Continuous Group of Transformations

Abstract

Let $B$ be Banach algebra and $M$ be topological space. If there exists homeomorphism \[ f:M\rightarrow N \] of topological space $M$ into convex set $N$ of the space $B^n$, then homeomorphism $f$ is called chart of the set $M$. The set $M$ is called simple $B$-manifold of class $C^k$ if for any two charts \[ f_1:M\rightarrow N_1\subseteq B^n \] \[ f_2:M\rightarrow N_2\subseteq B^n \] there exists diffeomorphism \[ f: B^n\rightarrow B^n \] of class $C^k$ such that \[ f_1\circ f=f_2 \] Topological space $M$ is called differential $B$-manifold of class $C^k$ if topological space $M$ is a union of simple $B$-manifolds $M_i$, $i\in I$, and intersection $M_i\cap M_j$ of simple $B$-manifolds $M_i$, $M_j$ is also simple $B$-manifold. Differential $B$-manifold $G$ equipped with group structure such that map \[ (f,g)\rightarrow fg^{-1} \] is differentiable is called Lie group. Module $T_eG$ equipped with product \[ [v,w]^c= R_{Ljm}^c\circ(v^m,w^j) -R_{Lmj}^c\circ(w^j,v^m) \in T_eG \] is Lie algebra $g_L$ of Lie group $G$.