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Alexandru Chirvasitu

Publications and source records attributed to Alexandru Chirvasitu.

At least 19 recordsLinked to original sources
arXiv2026

Quantum Hopf rigidity in symmetric monoidal categories

We prove that a coaction of a Hopf algebra on another preserving the latter's associative-algebra structure and comultiplication automatically also preserves the counit, antipode, and tensor-interchange map and in fact factors through the coaction dual to an action of an affine group scheme by Hopf-algebra automorphisms. This generalizes and unifies a number of results to the effect that quantum groups have only classical symmetries, due to Kasprzak et al., Budziński-Kasprzak and Brannan et al. The stated automatic classicality follows from the fact that the antipode, counit and tensorand-interchange of a Hopf algebra internal to any symmetric monoidal category are contained in the (non-symmetric) monoidal subcategory generated by the associative-algebra structure and the comultiplication, assuming a weak form of arrow-retraction invariance.

math.QA↗
arXiv2026

Spectral selections, commutativity preservation and Coxeter-Lipschitz maps

For a Coxeter system $(W,S)$ a self-map $τ$ of $W$ such that $τ_{σθ}\in \{τ_θ,\ στ_θ\}$ for all $θ\in W$ and all reflections $σ$ (analogous to being 1-Lipschitz with respect to the Bruhat order on $W$) is either constant or a right translation. A somewhat stronger version holds for $S_n$, where it suffices that $σ$ range over smaller, $θ$-dependent sets of reflections. These combinatorial results have a number of consequences concerning continuous spectrum- and commutativity-preserving maps $\mathrm{SU}(n)\to M_n$ defined on special unitary groups: every such map is a conjugation composed with (a) the identity; (b) transposition, or (c) a continuous diagonal spectrum selection. This parallels and recovers Petek's analogous statement for self-maps of the space $H_n\le M_n$ of self-adjoint matrices, strengthening it slightly by expanding the codomain to $M_n$.

math.SP↗
arXiv2026

Non-commutative classifying spaces of groups via quasi-topologies and pro-$C^*$-algebras

For a locally compact Hausdorff topological group $G$, we construct a universal pro-$C^*$-algebra $C(E^+G)$ as the non-commutative geometer's analogue of the total space $EG$ of the classifying principal $G$-bundle $EG\to BG$. The pro-$C^*$-algebra $C(EG)$ of (possibly unbounded) continuous functions on $EG$ is then recoverable as the abelianization of $C(E^+G)$. Along the way, we develop various aspects of the theory of quasi-topological $G$-spaces and $G$-pro-$C^*$-algebras.

math.OA↗
arXiv2026

Ergodic-transformation centralizers and essentially non-compact graphing symmetry

We prove that for every ergodic transformation $T$ on an infinite standard probability space both the automorphism group (i.e. centralizer) $\mathrm{Aut}(T)$ and its reversing automorphism group are realizable as symmetry groups of graphings. This is an analogue of Sabidussi's realization of arbitrary graph-automorphism groups, and provides numerous examples of graphing automorphism groups carrying no compatible compact topology, answering a question of Lovasz'. Another consequence of discussion and ensuing constructions is the existence of large mutually locally-globally equivalent graphing families with highly variable symmetry.

math.DS↗
arXiv2026

Prescribed Graphon Symmetries and Flavors of Rigidity

We prove that an arbitrary compact metrizable group can be realized as the automorphism group of a graphon; this is a continuous analogue to Frucht's theorem recovering arbitrary finite groups as automorphism groups of finite graphs. The paper also contains a number of results on the persistence of transitivity of a compact-group action upon passing to a limit of graphons. Call a compact group $\mathbb{G}$ graphon-rigid if, whenever it acts transitively on each member $Γ_n$ of a convergent sequence of graphons, it also acts transitively on the limit $\lim_n Γ_n$. We show that for a compact Lie group $\mathbb{G}$ graphon rigidity is equivalent to the identity component $\mathbb{G}_0$ being semisimple; as a partial converse to a result of Lovász and Szegedy, this is also equivalent to weak randomness: the property that the group have only finitely many irreducible representations in each dimension. Similarly, call a compact group $\mathbb{G}$ image-rigid if for every compact Lie group $\mathbb{H}$ the images of morphisms $\mathbb{G}\to \mathbb{H}$ form a closed set (of closed subgroups, in the natural topology). We prove that graphon rigidity implies image rigidity for compact groups that are either connected or profinite, and the two conditions are equivalent (and also equivalent to being torsion) for profinite abelian groups.

