### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 066, 17 pages      arXiv:1312.0362      https://doi.org/10.3842/SIGMA.2015.066

### Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras

Alexey A. Magazev, Vitaly V. Mikheyev and Igor V. Shirokov
Omsk State Technical University, 11 Mira Ave., Omsk, 644050, Russia

Received December 05, 2013, in final form July 25, 2015; Published online August 06, 2015

Abstract
Methods of construction of the composition function, left- and right-invariant vector fields and differential 1-forms of a Lie group from the structure constants of the associated Lie algebra are proposed. It is shown that in the second canonical coordinates these problems are reduced to the matrix inversions and matrix exponentiations, and the composition function can be represented in quadratures. Moreover, it is proven that the transition function from the first canonical coordinates to the second canonical coordinates can be found by quadratures.

Key words: Lie group; Lie algebra; left- and right-invariant vector fields; composition function; canonical coordinates.

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