arXiv subjects

Alexandru Chirvasitu

Publications and source records attributed to Alexandru Chirvasitu.

At least 55 records · Page 3Linked to original sources

arXiv2025

Polynomial identities and Azumaya loci for rational quantum spheres

We prove a number of structure and isomorphism results concerning the non-commutative Natsume-Olsen spheres $\mathbb{S}^{2n-1}_θ$ deformed along a skew-symmetric matrix $θ\in \mathbb{R}$. These include (a) the fact that two $C^*$-algebras of the form $\mathbb{S}^{3}_θ\otimes M_n$ are isomorphic precisely in the obvious cases; (b) the fact that $m$ and $n$ are recoverable from the isomorphism class of $C(\mathbb{S}^{2m-1}_θ)\otimes M_n$; (c) the PI character, PI degree and Azumaya loci of $C(\mathbb{S}^{2m-1}_θ)$ for rational $θ$, along with a realization of their centers as (function algebras of) branched cover of $\mathbb{S}^{2n-1}$ and (d) for rational $θ$ again, the topological finite generation of $C(\mathbb{S}^{2m-1}_θ)$ over their centers, with algebraic finite generation equivalent to being classical (equivalently, Azumaya).

math.QA↗

arXiv2025

Fano schemes of sub-maximal elementary symmetric functions

Denote by $E_r$ the $r^{th}$ elementary symmetric polynomial in $\dim V$ variables for a vector space $V$ over an infinite field $\Bbbk$. We describe the rational points on the Fano scheme $F_{d-1}(Z(E_{\dim V-1}))$ of projective $(d-1)$-spaces contained in the zero locus of $E_{\dim V-1}$. Isolated points exist precisely for $\dim V=2d$, in which case they are in bijection with the $1\cdot 3\cdots (2d-1)$ pairings on a $2d$-element set. This, in particular, confirming a conjecture of Ambartsoumian, Auel and Jebelli to the effect that (over $\mathbb{R}$) all isolated points are recoverable via integral star transforms with appropriate symbols.

math.AG↗

arXiv2025

Local presentability and monadicity of forgetful functors for operator algebraic categories

In recent work of Lindenhovius and Zamdzhiev, it was established that the category of complete operator spaces, with completely contractive linear maps as morphisms, is locally countably presentable. In this work, we extend their conclusion to the non-complete setting and prove that the categories of operator systems, (Archimedean) order unit spaces, and unital operator algebras are all locally countably presentable as well. This is established through an analysis of forgetful functors and the identification of Eilenberg-Moore categories. We provide a complete understanding of adjunction and monadicity for forgetful functors between these categories, together with the categories of $C^*$-algebras, Banach spaces, and normed spaces. In addition, for various subcategories of function-theoretic objects, we investigate completeness and local presentability through Kadison's duality theorem.

math.CT↗

arXiv2025

A variant of Šemrl's preserver theorem for singular matrices

For positive integers $1 \leq k \leq n$ let $M_n$ be the algebra of all $n \times n$ complex matrices and $M_n^{\le k}$ its subset consisting of all matrices of rank at most $k$. We first show that whenever $k>\frac{n}{2}$, any continuous spectrum-shrinking map $ϕ: M_n^{\le k} \to M_n$ (i.e. $\mathrm{sp}(ϕ(X)) \subseteq \mathrm{sp}(X)$ for all $X \in M_n^{\le k}$) either preserves characteristic polynomials or takes only nilpotent values. Moreover, for any $k$ there exists a real analytic embedding of $M_n^{\le k}$ into the space of $n\times n$ nilpotent matrices for all sufficiently large $n$. This phenomenon cannot occur when $ϕ$ is injective and either $k > n - \sqrt{n}$ or the image of $ϕ$ is contained in $M_n^{\le k}$. We then establish a main result of the paper -- a variant of Šemrl's preserver theorem for $M_n^{\le k}$: if $n \geq 3$, any injective continuous map $ϕ:M_n^{\le k} \to M_n^{\le k}$ that preserves commutativity and shrinks spectrum is of the form $ϕ(\cdot)=T(\cdot)T^{-1}$ or $ϕ(\cdot)=T(\cdot)^tT^{-1}$, for some invertible matrix $T\in M_n$. Moreover, when $k=n-1$, which corresponds to the set of singular $n\times n$ matrices, this result extends to maps $ϕ$ which take values in $M_n$. Finally, we discuss the indispensability of assumptions in our main result.

math.SP↗

arXiv2025

Generic infinite generation, fixed-point-poor representations and compact-element abundance in disconnected Lie groups

