-preserving quantum automorphism groups

Let B be a finite dimensional C ^\ast -algebra equipped with its canonical trace induced by the regular representation of B on itself. In this paper, we study various properties of the trace-preserving quantum automorphism group \mathbb G of B . We prove that the discrete dual quantum group \widehat{\mathbb G} has the property of rapid decay, the reduced von Neumann algebra L^\infty(\mathbb G) has the Haagerup property and is solid, and that L^\infty(\mathbb G) is (in most cases) a prime type II _1 -factor. As applications of these and other results, we deduce the metric approximation property, exactness, simplicity and uniqueness of trace for the reduced C^\ast -algebra C_r(\mathbb G) , and the existence of a multiplier-bounded approximate identity for the convolution algebra L^1(\mathbb G) .

Langlands duality for representations of quantum groups

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an “interpolating quantum group” depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups.

Reiter's properties [formula omitted] and [formula omitted] for locally compact quantum groups

A locally compact group G is amenable if and only if it has Reiter's property ( P p ) for p = 1 or, equivalently, all p ∈ [ 1 , ∞ ) , i.e., there is a net ( m α ) α of non-negative norm one functions in L p ( G ) such that lim α sup x ∈ K ‖ L x − 1 m α − m α ‖ p = 0 for each compact subset K ⊂ G ( L x − 1 m α stands for the left translate of m α by x −1 ). We extend the definitions of properties ( P 1 ) and ( P 2 ) from locally compact groups to locally compact quantum groups in the sense of J. Kustermans and S. Vaes. We show that a locally compact quantum group has ( P 1 ) if and only if it is amenable and that it has ( P 2 ) if and only if its dual quantum group is co-amenable. As a consequence, ( P 2 ) implies ( P 1 ) .

The Dual Quantum Group for the Quantum Group Analog of the Normalizer of SU(1, 1) in

The quantum group analogue of the normalizer of SU(1,1) in SL(2,C) is an important and non-trivial example of a non-compact quantum group. The general theory of locally compact quantum groups in the operator algebra setting implies the existence of the dual quantum group. The first main goal of the paper is to give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra U_q(su(1,1)). It turns out that U_q(su(1,1)) does not suffice to generate the dual quantum group. The dual quantum group is graded with respect to commutation and anticommutation with a suitable analogue of the Casimir operator characterized by an affiliation relation to a von Neumann algebra. This is used to obtain an explicit set of generators. Having the dual quantum group the left regular corepresentation of the quantum group analogue of the normalizer of SU(1,1) in SL(2,C) is decomposed into irreducible corepresentations. Upon restricting the irreducible corepresentations to U_q(su(1,1))-representation one finds combinations of the positive and negative discrete series representations with the strange series representations as well as combinations of the principal unitary series representations. The detailed analysis of this example involves analysis of special functions of basic hypergeometric type and, in particular, some results on these special functions are obtained, which are stated separately. The paper is split into two parts; the first part gives almost all of the statements and the results, and the statements in the first part are independent of the second part. The second part contains the proofs of all the statements.

On idempotent states on quantum groups

Idempotent states on a compact quantum group are shown to yield group-like projections in the multiplier algebra of the dual discrete quantum group. This allows to deduce that every idempotent state on a finite quantum group arises in a canonical way as the Haar state on a finite quantum hypergroup. A natural order structure on the set of idempotent states is also studied and some examples discussed.

CANONICAL AND LIE-ALGEBRAIC TWIST DEFORMATIONS OF κ-POINCARÉ AND CONTRACTIONS TO κ-GALILEI ALGEBRAS

We propose canonical and Lie-algebraic twist deformations of κ-deformed Poincaré Hopf algebra which leads to the generalized κ-Minkowski space–time relations. The corresponding deformed κ-Poincaré quantum groups are also calculated. Finally, we perform the nonrelativistic contraction limit to the corresponding twisted Galilean algebras and dual Galilean quantum groups.