\part{Advanced}

Derivation space


\chapter{Lie theory}




\begin{definition}[Lie group]
Lie group is a group $(M,\cdot)$ such that $M$ is also a smooth manifold and multiplication and the inverse operation are smooth functions.
\end{definition}


Lie derivative
Lie algebra


Lie subgroups

Closed-subgroup theorem
Let $G$ be a Lie group, then any closed subgroup $H$ is an embedded Lie group.


\section{Transformation groups}

Most Lie groups we will see are groups formed by special automorphisms that act on the manifold; these are known as \emph{transformation groups}.


Transformation groups are associated with a Lie group action

semiregular actions  are those whose orbits are embedded smooth manifolds that all have the same dimension. 

Alternatively, they are such that the stabilizer of each point on manifold is trivial.


\section{Vector field}


\subsection{Integral curve and flows}



Imagine a particle placed at point $p$ in a vector field; the vector field pushes the particle around, making it "flow" around the field. The "flow" of this particle is the integral curve that $p$ lies on.

\begin{definition}[$F$-integral curve]
Given a manifold $M$ with a smooth vector field $F$, a \emph{$F$-integral curve} is a smooth curve $\mathbf{r}$ whose derivative is the 
\[\mathbf{r}'(t)=F(\mathbf{r}(t))\]
\end{definition}


This motivated the definition of a \emph{flow}; a group action representing all integral curves on a vector field at once.
\begin{definition}[Flow]
	(left) group action of the group $(\mathbb{R},+)$ acting on a manifold $M$
\end{definition}

The situation we have described is the use of a flow to represent integral curves; this is known as a vector flow.

\begin{definition}[Vector flow]
Given a manifold $M$ with smooth vector field $F$ a vector flow $\psi$ is a flow on $M$ such that $\psi(\varepsilon,\mathbf{x})$ represents the $F$-integral cuve through $\mathbf{x}$
\end{definition}

Calculating vector flows essentially amounts to solving a system of IVP ODEs. since we can mathematically represent vector flows as integral curves passing through a manifold point $\mathbf{x}$ as $\psi (t,\mathbf{x}) = \mathbf{r}_{\mathbf{x}}(t)$  (where $\mathbf{r}_{\mathbf{x}}(0)=\mathbf{x}$), we obtain the following sets of IVP ODEs;

\[ \frac{partial \psi (t,\mathbf{x})}{\partial t} = F(\mathbf{r}_{\mathbf{x}}(t)) , \psi (0,\mathbf{x}) = \mathbf{x} \]

It is known that such IVPs always have a unique solution; hence we have the following theorem that permits the calculation of vector flows by solving the appropriate IVPs.

\begin{proposition}[Solving vector flows]
\end{proposition}

Note that the proposition also allows one to find the unique smooth vector fields associated with vector flows; simply differentiate the vector flow at $0$ to find the related vector field!





THe derivative of the flow (with respect to the real additive group) is the tangent vector at the point of the manifold.

\begin{theorem}
Let $p$ be a point on smooth manifold where vector field doesn't vanish, then there is a local coordinate chart such that the vector field is 1 in one direction and zero in the others in this chart.
\end{theorem}




\section{Local Lie group}

\begin{definition}[Local Lie group]
Pregroup 
\end{definition}

When one is dealing with only an open neighborhood of the identity, one can use local Lie groups instead of a (global) Lie group, so that one may work directly with the local coordinates of a chart covering said this neighborhood.


Local lie groups are always a neighborhood of some global Lie group


\subsection{Local transformation groups}

There is a local analogue of transformation groups too.