The semidirect product $\mathbb{G}=\mathbb{L}\rtimes \mathbb{K}$ attached to a compact-group action on a connected, simply-connected solvable Lie group has a dense set of compact elements precisely when the $s\in \mathbb{K}$ operating on $\mathbb{L}$ fixed-point-freely constitute a dense set. This (along with a number of alternative equivalent characterizations) extends the Wu's analogous result for connected Lie $\mathbb{K}$, and also provides ample supplies of examples of almost-connected Lie groups $\mathbb{G}$ which do not have dense sets of compact elements, even though their identity components $\mathbb{G}_0$ do. This corrects prior literature on the subject, claiming the property equivalent for $\mathbb{G}$ and $\mathbb{G}_0$. In a related discussion we characterize those connected Lie groups $\mathbb{G}$ with large sets of $d$-tuples generating dense subgroups $Γ\le \mathbb{G}$ for which the derived subgroup $Γ^{(1)}$ fails to be finitely-generated: $\mathbb{G}$ must either be non-trivial topologically perfect or have non-nilpotent maximal solvable quotient.

math.GR↗

arXiv2025

Pointwise-relatively-compact subgroups and trivial-weight-free representations

A pointwise-elliptic subset of a topological group is one whose elements all generate relatively-compact subgroups. A connected locally compact group has a dense pointwise-elliptic subgroup if and only if it is an extension by a compact normal subgroup of a semidirect product $\mathbb{L}\rtimes \mathbb{K}$ with connected, simply-connected Lie $\mathbb{L}$, compact Lie $\mathbb{K}$, with the commutator subgroup $\mathbb{K}'$ acting on the Lie algebra $Lie(\mathbb{L})$ with no trivial weights. This extends and recovers a result of Kabenyuk's, providing the analogous classification with $\mathbb{G}$ assumed Lie connected, topologically perfect, with no non-trivial central elliptic elements.

math.GR↗

arXiv2025

Pervasive ellipticity in locally compact groups

A topological group is (openly) almost-elliptic if it contains a(n open) dense subset of elements generating relatively-compact cyclic subgroups. We classify the (openly) almost-elliptic connected locally compact groups as precisely those with compact maximal semisimple quotient and whose maximal compact subgroups act trivial-weight-freely on the Euclidean quotients of the closed derived series' successive layers. In particular, an extension of a compact connected group $\mathbb{K}$ by a connected, simply-connected solvable Lie group $\mathbb{L}$ is (openly) almost-elliptic precisely when the weights of the $\mathbb{K}$-action on $Lie(\mathbb{L})$ afforded by the extension are all non-trivial.

math.GR↗

arXiv2025

Generically-constrained quantum isotropy

Let $V$ be a finite-dimensional unitary representation of a compact quantum group $\mathrm{G}$ and denote by $\mathrm{G}_W$ the isotropy subgroup of a linear subspace $W\le V$ regarded as a point in the Grassmannian $\mathbb{G}(V)$. We show that the space of those $W\in \mathbb{G}(V)$ for which $\mathrm{G}_W$ acts trivially on $W$ (or $V$) is open in the Zariski topology of the Weil restriction $\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}(V)$. More generally, this holds for the space of $W$ for which (a) the $\mathrm{G}_W$-action factors through its abelianization, or (b) the summands of the $\mathrm{G}_W$-representation on $W$ (or $V$) are otherwise dimensionally constrained. The results generalize analogous classical generic rigidity statements useful in establishing the triviality of the classical automorphism groups of random quantum graphs in the matrix algebra $M_n$, and can be put to similar use in fully non-commutative versions of those results (quantum graphs, quantum groups).

math.QA↗

arXiv2025

On the singularities of the exponential function of a semidirect product

We show that the Fréchet--Lie groups of the form $C^{\infty}(M)\rtimes \mathbb{R}$ resulting from smooth flows on compact manifolds $M$ fail to be locally exponential in several cases: when at least one non-periodic orbit is locally closed, or when the flow restricts to a linear one on an orbit closure diffeomorphic to a torus. As an application, we prove that the Bondi--Metzner--Sachs group of symmetries of an asymptotically flat spacetime is not locally exponential.

math.GR↗

arXiv2025

Fixed-point permanence under actions by finite quantum groups

Given an action by a finite quantum group $\mathbb{G}$ on a von Neumann algebra $M$, we prove that a number of familiar $W^*$ properties are equivalent for $M$ and the fixed-point algebra $M^{\mathbb{G}}$ (i.e. hold or not simultaneously for the two algebras); these include being hyperfinite, atomic, diffuse and of type $I$, $II$ or $III$. Moreover, in all cases the canonical central projections of $M$ and $M^{\mathbb{G}}$ cutting out the summand with the respective property coincide. The result generalizes its classical-$\mathbb{G}$ analogue due to Jones-Takesaki.

