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We prove that there exist Hopf algebras with surjective, non-bijective antipode which admit no non-trivial morphisms from Hopf algebras with bijective antipode; in particular, they are not quotients of such. This answers a question left open in prior work, and contrasts with the dual setup whereby a Hopf algebra has injective antipode precisely when it embeds into one with bijective antipode. The examples rely on the broader phenomenon of realizing pre-specified subspace lattices as comodule lattices: for a finite-dimensional vector space $V$ and a sequence $(\mathcal{L}_r)_r$ of successively finer lattices of subspaces thereof, assuming the minimal subquotients of the supremum $\bigvee_r \mathcal{L}_r$ are all at least 2-dimensional, there is a Hopf algebra equipping $V$ with a comodule structure in such a fashion that the lattice of comodules of the $r^{th}$ dual comodule $V^{r*}$ is precisely the given $\mathcal{L}_r$., Comment: 13 pages + references

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., Comment: v2 adds Theorem 2.13 and makes attendant changes; 24 pages + references

We prove a number of results to the general effect that, under obviously necessary numerical and determinant constraints, "most" morphisms between fixed bundles on a complex elliptic curve produce (co)kernels which can either be specified beforehand or else meet various rigidity constraints. Examples include: (a) for indecomposable $\mathcal{E}$ and $\mathcal{E'}$ with slopes and ranks increasing strictly in that order the space of monomorphisms whose cokernel is semistable and maximally rigid (i.e. has minimal-dimensional automorphism group) is open dense; (b) for indecomposable $\mathcal{K}$, $\mathcal{E}$ and stable $\mathcal{F}$ with slopes increasing strictly in that order and ranks and determinants satisfying the obvious additivity constraints the space of embeddings $\mathcal{K}\to \mathcal{E}$ whose cokernel is isomorphic to $\mathcal{F}$ is open dense; (c) the obvious mirror images of these results; (d) generalizations weakening indecomposability to semistability + maximal rigidity; (e) various examples illustrating the necessity of the assorted assumptions., Comment: 29 pages + references

We show that a continuously-normed Banach bundle $\mathcal{E}$ over a compact Hausdorff space $X$ whose space of sections is algebraically finitely-generated (f.g.) over $C(X)$ is locally trivial (and hence the section space is projective f.g over $C(X)$); this answers a question of I. Gogi\'c. As a preliminary we also provide sufficient conditions for a quotient bundle to be continuous phrased in terms of the Vietoris continuity of the unit-ball maps attached to the bundles. Related results include (a) the fact that the category of topologically f.g. continuous Banach bundles over $X$ is symmetric monoidal under the (fiber-wise-maximal) tensor product, (b) the full faithfulness of the global-section functor from topologically f.g. continuous bundles to $C(X)$-modules and (c) the consequent identification of the algebraically f.g. bundles as precisely the rigid objects in the aforementioned symmetric monoidal category., Comment: 22 pages + references; v2 corrects typos and makes small style changes

We characterize those (continuously-normed) Banach bundles $\mathcal{E}\to X$ with compact Hausdorff base whose spaces $\Gamma(\mathcal{E})$ of global continuous sections are topologically finitely-generated over the function algebra $C(X)$, answering a question of I. Gogi\'c's and extending analogous work for metrizable $X$. Conditions equivalent to topological finite generation include: (a) the requirement that $\mathcal{E}$ be locally trivial and of finite type along locally closed and relatively $F_{\sigma}$ strata in a finite stratification of $X$; (b) the decomposability of arbitrary elements in $\ell^p(\Gamma(\mathcal{E}))$, $1\le p<\infty$ as sums of $\le N$ products in $\ell^p(C(X))\cdot \Gamma(\mathcal{E})$ for some fixed $N$; (c) the analogous decomposability requirement for maximal Banach-module tensor products $F\widehat{\otimes}_{C(X)}\Gamma(\mathcal{E})$ or (d) equivalently, only for $F=\ell^1(C(X))$., Comment: v2 corrects minor typos; 12 pages + references

