The chain group $C(G)$ of a locally compact group $G$ has one generator $g_{rho}$ for each irreducible unitary $G$-representation $rho$, a relation $g_{rho}=g_{rho}g_{rho}$ whenever $rho$ is weakly contained in $rhootimes rho$, and $g_{rho^*}=g_{rho}
^{-1}$ for the representation $rho^*$ contragredient to $rho$. $G$ satisfies chain-center duality if assigning to each $g_{rho}$ the central character of $rho$ is an isomorphism of $C(G)$ onto the dual $widehat{Z(G)}$ of the center of $G$. We prove that $G$ satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. M{u}gers result compact groups satisfy chain-center duality.
The characteristic index of a locally compact connected group $G$ is the non-negative integer $d$ for which we have a homeomorphism $Gcong Ktimes mathbb{R}^d$ with $Kle G$ 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 s
aid 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 $cdots to F_3 to F_2to F_1$ 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 sen
se that the non-projective summands of $M^{otimes n}$ fall into finitely many orbits under the action of the syzygy operator $Omega$. Similarly, we prove that these dimension sequences are eventually linearly recursive when $M$ is what we term $Omega^{+}$-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.
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 $coprod_i x_iotimes f_i$ for objects $x_iin mathcal{C}$ and left adjoints $f_i$ from $mathc
al{C}$ 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.
For a finite-index $mathrm{II}_1$ subfactor $N subset M$, we prove the existence of a universal Hopf $ast$-algebra (or, a discrete quantum group in the analytic language) acting on $M$ in a trace-preserving fashion and fixing $N$ pointwise. We call t
his Hopf $ast$-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse.
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 equidist
ant 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 prove a number of results to the effect that generic quantum graphs (defined via operator systems as in the work of Duan-Severini-Winter / Weaver) have few symmetries: for a Zariski-dense open set of tuples $(X_1,cdots,X_d)$ of traceless self-adjo
int operators in the $ntimes n$ matrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters: $2le dle n^2-3$. Moreover, the automorphism group is generically abelian in the larger parameter range $1le dle n^2-2$. This then implies that for those respective parameters the corresponding random-quantum-graph model built on the GUE ensembles of $X_i$s (mimicking the ErdH{o}s-R{e}nyi $G(n,p)$ model) has trivial/abelian automorphism group almost surely.