math.OA↗

arXiv2025

Spans of quantum-inequality projections

A hereditarily atomic von Neumann algebra $A$ is a $W^*$ product of matrix algebras, regarded as the underlying function algebra of a quantum set. Projections in $A\overline{\otimes}A^{\circ}$ are interpreted as quantum binary relations on $A$, with the supremum of all $p\otimes (1-p)$ representing quantum inequality. We prove that the symmetrized weak$^*$-closed linear span of all such quantum-inequality projections is precisely the symmetric summand of the joint kernel of multiplication and opposite multiplication, a result valid without the symmetrization qualification for plain matrix algebras. The proof exploits the symmetries of the spaces involved under the compact unitary group of $A$, and related results include a classification of those von Neumann algebras (hereditarily atomic or not) for which the unitary group operates jointly continuously with respect to the weak$^*$ topology.

math.OA↗

arXiv2025

Continuous spectrum-shrinking maps and applications to preserver problems

For a positive integer $n$ let $\mathcal{X}_n$ be either the algebra $M_n$ of $n \times n$ complex matrices, the set $N_n$ of all $n \times n$ normal matrices, or any of the matrix Lie groups $\mathrm{GL}(n)$, $\mathrm{SL}(n)$ and $\mathrm{U}(n)$. We first give a short and elementary argument that for two positive integers $m$ and $n$ there exists a continuous spectrum-shrinking map $ϕ: \mathcal{X}_n \to M_m$ (i.e.\ $\mathrm{sp}(ϕ(X))\subseteq \mathrm{sp}(X)$ for all $X \in \mathcal{X}_n$) if and only if $n$ divides $m$. Moreover, in that case we have the equality of characteristic polynomials $k_{ϕ(X)}(\cdot) = k_{X}(\cdot)^\frac{m}{n}$ for all $X \in \mathcal{X}_n$, which in particular shows that $ϕ$ preserves spectra. Using this we show that whenever $n \geq 3$, any continuous commutativity preserving and spectrum-shrinking map $ϕ: \mathcal{X}_n \to M_n$ is of the form $ϕ(\cdot)=T(\cdot)T^{-1}$ or $ϕ(\cdot)=T(\cdot)^tT^{-1}$, for some $T\in \mathrm{GL}(n)$. The analogous results fail for the special unitary group $\mathrm{SU}(n)$ but hold for the spaces of semisimple elements in either $\mathrm{GL}(n)$ or $\mathrm{SL}(n)$. As a consequence, we also recover (a strengthened version of) Šemrl's influential characterization of Jordan automorphisms of $M_n$ via preserving properties.

math.SP↗

arXiv2025

Eigenvalue selectors for representations of compact connected groups

A representation $ρ$ of a compact group $\mathbb{G}$ selects eigenvalues if there is a continuous circle-valued map on $\mathbb{G}$ assigning an eigenvalue of $ρ(g)$ to every $g\in \mathbb{G}$. For every compact connected $\mathbb{G}$, we characterize the irreducible $\mathbb{G}$-representations which select eigenvalues as precisely those annihilating the intersection $Z_0(\mathbb{G})\cap \mathbb{G}'$ of the connected center of $\mathbb{G}$ with its derived subgroup. The result applies more generally to finite-spectrum representations isotypic on $Z_0(\mathbb{G})$, and recovers as applications (noted in prior work) the existence of a continuous eigenvalue selector for the natural representation of $\mathrm{SU}(n)$ and the non-existence of such a selector for $\mathrm{U}(n)$.

math.RT↗

arXiv2025

(Quantum) discreteness, spectrum compactness and uniform continuity

We prove a number of results linking properties of actions by compact groups (both quantum and classical) on Banach spaces, such as uniform continuity, spectrum finiteness and extensibility of the actions across several constructions. Examples include: (a) a unitary representation of a compact quantum group induces a continuous action on the $C^*$-algebra of bounded operators if and only if it has finitely many isotypic components, and hence is uniformly continuous; (b) a compact quantum group is finite if and only if its continuous actions on $C^*$-algebras lift to continuous actions on either the multiplier algebras or von Neumann envelopes thereof; (c) a (classical) locally compact group $\mathbb{G}$ is discrete if and only if the forgetful functor from $\mathbb{G}$-acted-upon compact $T_2$ spaces back to compact $T_2$ spaces creates coproducts; (d) a representation of a linearly reductive quantum group has finitely many isotypic components if and only if its restrictions to two topologically-generating quantum subgroups, one of which is normal, do; (e) equivalent characterizations of uniform continuity for actions of compact groups on Banach spaces, e.g. that such an action is uniformly continuous if and only if its restrictions to a pro-torus and to pro-$p$ subgroups are.