A slight extension of a construction due to Calabi-Rosenlicht (and later Gabard, Baillif and others) produces a typically non-metrizable $n$-manifold $\mathbb{P}$ by gluing two copies of the open upper half-space $\mathbb{H}_{++}$ in $\mathbb{R}^n$ along the disjoint union of the spaces of rays within $\mathbb{H}_{++}$ originating at points ranging over a subset $S\subseteq \mathbb{R}^{n-1}$ of the boundary $\mathbb{R}^{n-1}=\partial\overline{\mathbb{H}_{++}}$. The fundamental group $\pi_1(\mathbb{P})$ is free on the complement $S^{\times}$ of any singleton in $S\ne\emptyset$, and the main result below is that the first cohomotopy group $\pi^1(\mathbb{P})$, regarded as a space of functions $S^{\times}\to \mathbb{Z}$, is precisely the additive group of integer-valued Baire-1 functions on $S^{\times}$. This occasions a detour on characterizations (perhaps of independent interest) of Baire-1 real-valued functions on a metric space $(B,d)$ as various types of non-tangential boundary limits of continuous functions on $B\times \mathbb{R}_{>0}$., Comment: 12 pages + references

We prove a number of results of the following common flavor: for a category $\mathcal{C}$ of topological or uniform spaces with all manner of other properties of common interest (separation / completeness / compactness axioms), a group (or monoid) $\mathbb{G}$ equipped with various types of topological structure (topologies, uniformities) and the corresponding category $\mathcal{C}^{\mathbb{G}}$ of appropriately compatible $\mathbb{G}$-flows in $\mathcal{C}$, the forgetful functor $\mathcal{C}^{\mathbb{G}}\to \mathcal{C}$ is monadic. In all cases of interest the domain category $\mathcal{C}^{\mathbb{G}}$ is also cocomplete, so that results on adjunction lifts along monadic functors apply to provide equivariant completion and/or compactification functors. This recovers, unifies and generalizes a number of such results in the literature due to de Vries, Mart'yanov and others on existence of equivariant compactifications / completions and cocompleteness of flow categories., Comment: 15 pages + references

We study the local-triviality dimensions of actions on $C^*$-algebras, which are invariants developed for noncommutative Borsuk-Ulam theory. While finiteness of the local-triviality dimensions is known to guarantee freeness of an action, we show that free actions need not have finite weak local-triviality dimension. Moreover, the local-triviality dimensions of a continuous field may be greater than those of its individual fibers, and the dimensions may fail to vary continuously across the fibers. However, in certain circumstances upper semicontinuity of the weak local-triviality dimension is guaranteed. We examine these results and counterexamples with a focus on noncommutative tori and noncommutative spheres, both in terms of computation and theory., Comment: 21 pages + references

For any square-summable commuting family $(A_i)_{i\in I}$ of complex $n\times n$ matrices there is a normal commuting family $(B_i)_i$ no farther from it, in squared normalized $\ell^2$ distance, than the diameter of the numerical range of $\sum_i A_i^* A_i$. Specializing in one direction (limiting case of the inequality for finite $I$) this recovers a result of M. Fraas: if $\sum_{i=1}^{\ell} A_i^* A_i$ is scalar for commuting $A_i\in M_n(\mathbb{C})$ then the $A_i$ are normal; specializing in another (singleton $I$) retrieves the well-known fact that close-to-isometric matrices are close to isometries., Comment: 6 pages + references

Let $\mathbb{G}$ be a compact Hausdorff group acting on a compact Hausdorff space $X$, $\alpha$ an irreducible $\mathbb{G}$-representation, and $C(X)$ the $C^*$-algebra of complex-valued continuous functions on $X$. We prove that the isotypic component $C(X)_{\alpha}$ is finitely generated as a module over the invariant subalgebra $C(X/\mathbb{G})\subseteq C(X)$ precisely when the map sending $x\in X$ to the dimension of the space of vectors in $\alpha$ invariant under the isotropy group $\mathbb{G}_x$ is locally constant. This (a) specializes back to an observation of De Commer-Yamashita equating the finite generation of all $C(X)_{\alpha}$ with the Vietoris continuity of $x\mapsto \mathbb{G}_x$, and (b) recovers and extends Watatani's examples of infinite-index expectations resulting from non-free finite-group actions. We also show that the action of a compact group $\mathbb{G}$ on the maximal equivariant compactification on the disjoint union of its Lie-group quotients has tubes about all orbits precisely when $\mathbb{G}$ is Lie. This is the converse (via a canonical construction) of the well-known fact that actions of compact Lie groups on Tychonoff spaces admit tubes., Comment: 10 pages + references

