Universal Quantum Groups Research Articles

An intrinsic approach to relative braid group symmetries on ı$\imath$quantum groups

AbstractWe initiate a general approach to the relative braid group symmetries on (universal) quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining properties of quasi ‐matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on quantum groups are obtained. We establish a number of fundamental properties for these symmetries on quantum groups, strikingly parallel to their well‐known quantum group counterparts. We apply these symmetries to fully establish rank 1 factorizations of quasi ‐matrices, and this factorization property, in turn, helps to show that the new symmetries satisfy relative braid relations. As a consequence, conjectures of Kolb–Pellegrini and Dobson–Kolb are settled affirmatively. Finally, the above approach allows us to construct compatible relative braid group actions on modules over quantum groups for the first time.

Twisting of Graded Quantum Groups and Solutions to the Quantum Yang-Baxter Equation

AbstractLet H be a Hopf algebra that is $\mathbb Z$ ℤ -graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of H to be a Zhang twist of H. In particular, we introduce the notion of a twisting pair for H such that the Zhang twist of H by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin’s universal quantum groups associated with quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction.

The RFD and Kac quotients of the Hopf * -algebras of universal orthogonal quantum groups

We determine the Kac quotient and the RFD (residually finite dimensional) quotient for the Hopf *-algebras associated to universal orthogonal quantum groups.

Hopf coactions on odd spheres

We prove that the q-deformed unitary group, i.e., Uq(N), is the universal compact quantum group in the category of (compact) quantum groups which coact on the q-deformed odd sphere Sq2N−1 leaving the space spanned by the natural set of generators invariant and preserving the unique SUq(N) invariant functional on Sq2N−1. Using this, we identify Uq(N) as the quantum group of orientation preserving isometries (in the sense of Bhowmick and Goswami [5]) for a natural spectral triple associated with Sq2N−1 constructed by Chakraborty and Pal [9].

On quantum groups associated to a pair of preregular forms

We define the universal quantum group \mathcal{H} that preserves a pair of Hopf comodule maps, whose underlying vector space maps are preregular forms defined on dual vector spaces. This generalizes the construction of Bichon and Dubois-Violette (2013), where the target of these comodule maps are the ground field. We also recover the quantum groups introduced by Dubois-Violette and Launer (1990), by Takeuchi (1990), by Artin, Schelter, and Tate (1991), and by Mrozinski (2014), via our construction. As a consequence, we obtain an explicit presentation of a universal quantum group that coacts simultaneously on a pair of N-Koszul Artin–Schelter regular algebras with arbitrary quantum determinant.

Lévy–Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups

We study the first and second cohomology groups of the [Formula: see text]-algebras of the universal unitary and orthogonal quantum groups [Formula: see text] and [Formula: see text]. This provides valuable information for constructing and classifying Lévy processes on these quantum groups, as pointed out by Schürmann. In the case when all eigenvalues of [Formula: see text] are distinct, we show that these [Formula: see text]-algebras have the properties (GC), (NC) and (LK) introduced by Schürmann and studied recently by Franz, Gerhold and Thom. In the degenerate case [Formula: see text], we show that they do not have any of these properties. We also compute the second cohomology group of [Formula: see text] with trivial coefficients — [Formula: see text] — and construct an explicit basis for the corresponding second cohomology group for [Formula: see text] (whose dimension was known earlier, thanks to the work of Collins, Härtel and Thom).

L2-Betti numbers of rigid C∗-tensor categories and discrete quantum groups

We compute the $L^2$-Betti numbers of the free $C^*$-tensor categories, which are the representation categories of the universal unitary quantum groups $A_u(F)$. We show that the $L^2$-Betti numbers of the dual of a compact quantum group $G$ are equal to the $L^2$-Betti numbers of the representation category $Rep(G)$ and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of $L^2$-Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first $L^2$-Betti number in terms of a generating set of a $C^*$-tensor category.

Kirchberg's factorization property for discrete quantum groups

We show that the discrete duals of the universal unitary quantum groups and orthogonal quantum groups have Kirchberg's factorization property when n is different from 3.

Quantum Families of Invertible Maps and Related Problems

AbstractThe notion of families of quantum invertible maps (C*–algebra homomorphisms satisfying Podleś condition) is employed to strengthen and reinterpret several results concerning universal quantum groups acting on finite quantum spaces. In particular, Wang's 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 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.

Quantum automorphism groups of finite quantum groups are classical

In a recent paper of Bhowmick, Skalski and So{\l}tan 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.

Remarks on the quantum Bohr compactification

The category of locally compact quantum groups can be described as either Hopf $*$-homomorphisms between universal quantum groups, or as bicharacters on reduced quantum groups. We show how So{\l}tan's quantum Bohr compactification can be used to construct a ``compactification'' in this category. Depending on the viewpoint, different C$^*$-algebraic compact quantum groups are produced, but the underlying Hopf $*$-algebras are always, canonically, the same. We show that a complicated range of behaviours, with C$^*$-completions between the reduced and universal level, can occur even in the cocommutative case, thus answering a question of So{\l}tan. We also study such compactifications from the perspective of (almost) periodic functions. We give a definition of a periodic element in $L^\infty(\mathbb G)$, involving the antipode, which allows one to compute the Hopf $*$-algebra of the compactification of $\mathbb G$; we later study when the antipode assumption can be dropped. In the cocommutative case we make a detailed study of Runde's notion of a completely almost periodic functional-- with a slight strengthening, we show that for [SIN] groups this does recover the Bohr compactification of $\hat G$.