math.GR↗
arXiv2026

Associative/Lie radical (co)invariance under tracial (co)actions

We prove a number of Jacobson/solvable/nilpotent-radical (co)invariance results for finite-dimensional associative or Lie (co)module-algebras over Hopf algebras $H$, provided the characteristic, if positive, is large relative to the dimension of the algebra and the (co)actions satisfy trace-preservation conditions automatic when $H$ is involutory. This generalizes a number of such radical-invariance criteria in the literature, due to A. Gordienko, V. Linchenko, Pagon-Repov{š}-Zaicev and others, providing a uniform enriched-categorical framework for those results.

math.QA↗
arXiv2026

Coincident Poisson structures on principal-bundle moduli spaces

Consider a reductive complex algebraic group $\mathbb{G}$ equipped with an action by a linearly reductive affine group scheme $\mathbb{K}$. The extension of $\mathfrak{p}^*$ by $\mathfrak{p}$ induced by an $(\mathbb{K},\mathfrak{g})$-invariant symmetric non-degenerate bilinear form on $\mathfrak{g}:=Lie(\mathbb{G})$, for a $\mathbb{K}$-invariant parabolic $\mathbb{P}\le \mathbb{G}$, is $\mathbb{K}$-equivariantly isomorphic to the extension obtained via the standard bialgebra structure attached to a $\mathbb{K}$-invariant Cartan/Borel pair $\mathbb{H}\le \mathbb{B}\le \mathbb{P}\le \mathbb{G}$ and the same bilinear form. Associating bundle extensions on an elliptic curve $E$ to said $\mathfrak{p}$-module extensions, this identifies Poisson structures on the smooth locus of the principal-$\mathbb{P}$-bundle moduli space over $E$ respectively defined by Balduzzi (using the former extension) and Feigin-Odesskii (via the standard bialgebra structure). This in particular verifies Feigin-Odesskii's identification of the bialgebra-induced symplectic leaves with the loci of bundles mutually isomorphic after forgetting structure along $\mathbb{P}\le \mathbb{G}$.

math.AG↗
arXiv2026

On the category of Hopf braces

Hopf braces are the quantum analogues of skew braces and, as such, their cocommutative counterparts provide solutions to the quantum Yang-Baxter equation. We investigate various properties of categories related to Hopf braces. In particular, we prove that the category of Hopf braces is accessible while the category of cocommutative Hopf braces is even locally presentable. We also show that functors forgetting multiple antipodes and/or multiplications down to coalgebras are monadic. Colimits in the category of cocommutative Hopf braces are described explicitly and a free cocommutative Hopf brace on an arbitrary cocommutative Hopf algebra is constructed.

math.QA↗
arXiv2026

Rigid ternary relations in finite-dimensional Hilbert-space Grassmannians

For positive integers $1\le r<d<n$ consider subsets $S\subseteq \mathbb{G}(r,V)$ of the $r$-plane Grassmannian of an $n$-dimensional Hilbert space $V$ saturated in the sense that the $r$-plane $η''$ belongs to $S$ whenever it is the orthogonal projection of $η'\in S$ onto a $d$-plane through $η\in S$. Motivated by such closure operators' natural occurrence in projective-geometry and linear preserver problems, we classify said saturated sets as precisely the disjoint unions of Grassmannian spines, with cores standing in a relation of mutual separation that can be made precise (a spine being the set of $r$-planes containing a fixed core $k$-plane $π$ for some $0\le k\le r$). This generalizes the author's results describing saturated $r$-plane sets in the tame dimensional regime $2r\le d$, where the disjoint unions in question by necessity collapse to single spines.