For a compact group $\mathbb{G}$ acting continuously on a Banach Lie group $\mathbb{U}$, we prove that maps $\mathbb{G}\to \mathbb{U}$ close to being 1-cocycles for the action can be deformed analytically into actual 1-cocycles. This recovers Hyers-Ulam stability results of Grove-Karcher-Ruh (trivial $\mathbb{G}$-action, compact Lie $\mathbb{G}$ and $\mathbb{U}$) and de la Harpe-Karoubi (trivial $\mathbb{G}$-action, $\mathbb{U}:=$invertible elements of a Banach algebra). The obvious analogues for higher cocycles also hold for abelian $\mathbb{U}$., Comment: v2 makes small corrections, supplying a missing hypothesis in the main result; 5 pages + references

Fix a stable degree-$n$ rank-$k$ bundle $\mathcal{F}$ on a complex elliptic curve for (coprime) $1\le k

We (a) prove that continuous morphisms from locally compact groups to locally exponential (possibly infinite-dimensional) Lie groups factor through Lie quotients, recovering a result of Shtern's on factoring norm-continuous representations on Banach spaces; (b) characterize the maximal almost-periodicity of the identity component $\mathbb{G}_0\le \mathbb{G}$ of a locally compact group in terms of sufficiently discriminating families of continuous functions on $\mathbb{G}$ valued in Hausdorff spaces generalizing an analogous result by Kadison-Singer; (c) apply that characterization to recover the von Neumann kernel of $\mathbb{G}_0$ as the joint kernel of all appropriately bounded and continuous $\mathbb{G}$-representations on topological vector spaces extending Kallman's parallel statement for unitary representations, and (d) provide large classes of complete locally convex topological vector spaces (e.g. arbitrary products of Fr\'echet spaces) with the property that compact-group representations thereon whose vectors all have finite-dimensional orbits decompose as finite sums of isotypic components. This last result specializes to one of Hofmann-Morris on representations on products of real lines., Comment: 20 pages + references

For a compact group $\mathbb{G}$, the functor from unital Banach algebras with contractive morphisms to metric spaces with 1-Lipschitz maps sending a Banach algebra $A$ to the space of $\mathbb{G}$-representations in $A$ preserves filtered colimits. Along with this, we prove a number of analogues: one can substitute unitary representations in $C^*$-algebras, as well as semisimple finite-dimensional Banach algebras (or finite-dimensional $C^*$-algebras) for $\mathbb{G}$. These all mimic results on the metric-enriched finite generation/presentability of finite-dimensional Banach spaces due to Ad{\'a}mek and Rosick{\'y}. We also give an alternative proof of that finite presentability result, along with parallel results on functors represented by compact metric, metric convex, or metric absolutely convex spaces., Comment: 15 pages + references

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., Comment: 20 pages + references

We show that the endomorphisms of a compact connected group that extend to endomorphisms of every compact overgroup are precisely the trivial one and the inner automorphisms; this is an analogue, for compact connected groups, of results due to Schupp and Pettet on discrete groups (plain or finite). A somewhat more surprising result is that if $\mathbb{A}$ is compact connected and abelian, its endomorphisms extensible along morphisms into compact connected groups also include $-\mathrm{id}$ (in addition to the obvious trivial endomorphism and the identity). Connectedness cannot be dropped on either side in this last statement., Comment: 14 pages + references; v2 fixes a typo (resulting in a misstatement)

