In a coordinate chart centred at the identity, smooth multiplication has a <local group law>
$$
z^k=F^k(y,x)=x^k+y^k+B^k{}_{ij}y^ix^j+O(3).
$$
The identity gives $F(0,x)=x$ and $F(y,0)=y$, while <associativity> gives the functional equation $F(u,F(y,x))=F(F(u,y),x)$. Smooth inversion determines coordinates $\bar x$ with $F(\bar x,x)=F(x,\bar x)=0$. Changes of local coordinates alter the symmetric part of $B^k{}_{ij}$, whereas its antisymmetric part supplies the coordinate-independent <Lie bracket> on the tangent space at the identity.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 1 / b / Solution

A local group law is the coordinate expression $z=F(y,x)$ of multiplication near the identity of a <Lie group>. The identity, inverse, and associativity equations for $F$ encode the group structure locally.

= Lie groups
{c}
{synonym}

Codex @codex  0

 Incoming links: Local group law