Finite Quantum Groups Research Articles

The ergodic theorem for random walks on finite quantum groups

Necessary and sufficient conditions for a Markov chain to be ergodic are that the chain is irreducible and aperiodic. This result is manifest in the case of random walks on finite groups by a statement about the support of the driving probability: a random walk on a finite group is ergodic if and only if the support is not concentrated on a proper subgroup, nor on a coset of a proper normal subgroup. The study of random walks on finite groups extends naturally to the study of random walks on finite quantum groups, where a state on the algebra of functions plays the role of the driving probability. Necessary and sufficient conditions for ergodicity of a random walk on a finite quantum group are given on the support projection of the driving state.

From Hopf algebras to topological quantum groups. A short history, various aspects and some problems

Quantum groups have been studied within several areas of mathematics and mathematical physics. This has led to different approaches, each of them with their own techniques and conventions. Starting with Hopf algebras, where there is a general consensus, moving in the direction of topological quantum groups, where there is no such consensus, it is easy to get lost. Not only many difficulties have to be overcome, but also several choices must be made. The way this is done is often confusing. Some choices even turn out to be rather annoying. As an introductory lecture at the conference on Topological quantum groups and Hopf algebras in 2016, we have explained these choices, difficulties and annoyances, encountered on the road from Hopf algebras to topological quantum groups. In these notes, we discuss more aspects of the development of locally compact quantum groups. We not only explain some of these difficulties in greater detail, but we also give background information about the different steps, combined with some historical comments. We start with finite quantum groups and continue with discrete quantum groups, compact quantum groups and algebraic quantum groups. Multiplicative unitaries are an important side track before we finally arrive at locally compact quantum groups. Along the way, we also formulate some interesting remaining problems in the theory.

Trace of powers of representations of finite quantum groups

In this paper, we study (asymptotic) properties of the ∗-distribution of irreducible characters of finite quantum groups. We proceed in two steps, first examining the representation theory to determine irreducible representations and their powers, then we study the ∗-distribution of their trace with respect to the Haar measure. For the Sekine family, we look at the asymptotic distribution (as the dimension of the algebra goes to infinity).

Random Walks on Finite Quantum Groups

In this paper, we study convergence of random walks, on finite quantum groups, arising from linear combination of irreducible characters. We bound the distance to the Haar state and determine the asymptotic behavior, i.e., the limit state if it exists. We note that the possible limits are any central idempotent state. We also look at cutoff phenomenon in the Sekine finite quantum groups.

Infinitely divisible states on finite quantum groups

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 nge 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.

Idempotent states on Sekine quantum groups

Sekine quantum groups are a family of finite quantum groups. The main result of this article is to compute all the idempotent states on Sekine quantum groups, which completes the work of Franz and Skalski. This is achieved by solving a complicated system of equations using linear algebra and basic number theory. From this, we discover a new class of non-Haar idempotent states. The order structure of the idempotent states on Sekine quantum groups is also discussed. Finally we give a sufficient condition for the convolution powers of states on Sekine quantum group to converge.

Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups

A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses Fourier analysis on the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The Fourier analysis of quantum groups is remarkably similar to that of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for finite quantum groups. The Upper Bound Lemma is used to study the convergence of ergodic random walks on the dual group $$\widehat{S_n}$$ as well as on the truly quantum groups of Sekine.

Highest weight theory for finite-dimensional graded algebras with triangular decomposition

We show that the category of graded modules over a finite-dimensional graded algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category. When the algebra is self-injective, we show furthermore that this highest weight category has tilting modules in the sense of Ringel. This provides a new perspective on the representation theory of such algebras, and leads to several new structures attached to them. There are a wide variety of examples in algebraic Lie theory to which this applies: restricted enveloping algebras, Lusztig's small quantum groups, hyperalgebras, finite quantum groups, and restricted rational Cherednik algebras.

Thoma type results for discrete quantum groups

Thoma's theorem states that a group algebra [Formula: see text] is of type I if and only if [Formula: see text] is virtually abelian. We discuss here some similar questions for the quantum groups, our main result stating that, under suitable virtually abelianity conditions on a discrete quantum group [Formula: see text], we have a stationary model of type [Formula: see text], with [Formula: see text] being a finite quantum group, and with [Formula: see text] being a compact group. We discuss then some refinements of these results in the quantum permutation group case, [Formula: see text], by restricting the attention to the matrix models which are quasi-flat, in the sense that the images of the standard coordinates, known to be projections, have rank [Formula: see text].