Let $\mathbb{A}$ be a Dedekind domain and $T$ an endomorphism of a finitely-generated projective $\mathbb{A}$-module. If $T$ is an $s^{th}$ power in $\mathrm{End}_{\mathbb{A}}(M)$ for $s$ ranging over an infinite set $\mathcal{S}$ of positive integers, then (a) $T$ decomposes as a direct sum of the zero operator and an invertible operator on a summand of $M$ and (b) that summand is semisimple or of finite order if $\mathcal{S}$ is appropriately large (what this means depends on the structure of the additive and multiplicative groups of $\mathbb{A}$). This generalizes a result of M. Cavachi's to the effect that the only non-singular integer matrix that is an $s^{th}$ power in $M_n(\mathbb{Z})$ for all $s$ is the identity., Comment: 9 pages + references

Let $\mathcal{E}$ be a rank-2 vector bundle over an elliptic curve $E$, decomposable as a sum of line bundles of degrees $d'>d\ge 2$, and $\mathcal{L}$ the determinant of $\mathcal{E}$. The subspace $L(\mathcal{E})\subset \mathbb{P}^{n-1}\cong \mathbb{P}\mathrm{Ext}^1(\mathcal{L},\mathcal{O}_E)$ consisting of classes of extensions with middle term isomorphic to $\mathcal{E}$ is one of the symplectic leaves of a remarkable Poisson structure on $\mathbb{P}^{n-1}$ defined by Feigin-Odesskii/Polishchuk, and all symplectic leaves arise in this manner, as shown in earlier work that realizes $L(\mathcal{E})$ as the base space of a principal $\mathrm{Aut}(\mathcal{E})$-fibration. Here, we embed $L(\mathcal{E})$ into a larger, projective base space $\widetilde{L}(\mathcal{E})$ of a principal $\mathrm{Aut}(\mathcal{E})$-fibration whose total space consists of sections of $\mathcal{E}$. The embedding realizes $L(\mathcal{E})\subset \widetilde{L}(\mathcal{E})$ as a complement of an anticanonical divisor $Y$ (one of the main results), and we give an explicit description of the normalization of $Y$ as a projective-space bundle over a projective space. For $d=2$ $\widetilde{L}(\mathcal{E})$ is one of the three Hirzebruch surfaces $\Sigma_i$, $i=0,1,2$; we determine which occurs when and hence also the cases when $L(\mathcal{E})$ is affine. Separately, we prove that for $d<\frac n2$ the singular locus of the secant slice $\mathrm{Sec}_{d,z}(E)\subset \mathbb{P}^{n-1}$, the portion of the $d^{th}$ secant variety of $E$ consisting of points lying on spans of $d$-tuples with sum $z\in E$, is precisely $\mathrm{Sec}_{d-2}$. This strengthens result that $L(\mathcal{E})$ is smooth, appearing in prior joint work with R. Kanda and S.P. Smith., Comment: 32 pages + references

The bounded localization $\beta_b$ of a locally convex topology $\beta$ is defined as the finest locally convex topology agreeing with $\beta$ on all bounded sets. We show that the strict topology on the multiplier algebra of a bornological pro-$C^*$-algebras equals its own localization, generalizing the analogous result due to Taylor for multiplier algebras of plain $C^*$-algebras. We also (a) characterize the barreled commutative unital pro-$C^*$-algebras as those of continuous functions on functionally Hausdorff spaces whose relatively pseudocompact subsets are relatively compact, equipped with the topology of uniform convergence on compact subsets, and (b) describe a contravariant equivalence between the category of commutative unital pro-$C^*$-algebras and a category of Tychonoff (rather than functionally Hausdorff) topological spaces., Comment: 22 pages + references

