arXiv subjects

Alexandru Chirvasitu

Publications and source records attributed to Alexandru Chirvasitu.

At least 145 records · Page 8Linked to original sources

arXiv2018

The weak Hilbert-Smith conjecture from a Borsuk-Ulam-type conjecture

We prove a number of results surrounding the Borsuk-Ulam-type conjecture of Baum, Dąbrowski and Hajac (BDH, for short), to the effect that given a free action of a compact group $G$ on a compact space $X$, there are no $G$-equivariant maps $X*G\to X$ (with $*$ denoting the topological join). In particular, we prove the BDH conjecture for locally trivial principal $G$-bundles. The proof relies on the non-existence of $G$-equivariant maps $G^{*(n+1)}\to G^{*n}$, which in turn is a slight strengthening of an unpublished result of M. Bestvina and R. Edwards. Moreover, we show that the BDH conjecture partially settles a conjecture of Ageev. In turn, the latter implies the weak version Hilbert-Smith conjecture stating that no infinite compact zero-dimensional group can act freely on a manifold such that the orbit space is finite-dimensional.

math.GN↗

arXiv2018

Topological generation and matrix models for quantum reflection groups

We establish several new topological generation results for the quantum permutation groups $S^+_N$ and the quantum reflection groups $H^{s+}_N$. We use these results to show that these quantum groups admit sufficiently many "matrix models". In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain "flat" matrix models for $S_N^+$ are inner faithful.

math.OA↗

arXiv2018

Relative Fourier transforms and expectations on coideal subalgebras

For an algebraic compact quantum group $H$ we establish a bijection between the set of right coideal $*$-subalgebras $A\to H$ and that of left module quotient $*$-coalgebras $H\to C$. It turns out that the inclusion $A\to H$ always splits as a map of right $A$-modules and right $H$-comodules, and the resulting expectation $E:H\to A$ 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 $A\to H$ by means of a fiber functor from $H$-equivariant $A$-modules to Hilbert spaces, while the characterization of those $A\to H$ which admit positive expectations makes use of a Fourier transform turning elements of $H$ into functions on $C$.

math.QA↗

arXiv2018

Quasi-flat representations of uniform groups and quantum groups

Given a discrete group $Γ= $ and a number $K\in\mathbb N$, a unitary representation $ρ:Γ\to U_K$ is called quasi-flat when the eigenvalues of each $ρ(g_i)\in U_K$ are uniformly distributed among the $K$-th roots of unity. The quasi-flat representations of $Γ$ form altogether a parametric matrix model $π:Γ\to C(X,U_K)$. We compute here the universal model space $X$ for various classes of discrete groups, notably with results in the case where $Γ$ is metabelian. We are particularly interested in the case where $X$ is a union of compact homogeneous spaces, and where the induced representation $\tildeπ:C^*(Γ)\to C(X,U_K)$ is stationary in the sense that it commutes with the Haar functionals. We present several positive and negative results on this subject. We also discuss similar questions for the discrete quantum groups, proving a stationarity result for the discrete dual of the twisted orthogonal group $O_2^{-1}$.

math.QA↗

arXiv2017

Representation categories of Mackey Lie algebras as universal monoidal categories

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. We study a monoidal category $\mathbb{T}_α$ which is universal among all symmetric $\mathbb{K}$-linear monoidal categories generated by two objects $A$ and $B$ such that $A$ has a, possibly transfinite, filtration. We construct $\mathbb{T}_α$ as a category of representations of the Lie algebra $\mathfrak{gl}^M(V_*,V)$ consisting of endomorphisms of a fixed diagonalizable pairing $V_*\otimes V\to \mathbb{K}$ of vector spaces $V_*$ and $V$ of dimension $α$. Here $α$ is an arbitrary cardinal number. We describe explicitly the simple and the injective objects of $\mathbb{T}_α$ and prove that the category $\mathbb{T}_α$ is Koszul. We pay special attention to the case where the filtration on $A$ is finite. In this case $α=\aleph_t$ for $t\in\mathbb{Z}_{\geq 0}$.