math.CO↗
arXiv2026

Uniformity and isotypic smallness for quantum-group representations

Compact-group representations on Banach spaces are known to be norm-continuous precisely when they have finite spectra. For a quantum group with continuous-function algebra $\mathcal{C}(\mathbb{G})$ norm continuity can be cast analogously as the bounded weak$^*$-norm continuity of the representation's attached maps $\mathcal{C}(\mathbb{G})^*\to \mathrm{End}(E)$ and its mirror counterpart $E_{\le 1}\times E^*_{\le 1}\to \mathcal{C}(\mathbb{G})$. While the uniformity/isotypic finiteness equivalence no longer holds generally, it does (for the latter map) for compact quantum groups either coamenable or having dimension-bounded irreducible representations. This generalizes the aforementioned classical variant, providing two independent quantum-specific mechanisms of recovering it.

math.OA↗
arXiv2026

Eigenvalue collision and exotic preservers on semisimple operators

We classify $n\times n$-matrix-valued continuous commutativity and spectrum preservers defined on spaces of (a) normal, (b) semisimple and (c) arbitrary $n\times n$ matrices with spectra contained in sufficiently connected subsets $\mathcal{X}\subseteq \mathbb{C}$, generalizing a number of results due to Šemrl, Gogić, Tomašević and the author among others. In case (a) these are always conjugations or transpose conjugations, while in cases (b) and (c) qualitatively distinct possibilities arise depending on the local regularity of the complex-conjugation map close to coincident-eigenvalue loci of $\mathcal{X}^n$.

math.SP↗
arXiv2026

Action principality as a Lie-group certificate

A continuous action $\mathbb{G}\circlearrowright X$ of a topological group is principal if its isotropy groups are all conjugate to $\mathbb{H}\le \mathbb{G}$ and the quotient map $X\to X/\mathbb{G}$ is a locally trivial $\mathbb{G}/\mathbb{H}$-fiber bundle. We prove that compact groups whose identity component has metrizable abelianization are Lie provided their free actions on Tychonoff (equivalently, compact Hausdorff) spaces are all principal; this is a converse to Gleason's theorem. A variant confirms the conclusion for Tychonoff or compact Hausdorff actions with constant central isotropy by compact connected groups.

math.GR↗
arXiv2026

Equivariant compactifications, trivial embeddability and finite type

We characterize finite-type $\mathbb{G}$-principal $\mathbb{U}$-equivariant bundles on normal $\mathbb{U}$-spaces for compact Lie groups $\mathbb{U}$ and $\mathbb{G}$, in several ways, including (a) their extensibility across the $\mathbb{U}$-equivariant compactification $β_{\mathbb{U}}X$ and (b) their becoming finite-type upon extending the structure group along at least one $\mathbb{U}$-equivariant compact-Lie-group embedding $\mathbb{G}\le \mathbb{K}$. This generalizes non-equivariant results of Phillips and the author's characterizing finite-type matrix-algebra bundles, upon specializing $\mathbb{G}$ to projective unitary groups. When the $\mathbb{U}$-action on $X$ has virtually abelian isotropy, matrix-algebra equivariant bundles are also finite-type precisely when, locally over a finite open $\mathbb{U}$-cover, they are tensor factors of trivial matrix bundles. In a $K$-theoretic offshoot we prove that for $\mathbb{U}$-actions with finite isotropy groups on compact Hausdorff spaces $X$ equivariant vector bundles $\mathcal{E}\to X$ are factors of trivial bundles $K$-theoretically: there is a class $a\in K_{\mathbb{U}}(X)$ with $[\mathcal{E}]a$ the class of a bundle induced by a $\mathbb{U}$-representation (which furthermore can be chosen so as to restrict to isotropy groups to multiples of the regular representations). This generalizes a result of Donovan and Karoubi.

math.AT↗
arXiv2026

Irremediably singular quantum branched covers

We prove that the Natsume-Olsen non-commutative spheres $\mathbb{S}^{2n-1}_θ$ dualize for rational deformation parameters to provide examples of quantum branched covers over their respective centers' maximal spectra, embeddable into locally trivial $C^*$ bundles precisely when said spheres are classical; this, despite the bundles in question being of finite type over dimensional strata. The proof of the noted rigidity phenomenon relies on the quantum analogue of the 3-sphere's Heegaard splitting as a pushout of two solid tori.