Each symmetrically-normed ideal $\mathcal{I}$ of compact operators on a Hilbert space $H$ induces a multiplier topology $\mu^*_{\mathcal{I}}$ on the algebra $\mathcal{B}(H)$ of bounded operators. We show that under fairly reasonable circumstances those topologies precisely reflect, strength-wise, the inclusion relations between the corresponding ideals, including the fact that the topologies are distinct when the ideals are. Said circumstances apply, for instance, for the two-parameter chain of Lorentz ideals $\mathcal{L}^{p,q}$ interpolating between the ideals of trace-class and compact operators. This gives a totally ordered chain of distinct topologies $\mu^*_{p,q\mid 0}$ on $\mathcal{B}(H)$, with $\mu^*_{2,2\mid 0}$ being the $\sigma$-strong$^*$ topology and $\mu^*_{\infty,\infty\mid 0}$ the strict/Mackey topology. In particular, the latter are only two of a natural continuous family., Comment: 21 pages + references

For a completely Hausdorff quasi-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., Comment: 33 pages + references

Let $(0, a_1, \ldots, a_{d-1})_n$ denote the function $f_n(x_0, x_1, \ldots, x_{n-1})$ of degree $d$ in $n$ variables generated by the monomial $x_0x_{a_1} \cdots x_{a_{d-1}}$ and having the property that $f_n$ is invariant under cyclic permutations of the variables. Such a function $f_n$ is called monomial rotation symmetric (MRS). Much of this paper extends the work on quadratic MRS functions in a $2020$ paper of the authors to the case of binomial RS functions, that is sums of two quadratic MRS functions. There are also some results for the sum of any number of quadratic MRS functions., Comment: 20 pages + references

We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same codominion in any of these categories which contain it; (c) the invariance of the Hopf algebra or bialgebra (co)dominion construction under field extension, again mimicking the well-known corresponding algebraic-group result; (d) the fact that surjections of coalgebras, bialgebras or Hopf algebras are regular epimorphisms (i.e. coequalizers) provided the codomain is cosemisimple; (e) in particular, the fact that embeddings of compact quantum groups are equalizers in the category thereof, generalizing analogous results on (plain) compact groups; (f) coalgebra-limit preservation results for scalar-extension functors (e.g. extending scalars along a field extension $\Bbbk\le \Bbbk'$ is a right adjoint on the category of $\Bbbk$-coalgebras)., Comment: 24 pages + references

We prove that embeddings of compact groups are equalizers, and a number of results on pushouts (and more generally, amalgamated free products) in the category of compact groups. Call a family of compact-group embeddings $H\le G_i$ algebraically sound if the corresponding group-theoretic pushout embeds in its Bohr compactification. We (a) show that a family of normal embeddings is algebraically sound in the sense that $G_i$ admit embeddings $G_i\le G$ into a compact group which agree on $H$; (b) give equivalent characterizations of coherently embeddable families of normal embeddings in representation-theoretic terms, via Clifford theory; (c) characterize those compact connected Lie groups $H$ for which all finite families of normal embeddings $H\trianglelefteq G_i$ are coherently embeddable (not having central 2-tori is a sufficient, but not necessary condition), and (d) show that families of split embeddings of compact groups are always algebraically sound., Comment: 27 pages + references; the new version contains a number of additions and corrections

Let $U$ be a Banach Lie group and $G\le U$ a compact subgroup. We show that closed Lie subgroups of $U$ contained in sufficiently small neighborhoods $V\supseteq G$ are compact, and conjugate to subgroups of $G$ by elements close to $1\in U$; this generalizes a well-known result of Montgomery and Zippin's from finite- to infinite-dimensional Lie groups. Along the way, we also prove an approximate counterpart to Jordan's theorem on finite subgroups of general linear groups: finite subgroups of $U$ contained in sufficiently small neighborhoods $V\supseteq G$ have normal abelian subgroups of index bounded in terms of $G\le U$ alone. Additionally, various spaces of compact subgroups of $U$, equipped with the Hausdorff metric attached to a complete metric on $U$, are shown to be analytic Banach manifolds; this is the case for both (a) compact groups of a given, fixed dimension, or (b) compact (possibly disconnected) semisimple subgroups. Finally, we also prove that the operation of taking the centralizer (or normalizer) of a compact subgroup of $U$ is continuous (respectively upper semicontinuous) in the appropriate sense., Comment: 28 pages + references

