arXiv subjects

Alexandru Chirvasitu

Publications and source records attributed to Alexandru Chirvasitu.

158 records · Page 9Linked to original sources

arXiv2014

Reflexivity and dualizability in categorified linear algebra

The "linear dual" of a cocomplete linear category $\mathcal C$ is the category of all cocontinuous linear functors $\mathcal C \to \mathrm{Vect}$. We study the questions of when a cocomplete linear category is reflexive (equivalent to its double dual) or dualizable (the pairing with its dual comes with a corresponding copairing). Our main results are that the category of comodules for a countable-dimensional coassociative coalgebra is always reflexive, but (without any dimension hypothesis) dualizable if and only if it has enough projectives, which rarely happens. Along the way, we prove that the category $\mathrm{Qcoh}(X)$ of quasi-coherent sheaves on a stack $X$ is not dualizable if $X$ is the classifying stack of a semisimple algebraic group in positive characteristic or if $X$ is a scheme containing a closed projective subscheme of positive dimension, but is dualizable if $X$ is the quotient of an affine scheme by a virtually linearly reductive group. Finally we prove tensoriality (a type of Tannakian duality) for affine ind-schemes with countable indexing poset.

math.CT↗

arXiv2014

On quantum symmetries of compact metric spaces

An action of a compact quantum group on a compact metric space $(X,d)$ is (D)-isometric if the distance function is preserved by a diagonal action on $X\times X$. We show that an isometric action in this sense has the following additional property: The corresponding action on the algebra of continuous functions on $X$ by the convolution semigroup of probability measures on the quantum group contracts Lipschitz constants. It is, in other words, isometric in another sense due to H. Li, J. Quaegebeur and M. Sabbe; this partially answers a question of D. Goswami. We also introduce other possible notions of isometric quantum action, in terms of the Wasserstein $p$-distances between probability measures on $X$ for $p\ge 1$, used extensively in optimal transportation. It turns out all of these definitions of quantum isometry fit in a tower of implications, with the two above at the extreme ends of the tower. We conjecture that they are all equivalent.

math.OA↗

arXiv2014

Three results on representations of Mackey Lie algebras

I. Penkov and V. Serganova have recently introduced, for any non-degenerate pairing $W\otimes V\to\mathbb C$ of vector spaces, the Lie algebra $\mathfrak{gl}^M=\mathfrak{gl}^M(V,W)$ consisting of endomorphisms of $V$ whose duals preserve $W\subseteq V^*$. In their work, the category $\mathbb{T}_{\mathfrak{gl}^M}$ of $\mathfrak{gl}^M$-modules which are finite length subquotients of the tensor algebra $T(W\otimes V)$ is singled out and studied. In this note we solve three problems posed by these authors concerning the categories $\mathbb{T}_{\mathfrak{gl}^M}$. Denoting by $\mathbb{T}_{V\otimes W}$ the category with the same objects as $\mathbb{T}_{\mathfrak{gl}^M}$ but regarded as $V\otimes W$-modules, we first show that when $W$ and $V$ are paired by dual bases, the functor $\mathbb{T}_{\mathfrak{gl}^M}\to \mathbb{T}_{V\otimes W}$ taking a module to its largest weight submodule with respect to a sufficiently nice Cartan subalgebra of $V\otimes W$ is a tensor equivalence. Secondly, we prove that when $W$ and $V$ are countable-dimensional, the objects of $\mathbb{T}_{\mathrm{End}(V)}$ have finite length as $\mathfrak{gl}^M$-modules. Finally, under the same hypotheses, we compute the socle filtration of a simple object in $\mathbb{T}_{\mathrm{End}(V)}$ as a $\mathfrak{gl}^M$-module.

math.RT↗

arXiv2014

Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras

The question of whether or not a Hopf algebra $H$ is faithfully flat over a Hopf subalgebra $A$ has received positive answers in several particular cases: when $H$ (or more generally, just $A$) is commutative, or cocommutative, or pointed, or when $K$ contains the coradical of $H$. We prove the statement in the title, adding the class of cosemisimple Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting "exact sequence" $A\to H\to C$ is always a cosemisimple coalgebra, and that the expectation $H\to A$ is positive when $H$ is a CQG algebra.

math.RA↗

arXiv2013

Centers, cocenters and simple quantum groups

We define the notion of a (linearly reductive) center for a linearly reductive quantum group, and show that the quotient of a such a quantum group by its center is simple whenever its fusion semiring is free in the sense of Banica and Vergnioux. We also prove that the same is true of free products of quantum groups under very mild non-degeneracy conditions. Several natural families of compact quantum groups, some with non-commutative fusion semirings and hence very "far from classical", are thus seen to be simple. Examples include quotients of free unitary groups by their centers, recovering previous work, as well as quotients of quantum reflection groups by their centers.