math.RT↗

arXiv2017

Thoma type results for discrete quantum groups

Thoma's theorem states that a group algebra $C^*(Γ)$ is of type I if and only if $Γ$ 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 $Γ$, we have a stationary model of type $π:C^*(Γ)\to M_F(C(L))$, with $F$ being a finite quantum group, and with $L$ being a compact group. We discuss then some refinements of these results in the quantum permutation group case, $\widehatΓ\subset S_N^+$, 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 $\leq1$.

math.QA↗

arXiv2017

Compact Group Actions on Topological and Noncommutative Joins

We consider the Type 1 and Type 2 noncommutative Borsuk-Ulam conjectures of Baum, D$ą$browski, and Hajac: there are no equivariant morphisms $A \to A \circledast_δH$ or $H \to A \circledast_δH$, respectively, when $H$ is a nontrivial compact quantum group acting freely on a unital $C^*$-algebra $A$. Here $A \circledast_δH$ 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 $A \circledast_δH$. For the classical case $H = \mathcal{C}(G)$, $G$ 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.

math.OA↗

arXiv2017

Ergodic actions of the compact quantum group $O_{-1}(2)$

Among the ergodic actions of a compact quantum group $\mathbb{G}$ on possibly non-commutative spaces, those that are {\it embeddable} are the natural analogues of actions of a compact group on its homogeneous spaces. These can be realized as {\it coideal subalgebras} of the function algebra $\mathcal{O}(\mathbb{G})$ attached to the compact quantum group. We classify the embeddable ergodic actions of the compact quantum group $O_{-1}(2)$, basing our analysis on the bijective correspondence between all ergodic actions of the classical group $O(2)$ and those of its quantum twist resulting from the monoidal equivalence between their respective tensor categories of unitary representations. In the last section we give counterexamples showing that in general we cannot expect a bijective correspondence between embeddable ergodic actions of two monoidally equivalent compact quantum groups.

math.QA↗

arXiv2017

The prime spectra of relative stable module categories

For a finite group $G$ and an arbitrary commutative ring $R$, Broué has placed a Frobenius exact structure on the category of finitely generated $RG$-modules by taking the exact sequences to be those that split upon restriction to the trivial subgroup. The corresponding stable category is then tensor triangulated. In this paper we examine the case $R=S/t^n$, where $S$ is a discrete valuation ring having uniformising parameter $t$. We prove that the prime ideal spectrum (in the sense of Balmer) of this `relative' version of the stable module category of $RG$ is a disjoint union of $n$ copies of that for $kG$, where $k$ is the residue field of $S$.

math.RT↗

arXiv2017

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.

math.QA↗

arXiv2017

Topological automorphism groups of compact quantum groups

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 with the construction of a compact matrix quantum group whose fusion ring is not finitely generated, unlike the classical case.

math.OA↗

arXiv2016

Fundamental isomorphism theorems for quantum groups

The lattice of subgroups of a group is the subject of numerous results revolving around the central theme of decomposing the group into "chunks" (subquotients) that can then be compared to one another in various ways. Examples of results in this class would be the Noether isomorphism theorems, Zassenhaus' butterfly lemma, the Schreier refinement theorem for subnormal series of subgroups, the Dedekind modularity law, and last but not least the Jordan-Hölder theorem. We discuss analogues of the above-mentioned results in the context of locally compact quantum groups and linearly reductive quantum groups. The nature of the two cases is different: the former is operator algebraic and the latter Hopf algebraic, hence the corresponding two-part organization of our study. Our intention is that the analytic portion be accessible to the algebraist and vice versa. The upshot is that in the locally compact case one often needs further assumptions (integrability, compactness, discreteness). In the linearly reductive case on the other hand, the quantum versions of the results hold without further assumptions. Moreover the case of compact / discrete quantum groups is usually covered by both the linearly reductive and the locally compact framework, thus providing a bridge between the two.

math.QA↗

arXiv2016

On the Hopf (co)center of a Hopf algebra

The notion of Hopf center and Hopf cocenter of a Hopf algebra is investigated by the extension theory of Hopf algebras. We prove that each of them yields an exact sequence of Hopf algebras. Moreover the exact sequences are shown to satisfy the faithful (co)flatness condition. Hopf center and cocenter are computed for $\mathsf{U}_q(\mathfrak{g})$ and the Hopf algebra $\textrm{Pol}(\mathbb{G}_q)$, where $\mathbb{G}_q$ is the Drinfeld-Jimbo quantization of a compact semisimple simply connected Lie group $\mathbb{G}$ and $\mathfrak{g}$ is a simple complex Lie algebra.