Consider a compact group $G$ acting on a real or complex Banach Lie group $U$, by automorphisms in the relevant category, and leaving a central subgroup $K\le U$ invariant. We define the spaces ${}_KZ^n(G,U)$ of $K$-relative continuous cocycles as those maps ${G^n\to U}$ whose coboundary is a $K$-valued $(n+1)$-cocycle; this applies to possibly non-abelian $U$, in which case $n=1$. We show that the ${}_KZ^n(G,U)$ are analytic submanifolds of the spaces $C(G^n,U)$ of continuous maps $G^n\to U$ and that they decompose as disjoint unions of fiber bundles over manifolds of $K$-valued cocycles. Applications include: (a) the fact that ${Z^n(G,U)\subset C(G^n,U)}$ is an analytic submanifold and its orbits under the adjoint of the group of $U$-valued $(n-1)$-cochains are open; (b) hence the cohomology spaces $H^n(G,U)$ are discrete; (c) for unital $C^*$-algebras $A$ and $B$ with $A$ finite-dimensional the space of morphisms $A\to B$ is an analytic manifold and nearby morphisms are conjugate under the unitary group $U(B)$; (d) the same goes for $A$ and $B$ Banach, with $A$ finite-dimensional and semisimple; (e) and for spaces of projective representations of compact groups in arbitrary $C^*$ algebras (the last recovering a result of Martin's)., Comment: final version, to appear in SIGMA; 26 pages + references

This paper determines the symplectic leaves for a remarkable Poisson structure on $\mathbb{C}\mathbb{P}^{n-1}$ discovered by Feigin and Odesskii, and, independently, by Polishchuk. The Poisson bracket is determined by a holomorphic line bundle of degree $n \ge 3$ on a compact Riemann surface of genus one or, equivalently, by an elliptic normal curve $E\subseteq\mathbb{C}\mathbb{P}^{n-1}$. The symplectic leaves are described in terms of higher secant varieties to $E$., Comment: numerous changes throughout the paper, altering the structure of the main argument; 50 pages + references

An extended derivation (endomorphism) of a (restricted) Lie algebra $L$ is an assignment of a derivation (respectively) of $L'$ for any (restricted) Lie morphism $f:L\to L'$, functorial in $f$ in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of $L'$ to every $f$; and (b) if $L$ is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then $L$ is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms., Comment: 11 pages + references

Let $\mathbb{G}$ be a Lie group with solvable connected component and finitely-generated component group and $\alpha\in H^2(\mathbb{G},\mathbb{S}^1)$ a cohomology class. We prove that if $(\mathbb{G},\alpha)$ is of type I then the same holds for the finite central extensions of $\mathbb{G}$. In particular, finite central extensions of type-I connected solvable Lie groups are again of type I. This is by contrast with the general case, whereby the type-I property does not survive under finite central extensions. We also show that ad-algebraic hulls of connected solvable Lie groups operate on these even when the latter are not simply connected, and give a group-theoretic characterization of the intersection of all Euclidean subgroups of a connected, simply-connected solvable group $\mathbb{G}$ containing a given central subgroup of $\mathbb{G}$., Comment: 23 pages + references; minor changes and typo corrections

Let $\mathfrak{g}$ be a Lie algebra over an algebraically closed field $\Bbbk$ of characteristic zero. Define the universal grading group $\mathcal{C}(\mathfrak{g})$ as having one generator $g_{\rho}$ for each irreducible $\mathfrak{g}$-representation $\rho$, one relation $g_{\pi} = g_{\rho}^{-1}$ whenever $\pi$ is weakly contained in the dual representation $\rho^*$ (i.e. the kernel of $\pi$ in the enveloping algebra $U(\mathfrak{g})$ contains that of $\rho^*$), and one relation $g_{\rho} = g_{\rho'}g_{\rho"}$ whenever $\rho$ is weakly contained in $\rho'\otimes\rho"$. The main result is that attaching to an irreducible representation its central character gives an isomorphism between $\mathcal{C}(\mathfrak{g})$ and the dual $\mathfrak{z}^*$ of the center $\mathfrak{z}\le \mathfrak{g}$ when $\mathfrak{g}$ is (a) finite-dimensional solvable; (b) finite-dimensional semisimple. The group $\mathcal{C}(\mathfrak{g})$ is also trivial when the enveloping algebra $U(\mathfrak{g})$ has a faithful irreducible representation (which happens for instance for various infinite-dimensional algebras of interest, such as $\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$ and $\mathfrak{sp}(\infty)$). These are analogues of a result of M\"uger's for compact groups and a number of results by the author on locally compact groups, and provide further evidence for the pervasiveness of such center-reconstruction phenomena., Comment: 20 pages + references

