scholarly journals The path group construction of Lie group extensions

2008 ◽  
Vol 58 (7) ◽  
pp. 860-873 ◽  
Author(s):  
Cornelia Vizman
Keyword(s):  
Lie Group ◽  
Group Extensions ◽  
Group Construction

scholarly journals Topological Wiener–Wintner ergodic theorems via non-abelian Lie group extensions

2007 ◽  
Vol 27 (5) ◽  
pp. 1633-1650 ◽  
Author(s):  
SARA I. SANTOS ◽  
CHARLES WALKDEN
Keyword(s):  
Lie Group ◽  
Ergodic Theorems ◽  
Compact Lie Group ◽  
Group Extensions ◽  
Shift Map ◽  
Measure Preserving Transformations

AbstractWe generalize a series of topological Wiener–Wintner ergodic theorems due to Walters to the context of group extensions of measure-preserving transformations where the group is a non-abelian compact Lie group. Applications to random ergodic theorems for a shift map are given.

scholarly journals Transitivity of Heisenberg group extensions of hyperbolic systems

2011 ◽  
Vol 32 (1) ◽  
pp. 223-235 ◽  
Author(s):  
IAN MELBOURNE ◽  
VIOREL NIŢICĂ ◽  
ANDREI TÖRÖK
Keyword(s):  
Dynamical System ◽  
Heisenberg Group ◽  
Lie Group ◽  
Commutator Subgroup ◽  
Hyperbolic Systems ◽  
Nilpotent Lie Group ◽  
Topological Transitivity ◽  
Group Extensions ◽  
Topologically Transitive ◽  
Dimension 2

AbstractWe show that amongCrextensions (r>0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group ℋnof dimension 2n+1, those that avoid an obvious obstruction to topological transitivity are generically topologically transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with a compact commutator subgroup (for example ℋn/ℤ), among those that avoid the obvious obstruction, topological transitivity is open and dense.

scholarly journals Smooth 2-Group Extensions and Symmetries of Bundle Gerbes

2021 ◽  
Author(s):  
Severin Bunk ◽  
Lukas Müller ◽  
Richard J. Szabo
Keyword(s):  
Lie Group ◽  
Parallel Transport ◽  
Group Extension ◽  
Geometric Framework ◽  
Group Extensions ◽  
Bundle Gerbe ◽  
Bundle Gerbes ◽  
Definition Of ◽  
Simply Connected ◽  
Connected Lie Group

AbstractWe study bundle gerbes on manifolds M that carry an action of a connected Lie group G. We show that these data give rise to a smooth 2-group extension of G by the smooth 2-group of hermitean line bundles on M. This 2-group extension classifies equivariant structures on the bundle gerbe, and its non-triviality poses an obstruction to the existence of equivariant structures. We present a new global approach to the parallel transport of a bundle gerbe with connection, and use it to give an alternative construction of this smooth 2-group extension in terms of a homotopy-coherent version of the associated bundle construction. We apply our results to give new descriptions of nonassociative magnetic translations in quantum mechanics and the Faddeev–Mickelsson–Shatashvili anomaly in quantum field theory. We also propose a definition of smooth string 2-group models within our geometric framework. Starting from a basic gerbe on a compact simply-connected Lie group G, we prove that the smooth 2-group extensions of G arising from our construction provide new models for the string group of G.

Symmetry properties of fractional order transport equations

2012 ◽  
Vol 9 (1) ◽  
pp. 59-64
Author(s):  
R.K. Gazizov ◽  
A.A. Kasatkin ◽  
S.Yu. Lukashchuk
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Group ◽  
Initial Conditions ◽  
Group Analysis ◽  
Equivalence Transformation ◽  
Point Change ◽  
Infinitesimal Operator ◽  
Symmetry Properties ◽  
Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

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