arXiv subjects
Publications and source records attributed to Alexandru Chirvasitu.
The characteristic index of a locally compact connected group $G$ is the non-negative integer $d$ for which we have a homeomorphism $G\cong K\times \mathbb{R}^d$ with $K\le G$ maximal compact. We prove that the characteristic indices of closed connected subgroups are dominated by those of the ambient groups.
We prove that a family of at least two non-trivial, almost-connected locally compact groups cannot have a coproduct in the category of locally compact groups if at least one of the groups is connected; this confirms the intuition that coproducts in said category are rather hard to come by, save for the usual ones in the category of discrete groups. Along the way we also prove a number of auxiliary results on characteristic indices of locally compact or Lie groups as defined by Iwasawa: that characteristic indices can only decrease when passing to semisimple closed Lie subgroups, and also along dense-image morphisms.
We prove that for a connected, semisimple linear Lie group $G$ the spaces of generating pairs of elements or subgroups are well-behaved in a number of ways: the set of pairs of elements generating a dense subgroup is Zariski-open in the compact case, Euclidean-open in general, and always dense. Similarly, for sufficiently generic circle subgroups $H_i$, $i=1,2$ of $G$, the space of conjugates of $H_i$ that generate a dense subgroup is always Zariski-open and dense. Similar statements hold for pairs of Lie subalgebras of the Lie algebra $Lie(G)$.
Consider a diagram $\cdots \to F_3 \to F_2\to F_1$ of algebraic systems, where $F_n$ denotes the free object on $n$ generators and the connecting maps send the extra generator to some distinguished trivial element. We prove that (a) if the $F_i$ are free associative algebras over a fixed field then the limit in the category of graded algebras is again free on a set of homogeneous generators; (b) on the other hand, the limit in the category of associative (ungraded) algebras is a free formal power series algebra on a set of homogeneous elements, and (c) if the $F_i$ are free Lie algebras then the limit in the category of graded Lie algebras is again free.
The core of a finite-dimensional modular representation $M$ of a finite group $G$ is its largest non-projective summand. We prove that the dimensions of the cores of $M^{\otimes n}$ have algebraic Hilbert series when $M$ is Omega-algebraic, in the sense that the non-projective summands of $M^{\otimes n}$ fall into finitely many orbits under the action of the syzygy operator $Ω$. Similarly, we prove that these dimension sequences are eventually linearly recursive when $M$ is what we term $Ω^{+}$-algebraic. This partially answers a conjecture by Benson and Symonds. Along the way, we also prove a number of auxiliary permanence results for linear recurrence under operations on multi-variable sequences.
Given a smooth free action of a compact connected Lie group $G$ on a smooth compact manifold $M$, we show that the space of $G$-invariant Riemannian metrics on $M$ whose automorphism group is precisely $G$ is open dense in the space of all $G$-invariant metrics, provided the dimension of $M$ is "sufficiently large" compared to that of $G$. As a consequence, it follows that every compact connected Lie group can be realized as the automorphism group of some compact connected Riemannian manifold; this recovers prior work by Bedford-Dadok and Saerens-Zame under less stringent dimension conditions. Along the way we also show, under less restrictive conditions on both dimensions and actions, that the space of $G$-invariant metrics whose automorphism groups preserve the $G$-orbits is dense $G_δ$ in the space of all $G$-invariant metrics.
Let $k$ be a field. We show that locally presentable, $k$-linear categories $\mathcal{C}$ dualizable in the sense that the identity functor can be recovered as $\coprod_i x_i\otimes f_i$ for objects $x_i\in \mathcal{C}$ and left adjoints $f_i$ from $\mathcal{C}$ to $\mathrm{Vect}_k$ are products of copies of $\mathrm{Vect}_k$. This partially confirms a conjecture by Brandenburg, the author and T. Johnson-Freyd. Motivated by this, we also characterize the Grothendieck categories containing an object $x$ with the property that every object is a copower of $x$: they are precisely the categories of non-singular injective right modules over simple, regular, right self-injective rings of type I or III.
We prove a number of property (T) permanence results for locally compact quantum groups under exact sequences and the presence of invariant states, analogous to their classical versions. Along the way we characterize the existence of invariant weights on quantum homogeneous spaces of quotient type, and relate invariant states for LCQG actions on von Neumann algebras to invariant vectors in canonical unitary implementations, providing an application to amenability. Finally, we introduce a notion of lattice in a locally compact quantum group, noting examples provided by Drinfeld doubles of compact quantum groups. We show that property (T) lifts from a lattice to the ambient LCQG, just as it does classically, thus obtaining new examples of non-classical, non-compact, non-discrete LCQGs with property (T).
We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of points equidistant from pairs of points preceding them in the sequence. All of this provides evidence that Riemannian metric spaces admit what we term loose embeddings into finite-dimensional Euclidean spaces: continuous maps that preserve both equality as well as inequality. We also prove a local-to-global principle for Riemannian-metric-space loose embeddability: if every finite subspace thereof is loosely embeddable into a common $\mathbb{R}^N$, then the metric space as a whole is loosely embeddable into $\mathbb{R}^N$ in a weakened sense.