We prove that an arbitrary compact metrizable group can be realized as the automorphism group of a graphing; this is a continuous analogue to Frucht's theorem recovering arbitrary finite groups are automorphism groups of finite graphs. The paper also contains a number of results 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 $\Gamma_n$ of a convergent sequence of graphons, it also acts transitively on the limit $\lim_n \Gamma$. 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 Lov\'{a}sz 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., Comment: 24 pages + references

We prove a number of results involving categories enriched over \textsc{CMet}, the category of complete metric spaces with possibly infinite distances. The category \textsc{CPMet} of intrinsic complete metric spaces is locally $\aleph_1$-presentable, closed monoidal, and comonadic over \textsc{CMet}. We also prove that the category \textsc{CCMet} of convex complete metric spaces is not closed monoidal and characterize the isometry-$\aleph_0$-generated objects in \textsc{CMet}, \textsc{CPMet} and \textsc{CCMet}, answering questions by Di Liberti and Rosick\'{y}. Other results include the automatic completeness of a colimit of bi-Lipschitz morphisms of complete metric spaces and a characterization of those pairs (metric space, unital $C^*$-algebra) that have a tensor product in the \textsc{CMet}-enriched category of unital $C^*$-algebras., Comment: 36 pages + references

We prove that an action $\rho:A\to M(C_0(\mathbb{G})\otimes A)$ of a locally compact quantum group on a $C^*$-algebra has a universal equivariant compactification, and prove a number of other category-theoretic results on $\mathbb{G}$-equivariant compactifications: that the categories compactifications of $\rho$ and $A$ respectively are locally presentable (hence complete and cocomplete), that the forgetful functor between them is a colimit-creating left adjoint, and that epimorphisms therein are surjective and injections are regular monomorphisms. When $\mathbb{G}$ is regular coamenable we also show that the forgetful functor from unital $\mathbb{G}$-$C^*$-algebras to unital $C^*$-algebras creates finite limits and is comonadic, and that the monomorphisms in the former category are injective., Comment: minor edits/corrections after comments (e.g. added a regularity assumption in the last section); 29 pages + references

We prove that the forgetful functors from the categories of $C^*$- and $W^*$-algebras to Banach $*$-algebras, Banach algebras or Banach spaces are all monadic, answering a question of J.Rosick\'{y}, and that the categories of unital (commutative) $C^*$-algebras are not locally-isometry $\aleph_0$-generated either as plain or as metric-enriched categories, answering a question of I. Di Liberti and Rosick\'{y}. We also prove a number of negative presentability results for the category of von Neumann algebras: not only is that category not locally presentable, but in fact its only presentable objects are the two algebras of dimension $\le 1$. For the same reason, for a locally compact abelian group $\mathbb{G}$ the category of $\mathbb{G}$-graded von Neumann algebras is not locally presentable., Comment: minor typo fixes; 23 pages + references

We show that for a closed embedding $\mathbb{H}\le \mathbb{G}$ of locally compact quantum groups (LCQGs) with $\mathbb{G}/\mathbb{H}$ admitting an invariant probability measure, a unitary $\mathbb{G}$-representation is type-I if its restriction to $\mathbb{H}$ is. On a related note, we also prove that if an action $\mathbb{G}\circlearrowright A$ of an LCQG on a unital $C^*$-algebra admits an invariant state then the full group algebra of $\mathbb{G}$ embeds into the resulting full crossed product (and into the multiplier algebra of that crossed product if the original algebra is not unital). We also prove a few other results on crossed products of LCQG actions, some of which seem to be folklore; among them are (a) the fact that two mutually dual quantum-group morphisms produce isomorphic full crossed products, and (b) the fact that full and reduced crossed products by dual-coamenable LCQGs are isomorphic., Comment: 20 pages + references; added references to prior work that were previously lacking; strengthened the main results of Section 4

