arXiv subjects

Sergey Neshveyev

Publications and source records attributed to Sergey Neshveyev.

At least 37 records · Page 2Linked to original sources

arXiv2023

Non-Hausdorff etale groupoids and C*-algebras of left cancellative monoids

We study the question whether the representations defined by a dense subset of the unit space of a locally compact étale groupoid are enough to determine the reduced norm on the groupoid C$^*$-algebra. We present sufficient conditions for either conclusion, giving a complete answer when the isotropy groups are torsion-free. As an application we consider the groupoid $G(S)$ associated to a left cancellative monoid $S$ by Spielberg and formulate a sufficient condition, which we call C$^*$-regularity, for the canonical map $C^*_r(G(S))\to C^*_r(S)$ to be an isomorphism, in which case $S$ has a well-defined full semigroup C$^*$-algebra $C^*(S)=C^*(G(S))$. We give two related examples of left cancellative monoids $S$ and $T$ such that both are not finitely aligned and have non-Hausdorff associated étale groupoids, but $S$ is C$^*$-regular, while $T$ is not.

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arXiv2021

(Non)exotic completions of the group algebras of isotropy groups

Motivated by the problem of characterizing KMS states on the reduced C$^*$-algebras of étale groupoids, we show that the reduced norm on these algebras induces a C$^*$-norm on the group algebras of the isotropy groups. This C$^*$-norm coincides with the reduced norm for the transformation groupoids, but, as follows from examples of Higson-Lafforgue-Skandalis, it can be exotic already for groupoids of germs associated with group actions. We show that the norm is still the reduced one for some classes of graded groupoids, in particular, for the groupoids associated with partial actions of groups and the semidirect products of exact groups and groupoids with amenable isotropy groups.

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arXiv2021

Martin boundaries of the duals of free unitary quantum groups

Given a free unitary quantum group $G=A_u(F)$, with $F$ not a unitary $2$-by-$2$ matrix, we show that the Martin boundary of the dual of $G$ with respect to any $G$-$\hat G$-invariant, irreducible, finite range quantum random walk coincides with the topological boundary defined by Vaes and Vander Vennet. This can be thought of as a quantum analogue of the fact that the Martin boundary of a free group coincides with the space of ends of its Cayley tree.

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arXiv2021

On deformations of C*-algebras by actions of Kahlerian Lie groups

We show that two approaches to equivariant strict deformation quantization of C*-algebras by actions of negatively curved Kahlerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual 2-cocycles, are equivalent.

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arXiv2021

Probabilistic boundaries of finite extensions of quantum groups

Given a discrete quantum group $H$ with a finite normal quantum subgroup $G$, we show that any positive, possibly unbounded, harmonic function on $H$ with respect to an irreducible invariant random walk is $G$-invariant. This implies that, under suitable assumptions, the Poisson and Martin boundaries of $H$ coincide with those of $H/G$. A similar result is also proved in the setting of exact sequences of C$^*$-tensor categories. As an immediate application, we conclude that the boundaries of the duals of the group-theoretical easy quantum groups are classical.

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arXiv2021

Categorically Morita equivalent compact quantum groups

We give a dynamical characterization of categorical Morita equivalence between compact quantum groups. More precisely, by a Tannaka-Krein type duality, a unital C*-algebra endowed with commuting actions of two compact quantum groups corresponds to a bimodule category over their representation categories. We show that this bimodule category is invertible if and only if the actions are free, with finite dimensional fixed point algebras, which are in duality as Frobenius algebras in an appropriate sense. This extends the well-known characterization of monoidal equivalence in terms of bi-Hopf-Galois objects.

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arXiv2018

Ground states of groupoid C*-algebras, phase transitions and arithmetic subalgebras for Hecke algebras

We consider the Hecke pair consisting of the group $P^+_K$ of affine transformations of a number field $K$ that preserve the orientation in every real embedding and the subgroup $P^+_O$ consisting of transformations with algebraic integer coefficients. The associated Hecke algebra $C^*(P^+_K,P^+_O)$ has a natural time evolution $σ$, and we describe the corresponding phase transition for KMS$_β$-states and for ground states. From work of Yalkinoglu and Neshveyev it is known that a Bost-Connes type system associated to $K$ has an essentially unique arithmetic subalgebra. When we import this subalgebra through the isomorphism of $C^*(P^+_K,P^+_O)$ to a corner in the Bost-Connes system established by Laca, Neshveyev and Trifkovic, we obtain an arithmetic subalgebra of $C^*(P^+_K,P^+_O)$ on which ground states exhibit the `fabulous' property with respect to an action of the Galois group $Gal(K^{ab}/H_+(K))$, where $H_+(K)$ is the narrow Hilbert class field. In order to characterize the ground states of the $C^*$-dynamical system $(C^*(P^+_K,P^+_O),σ)$, we obtain first a characterization of the ground states of a groupoid $C^*$-algebra, refining earlier work of Renault. This is independent from number theoretic considerations, and may be of interest by itself in other situations.