The Quantum Group of a Preregular Multilinear Form

We describe the universal quantum group preserving a preregular multilinear form, by means of an explicit finite presentation of the corresponding Hopf algebra.

COMPLETELY POSITIVE MULTIPLIERS OF QUANTUM GROUPS

We show that any completely positive multiplier of the convolution algebra of the dual of an operator algebraic quantum group 𝔾 (either a locally compact quantum group, or a quantum group coming from a modular or manageable multiplicative unitary) is induced in a canonical fashion by a unitary corepresentation of 𝔾. It follows that there is an order bijection between the completely positive multipliers of L1(𝔾) and the positive functionals on the universal quantum group [Formula: see text]. We provide a direct link between the Junge, Neufang, Ruan representation result and the representing element of a multiplier, and use this to show that their representation map is always weak*–weak*-continuous.

Quantum symmetry groups of C *-algebras equipped with orthogonal filtrations

Motivated by the work of Goswami on quantum isometry groups of noncommutative manifolds we define the quantum symmetry group of a unital C*-algebra A equipped with an orthogonal filtration as the universal object in the category of compact quantum groups acting on A in a filtration preserving fashion. The existence of such a universal object is proved and several examples discussed. In particular we study the universal quantum group acting on the dual of the free group and preserving both the word length and the block length.

Free unitary groups are (almost) simple

We show that the quotients of Wang and Van Daele's universal quantum groups by their centers are simple in the sense that they have no normal quantum subgroups, thus providing the first examples of simple compact quantum groups with non-commutative fusion rings.

A theory of induction and classification of tensor C*-categories

This paper addresses the problem of describing the structure of tensor C*-categories \mathcal{M} with conjugates and irreducible tensor unit. No assumption on the existence of a braided symmetry or on amenability is made. Our assumptions are motivated by the remark that these categories often contain non-full tensor C*-subcategories with conjugates and the same objects admitting an embedding into the Hilbert spaces. Such an embedding defines a compact quantum group by Woronowicz duality. An important example is the Temperley–Lieb category canonically contained in a tensor C*-category generated by a single real or pseudoreal object of dimension ≥ 2 . The associated quantum groups are the universal orthogonal quantum groups of Wang and Van Daele. Our main result asserts that there is a full and faithful tensor functor from \mathcal{M} to a category of Hilbert bimodule representations of the compact quantum group. In the classical case, these bimodule representations reduce to the G -equivariant Hermitian bundles over compact homogeneous G -spaces, with G a compact group. Our structural results shed light on the problem of whether there is an embedding functor of \mathcal{M} into the Hilbert spaces. We show that this is related to the problem of whether a classical compact Lie group can act ergodically on a non-type I von Neumann algebra. In particular, combining this with a result of Wassermann shows that an embedding exists if \mathcal{M} is generated by a pseudoreal object of dimension 2.

Paths in quantum Cayley trees and [formula omitted]-cohomology

We study existence, uniqueness and triviality of path cocycles in the quantum Cayley graph of universal discrete quantum groups. In the orthogonal case we find that the unique path cocycle is trivial, in contrast with the case of free groups where it is proper. In the unitary case it is neither bounded nor proper. From this geometrical result we deduce the vanishing of the first L2-Betti number of Ao(In).

Grothendieck rings of universal quantum groups

We determine the Grothendieck ring of finite-dimensional comodules for the free Hopf algebra on a matrix coalgebra, and similarly for the free Hopf algebra with bijective antipode and other related universal quantum groups. The results turn out to parallel those for Wang and Van Daeleʼs universal compact quantum groups and Bichonʼs generalization of those results to universal cosovereign Hopf algebras: in all cases the rings are isomorphic to those of non-commutative polynomials over certain sets, these sets varying from case to case. In most cases we are able to give more precise information about the multiplication table of the Grothendieck ring.

Multipliers of locally compact quantum groups via Hilbert C *-modules

A result of Gilbert shows that every completely bounded multiplier $f$ of the Fourier algebra $A(G)$ arises from a pair of bounded continuous maps $\alpha,\beta:G \rightarrow K$, where $K$ is a Hilbert space, and $f(s^{-1}t) = (\beta(t)|\alpha(s))$ for all $s,t\in G$. We recast this in terms of adjointable operators acting between certain Hilbert C$^*$-modules, and show that an analogous construction works for completely bounded left multipliers of a locally compact quantum group. We find various ways to deal with right multipliers: one of these involves looking at the opposite quantum group, and this leads to a proof that the (unbounded) antipode acts on the space of completely bounded multipliers, in a way which interacts naturally with our representation result. The dual of the universal quantum group (in the sense of Kustermans) can be identified with a subalgebra of the completely bounded multipliers, and we show how this fits into our framework. Finally, this motivates a certain way to deal with two-sided multipliers.

Poisson boundary of the discrete quantum group

AbstractWe identify the Poisson boundary of the dual of the universal compact quantum group Au(F) with a measurable field of ITPFI (infinite tensor product of finite type I) factors.