math.QA↗
arXiv2026

Obstructed subhomogeneous-bundle extensions and embeddings

We address a number of problems concerning the (im)possibility of either extending locally trivial subbundles of possibly singular Banach/$C^*$ bundles globally, embedding subhomogeneous bundles into homogeneous ones, or recovering locally trivial compact-Lie-group-equivariant Banach or $C^*$ bundles as pullbacks along equivariant maps to compact spaces. The results include (1) the global extensibility of a locally trivial Banach/Hilbert/Banach-algebra/$C^*$ subbundle from a closed subspace of a paracompact space given appropriate homotopy constraints; (2) the homogeneous embeddability of equivariant subhomogeneous Banach/Hilbert bundles locally trivial along the singular locus under the same homotopy constraints, and (3) the characterization of finite-type equivariant locally trivial subhomogeneous $C^*$ bundles on normal spaces as precisely those (a) locally trivial as plain vector bundles, or (b) pulled back from the universal equivariant compactification or (c) pulled back from an equivariant map into a smooth manifold. The latter extends results of Phillips concerning non-equivariant matrix-algebra bundles restricted along the StoneČech compactification.

math.FA↗
arXiv2026

Injectivity paucity in AB5 categories of oversize chains

We construct examples of abelian categories with no non-zero injective (or projective) objects satisfying Grothendieck's AB5 condition. The procedure combines Rickard's examples of AB5 categories without products but some non-trivial injectives (also addressing an apparent gap in the literature) with a 2-functorial construct attaching to any category $\mathcal{C}$ that of $\mathcal{C}$-objects equipped with set-indexed families of endomorphisms.

math.CT↗
arXiv2026

Discontinuous actions on cones, joins, and $n$-universal bundles

We prove that locally countably-compact Hausdorff topological groups $\mathbb{G}$ act continuously on their iterated joins $E_n\mathbb{G}:=\mathbb{G}^{*(n+1)}$ (the total spaces of the Milnor-model $n$-universal $\mathbb{G}$-bundles) as well as the colimit-topologized unions $E\mathbb{G}=\varinjlim_n E_n\mathbb{G}$, and the converse holds under the assumption that $\mathbb{G}$ is first-countable. In the latter case other mutually equivalent conditions provide characterizations of local countable compactness: the fact that $\mathbb{G}$ acts continuously on its first self-join $E_1\mathbb{G}$, or on its cone $\mathcal{C}\mathbb{G}$, or the coincidence of the product and quotient topologies on $\mathbb{G}\times \mathcal{C}X$ for all spaces $X$ or, equivalently, for the discrete countably-infinite $X:=\aleph_0$. These can all be regarded as weakened versions of $\mathbb{G}$'s exponentiability, all to the effect that $\mathbb{G}\times -$ preserves certain colimit shapes in the category of topological spaces; the results thus extend the equivalence (under the separation assumption) between local compactness and exponentiability.

math.GN↗
arXiv2026

The combinatorics of permuting and preserving curve-bound spectra

We prove that continuous spectrum- and commutativity-preserving maps to $\mathcal{M}_n(\mathbb{C})$ from the space of normal (real or complex) $n\times n$, $n\ge 3$ matrices with spectra contained in a given continuous-injection interval image $Λ\subseteq \mathbb{C}$ or $\mathbb{R}$ are (a) conjugations; (b) transpose conjugations, or (c) orderings of spectra according to an orientation of $Λ$, with fixed eigenspaces. This generalizes results of Petek's (self-maps of real or complex Hermitian matrices) and the author's (complex Hermitian matrices as the domain, $\mathcal{M}_n(\mathbb{C})$ as the codomain). An application rules out possibility (c) for normal matrices with spectra constrained to a simple closed curve, extending a result by the author, Gogić and Tomašević to the effect that continuous commutativity and spectrum preservers on unitary groups are (transpose) conjugations. The involution preserving eigenspaces and complex-conjugating eigenvalues is a novel possibility beyond (a), (b) and (c) if the domain consists of all semisimple operators with $Λ$-bound spectra instead; its continuity (or lack thereof) and whether or not that map furthermore extends continuously to arbitrary $Λ$-constrained-spectrum matrices hinge on the geometry and regularity of $Λ$.

math.SP↗