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D.5.3 orbitparam_lib

Parametrizing orbits of unipotent actions

J. Boehm, boehm at mathematik.uni-kl.de
S. Papadakis, papadak at math.ist.utl.pt

This library implements the theorem of Chevalley-Rosenlicht as stated in Theorem 3.1.4 of [Corwin, Greenleaf]. Given a set of strictly upper triangular n x n matrices L_1,...,L_c which generate a Lie algebra as a vector space, and a vector v of size n, the function parametrizeOrbit constructs a parametrization of the orbit of v under the action of exp(<L_1,...,L_c>).

To compute exp of the Lie algebra elements corresponding to the parameters we require that the characteristic of the base field is zero or larger than n.

By determining the parameters from bottom to top
this allows you to find an element in the orbit with (at least) as many zeros as the dimension of the orbit.

Note: Theorem 3.1.4 of [Corwin, Greenleaf] uses strictly lower triangular matrices.

Laurence Corwin, Frederick P. Greenleaf: Representations of Nilpotent Lie Groups and their Applications: Volume 1, Part 1, Basic Theory and Examples, Cambridge University Press (2004).


D.5.3.1 tangentGens  Returns elements in the Lie algebra, which form a basis of the tangent space of the parametrization.
D.5.3.2 matrixExp  Matrix exp for nilpotent matrices.
D.5.3.3 matrixLog  Matrix log for unipotent matrices.
D.5.3.4 parametrizeOrbit  Returns parametrization of the orbit.
D.5.3.5 maxZeros  Determine an element in the orbit with the maximum number of zeroes.