$$\bar \partial $$ -Cohomology spaces associated with a certain group extension

1971 ◽  
pp. 138-168
Author(s):  
Ichiro Satake
Keyword(s):  
Group Extension ◽  
Cohomology Spaces

AUTOMORPHISMS OF BRAID GROUPS ON S2, T2, P2 AND THE KLEIN BOTTLE K

2008 ◽  
Vol 17 (01) ◽  
pp. 47-53 ◽  
Author(s):  
PING ZHANG
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Klein Bottle ◽  
Closed Surface ◽  
Group Extension ◽  
Braid Groups ◽  
Mapping Class ◽  
Group B ◽  
Central Automorphisms

It is shown that for the braid group Bn(M) on a closed surface M of nonnegative Euler characteristic, Out (Bn(M)) is isomorphic to a group extension of the group of central automorphisms of Bn(M) by the extended mapping class group of M, with an explicit and complete description of Aut (Bn(S2)), Aut (Bn(P2)), Out (Bn(S2)) and Out (Bn(P2)).

Hermann Weyl's Analysis of the “Problem of Space” and the Origin of Gauge Structures

2004 ◽  
Vol 17 (1-2) ◽  
pp. 165-197 ◽  
Author(s):  
Erhard Scholz
Keyword(s):  
General Relativity ◽  
Dirac Equation ◽  
A Priori ◽  
Group Extension ◽  
Central Idea ◽  
German Word ◽  
General Relativistic ◽  
Empirical Turn ◽  
Relativistic Framework ◽  
Internal Symmetries

Hermann Weyl (1885–1955) was one of the early contributors to the mathematics of general relativity. This article argues that in 1929, for the formulation of a general relativistic framework of the Dirac equation, he both abolished and preserved in modified form the conceptual perspective that he had developed earlier in his “analysis of the problem of space.” The ideas of infinitesimal congruence from the early 1920s were aufgehoben (in all senses of the German word) in the general relativistic framework for the Dirac equation. He preserved the central idea of gauge as a “purely infinitesimal” aspect of (internal) symmetries in a group extension schema. With respect to methodology, however, Weyl gave up his earlier preferences for relatively a-priori arguments and tried to incorporate as much empiricism as he could. This signified a clearly expressed empirical turn for him. Moreover, in this step he emphasized that the mathematical objects used for the representation of matter structures stood at the center of the construction, rather than interaction fields which, in the early 1920s, he had considered as more or less derivable from geometrico-philosophical considerations.

scholarly journals A note on central group extensions

1973 ◽  
Vol 15 (4) ◽  
pp. 428-429 ◽  
Author(s):  
G. J. Hauptfleisch
Keyword(s):  
Abelian Group ◽  
Homomorphic Image ◽  
Free Group ◽  
Central Extension ◽  
Dihedral Group ◽  
Group Extension ◽  
Central Group ◽  
Group Extensions

If A, B, H, K are abelian group and φ: A → H and ψ: B → K are epimorphisms, then a given central group extension G of H by K is not necessarily a homomorphic image of a group extension of A by B. Take for instance A = Z(2), B = Z ⊕ Z, H = Z(2), K = V4 (Klein's fourgroup). Then the dihedral group D8 is a central extension of H by K but it is not a homomorphic image of Z ⊕ Z ⊕ Z(2), the only group extension of A by the free group B.

Relative cohomology spaces for some osp(n|2)-modules

2018 ◽  
Vol 59 (10) ◽  
pp. 101704
Author(s):  
Didier Arnal ◽  
Mabrouk Ben Ammar ◽  
Wafa Mtaouaa ◽  
Zeineb Selmi
Keyword(s):  
Relative Cohomology ◽  
Cohomology Spaces

scholarly journals Tracking Maneuvering Group Target with Extension Predicted and Best Model Augmentation Method Adapted

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Linhai Gan ◽  
Gang Wang
Keyword(s):  
Target Tracking ◽  
Random Matrix ◽  
Poor Performance ◽  
Group Extension ◽  
New Approach ◽  
Basic Model ◽  
True Motion ◽  
Motion Modes ◽  
Model Set ◽  
Group Target

The random matrix (RM) method is widely applied for group target tracking. The assumption that the group extension keeps invariant in conventional RM method is not yet valid, as the orientation of the group varies rapidly while it is maneuvering; thus, a new approach with group extension predicted is derived here. To match the group maneuvering, a best model augmentation (BMA) method is introduced. The existing BMA method uses a fixed basic model set, which may lead to a poor performance when it could not ensure basic coverage of true motion modes. Here, a maneuvering group target tracking algorithm is proposed, where the group extension prediction and the BMA adaption are exploited. The performance of the proposed algorithm will be illustrated by simulation.

On Harmonic Theory in Flows

2003 ◽  
Vol 46 (4) ◽  
pp. 617-631 ◽  
Author(s):  
Hong Kyung Pak
Keyword(s):  
Contact Manifold ◽  
Natural Generalization ◽  
Basic Flow ◽  
Compact Manifolds ◽  
Cohomology Spaces ◽  
Special Case

AbstractRecently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of H-harmonic and H*-harmonic spaces associated to a Hörmander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.

On Graded Categorical Groups and Equivariant Group Extensions

2002 ◽  
Vol 54 (5) ◽  
pp. 970-997 ◽  
Author(s):  
A. M. Cegarra ◽  
J. M. Garćia-Calcines ◽  
J. A. Ortega
Keyword(s):  
Group Cohomology ◽  
Group Extension ◽  
Extension Problem ◽  
Homotopy Classification ◽  
Group Extensions ◽  
Categorical Groups

AbstractIn this article we state and prove precise theorems on the homotopy classification of graded categorical groups and their homomorphisms. The results use equivariant group cohomology, and they are applied to show a treatment of the general equivariant group extension problem.

scholarly journals Block Extensions, Local Categories and Basic Morita Equivalences

2020 ◽  
Vol 71 (2) ◽  
pp. 703-728
Author(s):  
Tiberiu Coconeţ ◽  
Andrei Marcus ◽  
Constantin-Cosmin Todea
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Cohomology Class ◽  
Group Extension ◽  
Modular System ◽  
Pointed Group ◽  
A Finite Group ◽  
Nilpotent Blocks ◽  
The Way

Abstract Let $(\mathcal{K},\mathcal{O},k)$ be a $p$-modular system where $p$ is a prime and $k$ algebraically closed, let $b$ be a $G$-invariant block of the normal subgroup $H$ of a finite group $G$, having defect pointed group $Q_\delta$ in $H$ and $P_\gamma$ in $G$ and consider the block extension $b\mathcal{O}G$. One may attach to $b$ an extended local category $\mathcal{E}_{(b,H,G)}$, a group extension $L$ of $Z(Q)$ by $N_G(Q_\delta )/C_H(Q)$ having $P$ as a Sylow $p$-subgroup, and a cohomology class $[\alpha ]\in H^2(N_G(Q_\delta )/QC_H(Q),k^\times )$. We prove that these objects are invariant under the $G/H$-graded basic Morita equivalences. Along the way, we give alternative proofs of the results of Külshammer and Puig (1990), and Puig and Zhou (2012) on extensions of nilpotent blocks. We also deduce by our methods a result of Zhou (2016) on $p^{\prime}$-extensions of inertial blocks.

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