Table of Lie groups

This article gives a table of some common Lie groups and their associated Lie algebras. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple). For more examples of Lie groups and other related topics see the list of simple Lie groups; the Bianchi classification of groups of up to three dimensions; see classification of low-dimensional real Lie algebras for up to four dimensions; and the list of Lie group topics. Real Lie groups and their algebras Column legend

Cpt: Is this group G compact? (Yes or No)
\pi_0: Gives the group of components of G. The order of the component group gives the number of connected components. The group is connected if and only if the component group is trivial (denoted by 0).
\pi_1: Gives the fundamental group o

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Feride Tiglay

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar, equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler-Poincar, equations that have not yet been considered in the literature as well as integrable equations like Camassa-Holm, Degasperis-Procesi, mu CH and mu DP equations, and the geodesic equations with respect to right-invariant Sobolev metrics on the group of diffeomorphisms of the circle.

Springer Netherlands2011

Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions

Feride Tiglay

We study a family of equations defined on the space of tensor densities of weight lambda on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct "peakon" and "multi-peakon" solutions for all lambda not equal 0, 1, and "shock-peakons" for lambda = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold's approach to Euler equations on Lie groups.

Springer Verlag2010

Toda frames, harmonic maps and extended Dynkin diagrams

Katharine Felicity Turner

We consider a natural subclass of harmonic maps from a surface into G/T, namely cyclic primitive maps. Here G is any simple real Lie group (not necessarily compact), T is a Cartan subgroup and both are chosen so that there is a Coxeter automorphism on G(C)/T-C which restricts to give a k-symmetric space structure on G/T. When G is compact, any Coxeter automorphism restricts to the real form. It was shown in [3] that cyclic primitive immersions into compact G/T correspond to solutions of the affine Toda field equations and all those of a genus one surface can be constructed by integrating a pair of commuting vector fields on a finite dimensional vector subspace of a Lie algebra. We generalise these results, removing the assumption that G is compact. The first major obstacle is that a Coxeter automorphism may not restrict to a non-compact real form. We characterise, in terms of extended Dynkin diagrams, those simple real Lie groups G and Cartan subgroups T such that G/T has a k-symmetric space structure induced from a Coxeter automorphism. A Coxeter automorphism preserves the real Lie algebra g if and only if any corresponding Cartan involution defines a permutation of the extended Dynkin diagram for g(C) = g circle times C; we show that every involution of the extended Dynkin diagram for a simple complex Lie algebra g(C) is induced by a Cartan involution of a real form of sf. (C) 2017 Published by Elsevier B.V.

Elsevier2017

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