Let $\mathbb{H}\trianglelefteq\mathbb{G}$ be a closed normal subgroup of a locally compact quantum group. We introduce a strictly positive group-like element affiliated with $L^{\infty}(\mathbb{G})$ that, roughly, measures the failure of $\mathbb{G}$ to act measure-preservingly on $\mathbb{H}$ by conjugation. The triviality of that element is equivalent to the condition that $\mathbb{G}$ and $\mathbb{G}/\mathbb{H}$ have the same modular element, by analogy with the classical situation. This condition is automatic if $\mathbb{H}\le \mathbb{G}$ is central, and in general implies the unimodularity of $\mathbb{H}$. We also describe a bijection between strictly positive group-like elements $\delta$ affiliated with $C_0(\mathbb{G})$ and quantum-group morphisms $\mathbb{G}\to (\mathbb{R},+)$, with the closed image of the morphism easily described in terms of the spectrum of $\delta$. This then implies that property-(T) locally compact quantum groups admit no non-obvious strictly positive group-like elements.

We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding $\mathbb{N}\trianglelefteq\mathbb{E}$ of locally compact groups and a twisted action $(\alpha,\tau)$ thereof on a (post)liminal $C^*$-algebra $A$ the twisted crossed product $A\rtimes_{\alpha,\tau}\mathbb{E}$ is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup $\mathbb{N}\trianglelefteq \mathbb{E}$ is type-I as soon as $\mathbb{E}$ is. This happens for instance if $\mathbb{N}$ is discrete and $\mathbb{E}$ is Lie, or if $\mathbb{N}$ is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions. In the same spirit, call a locally compact group $\mathbb{G}$ type-I-preserving if all semidirect products $\mathbb{N}\rtimes \mathbb{G}$ are type-I as soon as $\mathbb{N}$ is, and {\it linearly} type-I-preserving if the same conclusion holds for semidirect products $V\rtimes\mathbb{G}$ arising from finite-dimensional $\mathbb{G}$-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie., Comment: 30 pages + references; major revision + added material; addresses a serious issue in one of the main results in the previous version

We prove a number of structural and representation-theoretic results on linearly reductive quantum groups, i.e. objects dual to that of cosemisimple Hopf algebras: (a) a closed normal quantum subgroup is automatically linearly reductive if its squared antipode leaves invariant each simple subcoalgebra of the underlying Hopf algebra; (b) for a normal embedding $\mathbb{H}\trianglelefteq \mathbb{G}$ there is a Clifford-style correspondence between two equivalence relations on irreducible $\mathbb{G}$- and, respectively, $\mathbb{H}$-representations; and (c) given an embedding $\mathbb{H}\le \mathbb{G}$ of linearly reductive quantum groups the Pontryagin dual of the relative center $Z(\mathbb{G})\cap \mathbb{H}$ can be described by generators and relations, with one generator $g_V$ for each irreducible $\mathbb{G}$-representation $V$ and one relation $g_U=g_Vg_W$ whenever $U$ and $V\otimes W$ are not disjoint over $\mathbb{H}$. This latter center-reconstruction result generalizes and recovers M\"uger's compact-group analogue and the author's quantum-group version of that earlier result by setting $\mathbb{H}=\mathbb{G}$., Comment: 14 pages + references

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 $\rho'\otimes \rho"$, 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}ger's result compact groups satisfy chain-center duality., Comment: removed a small amount of unnecessary material; 26 pages + references

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., Comment: 10 pages + references

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., Comment: 15 pages + references

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 $Lie(G)$., Comment: 20 pages + references

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., Comment: 11 pages

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 $\Omega$. 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., Comment: 30 pages + references

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., Comment: 11 pages + references