\contentsline {part}{I\hspace {1em}Fundamentals}{1}{}%
\contentsline {chapter}{\numberline {1}Lie groups}{5}{}%
\contentsline {section}{\numberline {1.1}Continuous transformation groups}{6}{}%
\contentsline {section}{\numberline {1.2}Lie groups}{6}{}%
\contentsline {section}{\numberline {1.3}Transformation groups}{7}{}%
\contentsline {section}{\numberline {1.4}Local Lie group}{7}{}%
\contentsline {subsection}{\numberline {1.4.1}Local transformation groups}{8}{}%
\contentsline {section}{\numberline {1.5}Classical Lie groups}{8}{}%
\contentsline {chapter}{\numberline {2}Lie algebrae}{9}{}%
\contentsline {subsection}{\numberline {2.0.1}Generic Lie algebrae}{11}{}%
\contentsline {subsection}{\numberline {2.0.2}Invariant vector fields}{11}{}%
\contentsline {subsection}{\numberline {2.0.3}Vector flows on infinitesimal generators}{12}{}%
\contentsline {subsection}{\numberline {2.0.4}Exponential map}{12}{}%
\contentsline {section}{\numberline {2.1}Lie subalgebrae}{13}{}%
\contentsline {subsection}{\numberline {2.1.1}Lie algebrae of local Lie groups}{14}{}%
\contentsline {section}{\numberline {2.2}Classical Lie algebrae}{14}{}%
\contentsline {chapter}{\numberline {3}Smooth transformation groups}{15}{}%
\contentsline {section}{\numberline {3.1}Invariant functions}{15}{}%
\contentsline {subsection}{\numberline {3.1.1}Invariant subsets}{16}{}%
\contentsline {section}{\numberline {3.2}Further results?}{16}{}%
\contentsline {chapter}{\numberline {4}Applications to differential equations}{17}{}%
\contentsline {section}{\numberline {4.1}Prolongations}{17}{}%
\contentsline {subsection}{\numberline {4.1.1}Prolongation of IGs}{17}{}%
\contentsline {subsection}{\numberline {4.1.2}Prolongation formula}{18}{}%
\contentsline {subsection}{\numberline {4.1.3}Properties of prolonged IGs}{18}{}%
\contentsline {chapter}{\numberline {5}Solvable Lie groups}{19}{}%