math.OA↗

arXiv2025

Quantum group coproducts and universality under scalar extensions

We characterize the families of bialgebras or Hopf algebras over fields for which the product in the corresponding category is finite-dimensional, answering a question of M. Lorenz: if the ground field is infinite then bialgebra or Hopf products are finite-dimensional precisely when the factors are, with at most one of dimension $>1$; over finite fields the necessary and sufficient condition is instead that factors be finite-dimensional with at most finitely many of dimension $>1$; finally, these statements hold for coalgebras as well, provided the family is finite. We also characterize (a) finite field extensions as precisely those whose underlying scalar extension functor preserves coalgebra, or bialgebra, or Hopf algebra products (correcting an error in the literature); (b) algebraic field extensions as those along which finite coalgebra (bialgebra, Hopf algebra) products are preserved; and (c) again algebraic field extensions as precisely those which intertwine cofree coalgebras on vector spaces, or cofree bialgebras (Hopf algebras) on algebras.

math.QA↗

arXiv2024

Single and multi-valued Hilbert-bundle renormings

We prove that subhomogeneous continuous Banach bundles over compact metrizable spaces are equivalent to Hilbert bundles, while examples show that the metrizability assumption cannot be dropped completely. This complements the parallel statement for homogeneous bundles without the metrizability assumption, and generalizes the analogous result to the effect that subhomogeneous $C^*$ bundles over compact metrizable spaces admit finite-index expectations.

math.FA↗

arXiv2024

Proper actions and supported-section-valued cohomology

Consider a proper action of $\mathbb{Z}^d$ on a smooth (perhaps non-paracompact) manifold $M$. The $p^{th}$ cohomology $H^p(\mathbb{Z}^d,\ Γ_{\mathrm{c}}(\mathcal{F}))$ valued in the space of compactly-supported sections of a natural sheaf $\mathcal{F}$ on $M$ (such as those of smooth function germs, smooth $k$-form germs, etc.) vanishes for $p\ne d$ (the cohomological dimension of $\mathbb{Z}^d$) and, at $d$, equals the space of compactly-supported sections of the descent ($\mathbb{G}$-invariant push-forward) $\mathcal{F}/\mathbb{Z}^d$ to the orbifold quotient $M/\mathbb{Z}^d$. We prove this and analogous results on $\mathbb{Z}^d$ cohomology valued in $Φ$-supported sections of an equivariant appropriately soft sheaf $\mathcal{F}$ in a broader context of $\mathbb{Z}^d$-actions proper with respect to a paracompactifying family of supports $Φ$, in the sense that every member of $Φ$ has a neighborhood small with respect to the action in Palais' sense.

math.GR↗

arXiv2024

Deformations of nearby subgroups and approximate Jordan constants

Let $\mathbb{U}$ be a Banach Lie group and $S\subseteq \mathbb{U}$ an ad-bounded subset thereof, in the sense that there is a uniform bound on the adjoint operators induced by elements of $S$ on the Lie algebra of $\mathbb{U}$. We prove that (1) $S$-valued continuous maps from compact groups to $\mathbb{U}$ sufficiently close to being morphisms are uniformly close to morphisms; and (2) for any Lie subgroup $\mathbb{G}\le \mathbb{U}$ there is an identity neighborhood $U\ni 1\in \mathbb{U}$ so that $\mathbb{G}\cdot U\cap S$-valued morphisms (embeddings) from compact groups into $\mathbb{U}$ are close to morphisms (respectively embeddings) into $\mathbb{G}$. This recovers and generalizes results of Turing's to the effect that (a) Lie groups arbitrarily approximable by finite subgroups have abelian identity component and (b) if a Lie group is approximable in this fashion and has a faithful $d$-dimensional representation then it is also so approximable by finite groups with the same property. Another consequence is a strengthening of a prior result stating that finite subgroups in a Banach Lie group sufficiently close to a given compact subgroup thereof admit a finite upper bound on the smallest indices of their normal abelian subgroups (an approximate version of Jordan's theorem on finite subgroups of linear groups).

math.GR↗