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arXiv2018

KMS states on Nica-Toeplitz C*-algebras

Given a quasi-lattice ordered group $(G,P)$ and a compactly aligned product system $X$ of essential C$^*$-correspondences over the monoid $P$, we show that there is a bijection between the gauge-invariant KMS$_β$-states on the Nica-Toeplitz algebra $\mathcal{NT}(X)$ of $X$ with respect to a gauge-type dynamics, on one side, and the tracial states on the coefficient algebra $A$ satisfying a system (in general infinite) of inequalities, on the other. This strengthens and generalizes a number of results in the literature in several directions: we do not make any extra assumptions on $P$ and $X$, and our result can, in principle, be used to study KMS-states at any finite inverse temperature $β$. Under fairly general additional assumptions we show that there is a critical inverse temperature $β_c$ such that for $β>β_c$ all KMS$_β$-states are of Gibbs type, hence gauge-invariant, in which case we have a complete classification of KMS$_β$-states in terms of tracial states on $A$, while at $β=β_c$ we have a phase transition manifesting itself in the appearance of KMS$_β$-states that are not of Gibbs type. In the case of right-angled Artin monoids we show also that our system of inequalities for traces on $A$ can be reduced to a much smaller system, a finite one when the monoid is finitely generated. Most of our results generalize to arbitrary quasi-free dynamics on $\mathcal{NT}(X)$.

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arXiv2018

A few remarks on the tube algebra of a monoidal category

We prove two results on the tube algebras of rigid C$^*$-tensor categories. The first is that the tube algebra of the representation category of a compact quantum group $G$ is a full corner of the Drinfeld double of $G$. As an application we obtain some information on the structure of the tube algebras of the Temperley-Lieb categories $TL(d)$ for $d>2$. The second result is that the tube algebras of weakly Morita equivalent C$^*$-tensor categories are strongly Morita equivalent. The corresponding linking algebra is described as the tube algebra of the $2$-category defining the Morita context.

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arXiv2018

Graded twisting of comodule algebras and module categories

Continuing our previous work on graded twisting of Hopf algebras and monoidal categories, we introduce a graded twisting construction for equivariant comodule algebras and module categories. As an example we study actions of quantum subgroups of $G\subset SL_{-1}(2)$ on $K_{-1}[x,y]$ and show that in most cases the corresponding invariant rings $K_{-1}[x,y]^G$ are invariant rings $K[x,y]^{G'}$ for the action of a classical subgroup $G'\subset SL(2)$. As another example we study Poisson boundaries of graded twisted categories and show that under the assumption of weak amenability they are graded twistings of the Poisson boundaries.

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arXiv2018

Ribbon braided module categories, quantum symmetric pairs and Knizhnik-Zamolodchikov equations

Let $\mathfrak u$ be a compact semisimple Lie algebra, and $σ$ be a Lie algebra involution of $\mathfrak u$. Let Rep$_q(\mathfrak u)$ be the ribbon braided tensor C*-category of $U_q(\mathfrak u)$-representations for $0<q<1$. We introduce three module C*-categories over Rep$_q(\mathfrak u)$ starting from the input data $(\mathfrak u,σ)$. The first construction is based on the theory of cyclotomic KZ-equations. The second construction uses the notion of quantum symmetric pair as developed by G. Letzter. The third construction uses a variation of Drinfeld twisting. In all three cases the module C*-category is ribbon twist-braided in the sense of A. Brochier---this is essentially due to B. Enriquez in the first case, is proved by S. Kolb in the second case, and is closely related to work of J. Donin, P. Kulish, and A. Mudrov in the third case. We formulate a conjecture concerning equivalence of these ribbon twist-braided module C*-categories, and confirm it in the rank one case.

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arXiv2017

Noncommutative Borsuk-Ulam-type conjectures revisited

Let $H$ be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra $A$. It was recently conjectured that there does not exist an equivariant $*$-homomorphism from $A$ (type-I case) or $H$ (type-II case) to the equivariant noncommutative join C*-algebra $A\circledast^δH$. When $A$ is the C*-algebra of functions on a sphere, and $H$ is the C*-algebra of functions on ${\mathbb Z}/2{\mathbb Z}$ acting antipodally on the sphere, then the conjecture of type I becomes the celebrated Borsuk-Ulam theorem. Following recent work of Passer, we prove the conjecture of type I for compact quantum groups admitting a non-trivial torsion character. Next, we prove that, if a compact quantum group admits a representation whose \mbox{$K_1$-class} is non-trivial and $A$ admits a character, then a stronger version of the type-II conjecture holds: the finitely generated projective module associated with $A\circledast^δH$ via this representation is not stably free. In particular, we apply this result to the $q$-deformations of compact connected semisimple Lie groups and to the reduced group C*-algebras of free groups on $n>1$ generators.