We show that countable metric spaces always have quantum isometry groups, thus extending the class of metric spaces known to possess such universal quantum-group actions. Motivated by this existence problem we define and study the notion of loose embeddability of a metric space $(X,d_X)$ into another, $(Y,d_Y)$: the existence of an injective continuous map that preserves both equalities and inequalities of distances. We show that $0$-dimensional compact metric spaces are "generically" loosely embeddable into the real line, even though not even all countable metric spaces are.
We explore the recently introduced local-triviality dimensions by studying gauge actions on graph $C^*$-algebras, as well as the restrictions of the gauge action to finite cyclic subgroups. For $C^*$-algebras of finite acyclic graphs and finite cycles, we characterize the finiteness of these dimensions, and we further study the gauge actions on many examples of graph $C^*$-algebras. These include the Toeplitz algebra, Cuntz algebras, and $q$-deformed spheres.
We prove that pushouts $A*_CB$ of residually finite-dimensional (RFD) $C^*$-algebras over central subalgebras are always residually finite-dimensional provided the fibers $A_p$ and $B_p$, $p\in \mathrm{spec}~C$ are RFD, recovering and generalizing results by Korchagin and Courtney-Shulman. This then allows us to prove that certain central pushouts of amenable groups have RFD group $C^*$-algebras. Along the way, we discuss the problem of when, given a central group embedding $H\le G$, the resulting $C^*$-algebra morphism is a continuous field: this is always the case for amenable $G$ but not in general.
We prove that (a) discrete compact quantum groups (or more generally locally compact, under additional hypotheses) with coamenable dual are continuous fields over their central closed quantum subgroups, and (b) the same holds for free products of discrete quantum groups with coamenable dual amalgamated over a common central subgroup. Along the way we also show that free products of continuous fields of $C^*$-algebras are again free via a Fell-topology characterization for $C^*$-field continuity, recovering a result of Blanchard's in a somewhat more general setting.
This is a survey on the transitive quantum groups $G\subset S_N^+$, and on the flat matrix models $π:C(G)\to M_N(C(X))$ for the corresponding Hopf algebras. We review the known results on the subject, with a number of improvements, technical generalizations, and some new results as well, and we discuss the open problems.
We study the graph isomorphism game that arises in quantum information theory from the perspective of bigalois extensions of compact quantum groups. We show that every algebraic quantum isomorphism between a pair of (quantum) graphs $X$ and $Y$ arises as a quotient of a certain measured bigalois extension for the quantum automorphism groups $G_X$ and $G_Y$ of the graphs $X$ and $Y$. In particular, this implies that the quantum groups $G_X$ and $G_Y$ are monoidally equivalent. We also establish a converse to this result, which says that every compact quantum group $G$ monoidally equivalent to $G_X$ is of the form $G_Y$ for a suitably chosen quantum graph $Y$ that is quantum isomorphic to $X$. As an application of these results, we deduce that the $\ast$-algebraic, C$^\ast$-algebraic, and quantum commuting (qc) notions of a quantum isomorphism between classical graphs $X$ and $Y$ all coincide. Using the notion of equivalence for non-local games, we deduce the same result for other synchronous non-local games, including the synBCS game and certain related graph homomorphism games.
Let $V_*\otimes V\rightarrow\mathbb{C}$ be a non-degenerate pairing of countable-dimensional complex vector spaces $V$ and $V_*$. The Mackey Lie algebra $\mathfrak{g}=\mathfrak{gl}^M(V,V_*)$ corresponding to this paring consists of all endomorphisms $φ$ of $V$ for which the space $V_*$ is stable under the dual endomorphism $φ^*: V^*\rightarrow V^*$. We study the tensor Grothendieck category $\mathbb{T}$ generated by the $\mathfrak{g}$-modules $V$, $V_*$ and their algebraic duals $V^*$ and $V^*_*$. This is an analogue of categories considered in prior literature, the main difference being that the trivial module $\mathbb{C}$ is no longer injective in $\mathbb{T}$. We describe the injective hull $I$ of $\mathbb{C}$ in $\mathbb{T}$, and show that the category $\mathbb{T}$ is Koszul. In addition, we prove that $I$ is endowed with a natural structure of commutative algebra. We then define another category $_I\mathbb{T}$ of objects in $\mathbb{T}$ which are free as $I$-modules. Our main result is that the category ${}_I\mathbb{T}$ is also Koszul, and moreover that ${}_I\mathbb{T}$ is universal among abelian $\mathbb{C}$-linear tensor categories generated by two objects $X$, $Y$ with fixed subobjects $X'\hookrightarrow X$, $Y'\hookrightarrow Y$ and a pairing $X\otimes Y\rightarrow \text{\textbf{1}}$ where \textbf{1} is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $\mathbb{T}$ and ${}_I\mathbb{T}$.
We prove a number of results having to do with equipping type-I $\mathrm{C}^*$-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the $\mathrm{C}^*$-algebra in question is an extension of a non-zero finite direct sum of elementary $\mathrm{C}^*$-algebras by a commutative unital $\mathrm{C}^*$-algebra then it must be finite-dimensional.
We prove that irreducible symmetric domains are uniquely determined by the homotopy equivalence classes of their Shilov boundaries.