<span title="Home" class="fa-solid-900 icon"></span> Home

Ciro Santilli's Homepage

Mathematics

Area of mathematics

Geometry

Differential geometry

Lie group

Important Lie group

Matrix Lie group

Lie algebra of a matrix Lie group

The one parameter subgroup of a <Lie group> for a given element $M$ of its <Lie algebra> is a <subgroup> of $G$ given by:
$$
{ e^{tM} \in G | t \in \R }
$$

Intuitively, $M$ is a direction, and $t$ is how far we move along a given direction. This intuition is especially vivid in for example in the case of the <Lie algebra of SO(3)>, the <rotation group>.

One parameter subgroups can be seen as the continuous analogue to the <cycle of an element of a group>.


One parameter subgroup

Cycle of an element of a group

{wiki}

A one-parameter subgroup of a Lie group $G$ is a smooth homomorphism $\gamma:(\mathbb R,+)\to G$. Every such subgroup has the form $\gamma(t)=\exp(tX)$ for a unique $X$ in the Lie algebra of $G$.


One-parameter subgroup

One parameter subgroup

 Ancestors (9)

 Incoming links (2)

 Discussion (0)

 Articles by others on the same topic (1)

One parameter subgroup

 Ancestors (9)

 Incoming links (2)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (1)

 Discussion (0)