Potential Spaces on Lie Groups

Scale Spaces on Lie Groups

In the standard scale space approach one obtains a scale space representation u : R R → R of an image f ∈ L2(R) by means of an evolution equation on the additive group (R,+). However, it is common to apply a wavelet transform (constructed via a representation U of a Lie-group G and admissible wavelet ψ) to an image which provides a detailed overview of the group structure in an image. The resul...

SEMINAR ON LIE GROUPS 1. Lie Groups

Example 1.3. (R,+) Example 1.4. S or T n = S × ...× S Example 1.5. Gl (n,F) ⊆ F, where F = R or C Example 1.6. E3 = isometries of R (2 connected components) Let the orthogonal group O3 < E3 be the subgroup that fixes the origin, and let the special orthogonal group SO (3) = SO3 < O3 be the orientation-preserving elements of O3. Visualizing SO (3): Let u be a vector of length l in R, correspondi...

Lie Algebras, 2-Groups and Cotriangular Spaces

We describe the construction of a Lie algebra from a partial linear space with oriented lines of size 3, generalizing a construction by Kaplansky. We determine all suitable partial linear spaces and the resulting Lie algebras. 1 Lie oriented partial linear spaces A partial linear space is an incidence structure (P,L) with points and lines, such that the point-line incidence graph does not conta...

Lie Groups. Representation Theory and Symmetric Spaces

Euler-Lagrange equations and geometric mechanics on Lie groups with potential

Abstract. Let G be a Banach Lie group modeled on the Banach space, possibly infinite dimensional, E. In this paper first we introduce Euler-Lagrange equations on the Lie group G with potential and right invariant metric. Euler-Lagrange equations are natural extensions of the geodesic equations on manifolds and Lie groups. In the second part, we study the geometry of the mechanical system of a r...