math.QA↗

arXiv2016

Parkable convex sets and finite-dimensional Hilbert spaces

A subset of a convex body $B$ containing the origin in a Euclidean space is {\it parkable in $B$} if it can be translated inside $B$ in such a manner that the translate the origin. We provide characterizations of ellipsoids and of centrally symmetric convex bodies in Euclidean spaces of dimension $\ge 3$ based on the notion of parkability, answering several questions posed by G. Bergman. The techniques used, which are based on characterizations of Hilbert spaces among finite-dimensional Banach spaces in terms of their lattices of subspaces and algebras of endomorphisms, also apply to improve a result of W. Blaschke characterizing ellipsoids in terms of boundaries of illumination.

math.MG↗

arXiv2016

Ordered tensor categories and representations of the Mackey Lie algebra of infinite matrices

We introduce (partially) ordered Grothendieck categories and apply results on their structure to the study of categories of representations of the Mackey Lie algebra of infinite matrices $\mathfrak{gl}^M\left(V,V_*\right)$. Here $\mathfrak{gl}^M\left(V,V_*\right)$ is the Lie algebra of endomorphisms of a nondegenerate pairing of countably infinite-dimensional vector spaces $V_*\otimes V\to\mathbb{K}$, where $\mathbb{K}$ is the base field. Tensor representations of $\mathfrak{gl}^M\left(V,V_*\right)$ are defined as arbitrary subquotients of finite direct sums of tensor products $(V^*)^{\otimes m}\otimes (V_*)^{\otimes n}\otimes V^{\otimes p}$ where $V^*$ denotes the algebraic dual of $V$. The category $\mathbb{T}^3_{\mathfrak{gl}^M\left(V,V_*\right)}$ which they comprise, extends a category $\mathbb{T}_{\mathfrak{gl}^M\left(V,V_*\right)}$ previously studied in [4, 12,17], and our main result is that $\mathbb{T}^3_{\mathfrak{gl}^M\left(V,V_*\right)}$ is a finite-length, Koszul self-dual, tensor category with a certain universal property that makes it into a "categorified algebra" defined by means of a handful of generators and relations. This result uses essentially the general properties of ordered Grothendieck categories, which yield also simpler proofs of some facts about the category $\mathbb{T}_{\mathfrak{gl}^M\left(V,V_*\right)}$ established in [12]. Finally, we discuss the extension of $\mathbb{T}^3_{\mathfrak{gl}^M\left(V,V_*\right)}$ by the algebraic dual $(V_*)^*$ of $V_*$.

math.RT↗

arXiv2015

Quantum rigidity of negatively curved manifolds

We show that an isometric action of a compact quantum group on the underlying geodesic metric space of a compact connected Riemannian manifold $(M,g)$ with strictly negative curvature is automatically classical, in the sense that it factors through the action of the isometry group of $(M,g)$. This partially answers a question by D. Goswami.

math.QA↗

arXiv2014

Residually finite quantum group algebras

We show that provided $n\ne 3$, the involutive Hopf *-algebra $A_u(n)$ 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 $A_u(n)$ is maximal almost periodic (i.e. it embeds in its quantum Bohr compactification), answering a question posed by P. So tan. We also prove analogous results for the involutive Hopf *-algebra $B_u(n)$ coacting universally on an $n$-dimensional Hilbert space equipped with a non-degenerate bilinear form.

math.QA↗

arXiv2014

Tensor functors between categories of quasi-coherent sheaves

For a quasi-compact quasi-separated scheme X and an arbitrary scheme Y we show that the pullback construction implements an equivalence between the discrete category of morphisms Y --> X and the category of cocontinuous tensor functors Qcoh(X) --> Qcoh(Y). This is an improvement of a result by Lurie and may be interpreted as the statement that algebraic geometry is 2-affine. Moreover, we prove the analogous version of this result for Durov's notion of generalized schemes over F_1.

math.AG↗