math.QA↗

arXiv2013

Dedekind complete posets from sheaves on von Neumann algebras

We show that for any two von Neumann algebras $M$ and $N$, the space of non-unital normal homomorphisms $N\to M$ with finite support, modulo conjugation by unitaries in $M$, is Dedekind complete with respect to the partial order coming from the addition of homomorphisms with orthogonal ranges; this ties in with work by Brown and Capraro, where the corresponding objects are given Banach space structures under various niceness conditions on $M$ and $N$. More generally, we associate to $M$ a Grothendieck site-type category, and show that presheaves satisfying a sheaf-like condition on this category give rise to Dedekind complete lattices upon modding out unitary conjugation. Examples include the above-mentioned spaces of morphisms, as well as analogous spaces of completely positive or contractive maps. We also study conditions under which these posets can be endowed with cone structures extending their partial additions.

math.OA↗

arXiv2012

Remarks on Quantum Symmetric Algebras

We examine the quantum symmetric and exterior algebras of finite-dimensional \uqg-modules first systematically studied by Berenstein and Zwicknagl, and resolve some questions that they raised. We show that the difference (in the Grothendieck group) between the quantum symmetric and exterior cubes of a finite-dimensional module is the same as it is classically. Furthermore, we show that quantum symmetric algebras are commutative in an appropriate sense, and are universal with this property. We make extensive use of the coboundary structure on the module category.

math.QA↗

arXiv2012

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.

math.QA↗

arXiv2012

Categorical aspects of compact quantum groups

We show that either of the two reasonable choices for the category of compact quantum groups is nice enough to allow for a plethora of universal constructions, all obtained "by abstract nonsense" via the adjoint functor theorem. This approach both recovers constructions which have appeared in the literature, such as the quantum Bohr compactification of a locally compact semigroup, and provides new ones, such as the coproduct of a family of compact quantum groups, and the compact quantum group freely generated by a locally compact quantum space. In addition, we characterize epimorphisms and monomorphisms in the category of compact quantum groups.

math.QA↗

arXiv2012

The fundamental pro-groupoid of an affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don't remember $\Forget$? Working over an arbitrary commutative ring $R$, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid $π_1(\spec(R))$, i.e.\ the separable absolute Galois group of $R$ when it is a field. This gives a new definition for étale $π_1(\spec(R))$ in terms of the category of $R$-modules rather than the category of étale covers. More generally, we introduce a new notion of "commutative 2-ring" that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of $π_1$ for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the étale fundamental group of a scheme preserves finite products but not all products.

math.CT↗

arXiv2010

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 be parallel to those for Wang and Van Daele's deformed universal compact quantum groups and Bichon's 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.

math.RA↗

arXiv2010

Subcoalgebras and endomorphisms of free Hopf algebras

For a matrix coalgebra $C$ over some field, we determine all small subcoalgebras of the free Hopf algebra on $C$, the free Hopf algebra with a bjective antipode on $C$, and the free Hopf algebra with antipode $S$ satisfying $S^{2d}={\rm id}$ on $C$ for some fixed $d$. We use this information to find the endomorphisms of these free Hopf algebras, and to determine the centers of the categories of Hopf algebras, Hopf algebras with bijective antipode, and Hopf algebras with antipode of order dividing 2d.

math.RA↗

arXiv2009

On epimorphisms and monomorphisms of Hopf algebras

We provide examples of non-surjective epimorphisms $H\to K$ in the category of Hopf algebras over a field, even with the additional requirement that $K$ have bijective antipode, by showing that the universal map from a Hopf algebra to its enveloping Hopf algebra with bijective antipode is an epimorphism in $\halg$, although it is known that it need not be surjective. Dual results are obtained for the problem of whether monomorphisms in the category of Hopf algebras are necessarily injective. We also notice that these are automatically examples of non-faithfully flat and respectively non-faithfully coflat maps of Hopf algebras.

math.RA↗

arXiv2009

When is the diagonal functor Frobenius?

Given a complete, cocomplete category $\mathcal C$, we investigate the problem of describing those small categories $I$ such that the diagonal functor $Δ:\mathcal C\to {\rm Functors}(I,\mathcal C)$ is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors $I\to\mathcal C$ are naturally isomorphic. We find necessary conditions on $I$ for a certain class of categories $\mathcal C$, and, as an application, we give both necessary and sufficient conditions in the two special cases $\mathcal C={\bf Set}$ or $_R\mathcal M$, the category of left modules over a ring $R$.

math.CT↗