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arXiv2017

Relations in quantized function algebras

We develop a method to give presentations of quantized function algebras of complex reductive groups. In particular, we give presentations of quantized function algebras of automorphism groups of finite dimensional simple complex Lie algebras.

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arXiv2017

Poisson boundaries of monoidal categories

Given a rigid C*-tensor category C with simple unit and a probability measure $μ$ on the set of isomorphism classes of its simple objects, we define the Poisson boundary of $(C,μ)$. This is a new C*-tensor category P, generally with nonsimple unit, together with a unitary tensor functor $Π: C \to P$. Our main result is that if P has simple unit (which is a condition on some classical random walk), then $Π$ is a universal unitary tensor functor defining the amenable dimension function on C. Corollaries of this theorem unify various results in the literature on amenability of C*-tensor categories, quantum groups, and subfactors.

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arXiv2016

Drinfeld center and representation theory for monoidal categories

Motivated by the relation between the Drinfeld double and central property (T) for quantum groups, given a rigid C*-tensor category C and a unitary half-braiding on an ind-object, we construct a *-representation of the fusion algebra of C. This allows us to present an alternative approach to recent results of Popa and Vaes, who defined C*-algebras of monoidal categories and introduced property (T) for them. As an example we analyze categories C of Hilbert bimodules over a II$_1$-factor. We show that in this case the Drinfeld center is monoidally equivalent to a category of Hilbert bimodules over another II$_1$-factor obtained by the Longo-Rehren construction. As an application, we obtain an alternative proof of the result of Popa and Vaes stating that property (T) for the category defined by an extremal finite index subfactor $N \subset M$ is equivalent to Popa's property (T) for the corresponding SE-inclusion of II$_1$-factors. In the last part of the paper we study Müger's notion of weakly monoidally Morita equivalent categories and analyze the behavior of our constructions under the equivalence of the corresponding Drinfeld centers established by Schauenburg. In particular, we prove that property (T) is invariant under weak monoidal Morita equivalence.

math.OA↗

arXiv2016

Graded twisting of categories and quantum groups by group actions

Given a Hopf algebra $A$ graded by a discrete group together with an action of the same group preserving the grading, we define a new Hopf algebra, which we call the graded twisting of $A$. If the action is adjoint, this new Hopf algebra is a twist of $A$ by a pseudo-$2$-cocycle. Analogous construction can be carried out for monoidal categories. As examples we consider graded twistings of the Hopf algebras of nondegenerate bilinear forms, their free products, hyperoctahedral quantum groups and $q$-deformations of compact semisimple Lie groups. As applications, we show that the analogues of the Kazhdan-Wenzl categories in the general semisimple case cannot be always realized as representation categories of compact quantum groups, and for genuine compact groups, we analyze quantum subgroups of the new twisted compact quantum groups, providing a full description when the twisting group is cyclic of prime order.

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arXiv2016

Towards a classification of compact quantum groups of Lie type

This is a survey of recent results on classification of compact quantum groups of Lie type, by which we mean quantum groups with the same fusion rules and dimensions of representations as for a compact connected Lie group $G$. The classification is based on a categorical duality for quantum group actions recently developed by De Commer and the authors in the spirit of Woronowicz's Tannaka--Krein duality theorem. The duality establishes a correspondence between the actions of a compact quantum group $H$ on unital C$^*$-algebras and the module categories over its representation category Rep $H$. This is further refined to a correspondence between the braided-commutative Yetter--Drinfeld $H$-algebras and the tensor functors from Rep $H$. Combined with the more analytical theory of Poisson boundaries, this leads to a classification of dimension-preserving fiber functors on the representation category of any coamenable compact quantum group in terms of its maximal Kac quantum subgroup, which is the maximal torus for the $q$-deformation of $G$ if $q\ne1$. Together with earlier results on autoequivalences of the categories Rep $G_q$, this allows us to classify up to isomorphism a large class of quantum groups of $G$-type for compact connected simple Lie groups $G$. In the case of $G=SU(n)$ this class exhausts all non-Kac quantum groups.

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arXiv2015

Classification of non-Kac compact quantum groups of SU(n) type

We classify up to isomorphism all non-Kac compact quantum groups with the same fusion rules and dimension function as $SU(n)$. For this we first prove, using categorical Poisson boundary, the following general result. Let $G$ be a coamenable compact quantum group and $K$ be its maximal quantum subgroup of Kac type. Then any dimension-preserving unitary fiber functor $Rep\ G \to Hilb_f$ factors, uniquely up to isomorphism, through $Rep\ K$. Equivalently, we have a canonical bijection $H^2(\hat G; T) \cong H^2(\hat K; T)$. Next, we classify autoequivalences of the representation categories of twisted $q$-deformations of compact simple Lie groups.

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