Quasi-idempotent Rota–Baxter operators arising from quasi-idempotent elements

In this short note, we construct quasi-idempotent Rota-Baxter operators by quasi-idempotent elements and show that every finite dimensional Hopf algebra admits nontrivial Rota-Baxter algebra structures and tridendriform algebra structures. Several concrete examples are provided, including finite quantum groups and Iwahori-Hecke algebras.

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.

$L_p$-improving convolution operators on finite quantum groups

We characterize positive convolution operators on a finite quantum group $\mathbb{G}$ which are $L_{p}$-improving. More precisely, we prove that the convolution operator $T_{\varphi}:x\mapsto\varphi\star x$ given by a state $\varphi$ on $C(\mathbb{G})$ satisfies \[ \exists1<p<2,\quad\|T_{\varphi}:L_{p}(\mathbb{G})\to L_{2}(\mathbb{G})\|=1 \] if and only if the Fourier series $\hat{\varphi}$ satisfy $\|\hat{\varphi}(\alpha)\|<1$ for all nontrivial irreducible unitary representations $\alpha$, if and only if the state $(\varphi\circ S)\star\varphi$ is non-degenerate (where $S$ is the antipode). We also prove that these $L_{p}$-improving properties are stable under taking free products, which gives a method to construct $L_{p}$-improving multipliers on infinite compact quantum groups. Our methods for non-degenerate states yield a general formula for computing idempotent states associated to Hopf images, which generalizes earlier work of Banica, Franz and Skalski.

Relative Yetter-Drinfeld modules and comodules over braided groups

Let H1 be a quantum group and f : H1⟶H2 a Hopf algebra homomorphism. Assume that B is some braided group obtained by Majid’s transmutation process. We first show that there is a tensor equivalence between the category of comodules over the braided group B and that of relative Yetter-Drinfeld modules. Next, we prove that the Drinfeld centers of the two categories mentioned above are equivalent to the category of modules over some quantum double, namely, the category of ordinary Yetter-Drinfeld modules over some Radford’s biproduct Hopf algebra. Importantly, the above results not only hold for a finite dimensional quantum group but also for an infinite dimensional one.

Residually finite quantum group algebras

We show that provided n≠3, the involutive Hopf ⁎-algebra Au(n) coacting universally on an n-dimensional Hilbert space has enough finite-dimensional representations in the sense that every non-zero element acts non-trivially in some finite-dimensional ⁎-representation. This implies that the discrete quantum group with group algebra Au(n) is maximal almost periodic (i.e. it embeds in its quantum Bohr compactification), answering a question posed by P. Sołtan in [21].We also prove analogous results for the involutive Hopf ⁎-algebra Bu(n) coacting universally on an n-dimensional Hilbert space equipped with a non-degenerate bilinear form.

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.

Quantum group of automorphisms of a finite quantum group

A notion of a quantum automorphism group of a finite quantum group, generalising that of a classical automorphism group of a finite group, is proposed and a corresponding existence result proved.

Poincaré–Birkhoff–Witt bases and Khovanov–Lauda–Rouquier algebras

We generalize Lusztig’s geometric construction of the Poincaré–Birkhoff–Witt (PBW) bases of finite quantum groups of type ADE under the framework of Varagnolo and Vasserot. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the Khovanov–Lauda–Rouquier (KLR) algebras. This enables us to prove Lusztig’s conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases. In addition, we verify Kashiwara’s problem on the finiteness of the global dimensions of the KLR algebras of type ADE.

Finite quantum groups and quantum permutation groups

We give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups. We also give examples of finite quantum groups which are not quantum permutation groups: one such example occurs as a split abelian extension associated to the exact factorization S4=Z4S3 and has dimension 24. We show that, in fact, this is the smallest possible dimension that a non-quantum permutation group can have.

On square roots of the Haar state on compact quantum groups

The paper is concerned with the extension of the classical study of probability measures on a compact group which are square roots of the Haar measure, due to Diaconis and Shahshahani, to the context of compact quantum groups. We provide a simple characterisation for compact quantum groups which admit no non-trivial square roots of the Haar state in terms of their corepresentation theory. In particular it is shown that such compact quantum groups are necessarily of Kac type and their subalgebras generated by the coefficients of a fixed two-dimensional irreducible corepresentation are isomorphic (as finite quantum groups) to the algebra of functions on the group of unit quaternions. An example of a quantum group whose Haar state admits no nontrivial square root and which is neither commutative nor cocommutative is given.

Book Review: Finite dimensional algebras and quantum groups

Book Review: Finite dimensional algebras and quantum groups