Search

The notion of conditional coproduct of a family of abelian pro-Lie groups in the category of abelian pro-Lie groups is introduced. It is shown that the cartesian product of an arbitrary family of abelian pro-Lie groups can be characterized by the universal property of the conditional coproduct., Comment: 6 pages

We study universal enveloping Hopf algebras of Lie algebras in the category of weakly complete vector spaces over the real and complex field., Comment: v2 completely rewritten and corrected. v3 title corrected. To appear in Journal of Lie Theory

We study group algebras for compact groups in the category of real and complex weakly complete vector spaces. We also show that the group algebra is a quotient of the weakly complete universal enveloping algebra of the Lie algebra of the compact group. We relate this to Tannaka duality, and to functorial properties of the group algebra. We determine the structure of the group algebra in terms of the irreducible representation, both in the real and the complex case. The particular case of a compact abelian group is worked out in detail., Comment: v1, 2 appeared as an Oberwolfach Preprint. v3 considerably expanded and extended. v4 final version, to appear in Journal of Lie Theory

A weakly complete vector space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ is isomorphic to $\mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is illustrated by the fact that the underlying vector space of the Lie algebra of any pro-Lie group is weakly complete. In this study, weakly complete real or complex associative algebras are studied because they are necessarily projective limits of finite dimensional algebras. The group of units $A^{-1}$ of a weakly complete algebra $A$ is a pro-Lie group with the associated topological Lie algebra $A_{\rm Lie}$ of $A$ as Lie algebra and the globally defined exponential function $\exp\colon A\to A^{-1}$ as the exponential function of $A^{-1}$. With each topological group, a weakly complete group algebra $\mathbb{K}[G]$ is associated functorially so that the functor $G\mapsto \mathbb{K}[G]$ is left adjoint to $A\mapsto A^{-1}$. The group algebra $\mathbb{K}[G]$ is a weakly complete Hopf algebra. If $G$ is compact, the $\mathbb{R}[G]$ contains $G$ as the set of grouplike elements. The category of all real Hopf algebras $A$ with a compact group of grouplike elements whose linear span is dense in $A$ is shown to be equivalent to the category of compact groups. The group algebra $A=\mathbb{R}[G]$ of a compact group $G$ contains a copy of the Lie algebra $\mathcal{L}(G)$ in $A_{\rm Lie}$; it also contains a copy of the Radon measure algebra $M(G,\mathbb{R})$. The dual of the group algebra $\mathbb{R}[G]$ is the Hopf algebra ${\mathcal R}(G,\mathbb{R})$ of representative functions of $G$. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis of a duality ${\mathcal R}(G,\mathbb{R})\leftrightarrow \mathbb{R}[G]$ and thus yields a new aspect of Tannaka duality., Comment: This is version 2. Only spelling mistakes and formal flaws have been corrected. Some explanations have been added. The content of the mathematical results has not been changed

The class of locally compact near abelian groups is introduced and investigated as a class of metabelian groups formalizing and applying the concept of scalar multiplication. The structure of locally compact near abelian groups and its close connections to prime number theory are discussed and elucidated by graph theoretical tools. These investigations require a thorough reviewing and extension to present circumstances of various aspects of the general theory of locally compact groups such as -the Chabauty space of closed subgroups with its natural compact Hausdorff topology, -a very general Sylow subgroup theory for periodic groups including their Hall systems, -the scalar automorphisms of locally compact abelian groups, -the theory of products of closed subgroups and their relation to semidirect products, and -inductively monothetic groups are introduced and classified. As applications, firstly, a complete classification is given of locally compact topologically quasihamiltonian groups, which has been initiated by F. K\"ummich, and, secondly, Yu. Mukhin's classification of locally compact topologically modular groups is retrieved and further illuminated., Comment: 32 pages

High-mass stars exhibit a significantly higher multiplicity frequency than low-mass stars, likely reflecting differences in how they formed. Theory suggests that high-mass binaries may form by the fragmentation of self-gravitational discs or by alternative scenarios such as disc-assisted capture. Near-infrared interferometric observations reveal the high-mass young stellar object IRAS 17216-3801 to be a close high-mass protobinary with a separation of 0.058 arcseconds (~ 170 au). This is the closest high-mass protobinary system imaged to date. We also resolve near- infrared excess emission around the individual stars, which is associated with hot dust in circumstellar discs. These discs are strongly misaligned with respect to the binary separation vector, indicating that tidal forces have not yet had time to realign. We measure a higher accretion rate towards the circumsecondary disc, confirming a hydrodynamic effect where the secondary star disrupts the primary star’s accretion stream and effectively limits the mass that the primary star can accrete. NACO L′-band imaging may also have resolved the circumbinary disc that feeds the accretion onto the circumstellar discs. This discovery demonstrates the unique capabilities of the VLTI, creating exciting new opportunities to study the dynamical processes that govern the architecture of close multiple systems., Published in The Messenger vol. 170, pp. 45-49, December 2017.

Let $G$ be a locally compact topological group, $G_0$ the connected component of its identity element, and comp(G) the union of all compact subgroups. A topological group will be called inductively monothetic if any subgroup generated (as a topological group) by finitely many elements is generated (as a topological group) by a single element. The space SUB(G) of all closed subgroups of $G$ carries a compact Hausdorff topology called the Chabauty topology. Let $F_1(G)$, respectively, $R_1(G)$, denote the subspace of all discrete subgroups isomorphic to $\mathbb Z$, respectively, all subgroups isomorphic to $\mathbb R$. It is shown that a necessary and sufficient condition for $G\in\overline{F_1(G)}$ to hold is that $G$ is abelian, and either that $G\cong \mathbb R\times {\rm comp}(G)$ and $G/G_0$ is inductively monothetic, or else that $G$ is discrete and isomorphic to a subgroup of $\mathbb Q$. It is further shown that a necessary and sufficient condition for $G\in\overline{R_1(G)}$ to hold is that $G\cong\mathbb R\times C$ for a compact connected abelian group $C$.

We contribute some information towards finding a general algorithm for constructing, for a given profinite group, $G$, a compact connected space, $X$, such that the full homeomorphism group, $H(X)$, with the compact-open topology is isomorphic to $G$ as a topological group. It is proposed that one should find a compact topological oriented graph $\Gamma$ such that $G\cong Aut(\Gamma)$. The replacement of the edges of $\Gamma$ by rigid continua should work as is exemplified in various instances where discrete graphs were used. It is shown here that the strategy can be implemented for profinite monothetic groups $G$.

It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.

Surfactant-mediated epitaxy (SME) allows the growth of smooth, continuous, relaxed, and principally defect free Ge films directly on Si(111); however, the very high surfactant doping level in the range of the solid solubility limit made them unacceptable for most device applications. By using high temperature SME we have reduced the Sb surfactant background doping level by more than three orders of magnitude. This is attributed to an enhanced surfactant segregation without kinetic limitations. The low Sb incorporation has been determined by an electrical characterization: An electron concentration of 1.1×1016 cm−3 and a very high electron Hall mobility of 3100 cm2/V s at 300 K (12 300 cm2/V s at 77 K) suggest an interesting potential of SME grown Ge films for future device applications.

n/a

n/a

n/a