1-parameter group

A 1-parameter group is a continuous group homomorphism

𝜑 ? : ℝ + \to 𝐺

where $𝐺$ is a topological group. lie Often, we abuse terminology and conflate $𝜑$ with $i m 𝜑$ .

Local 1-parameter group

A local 1-parameter group is a weakening akin to a local group.¹ We restrict to a flow domain $(𝜖 -, 𝜖 +)$ for $𝜖 - < 0 < 𝜖 +$ , and take

𝜑 ? : (𝜖 -, 𝜖 +) \to 𝐺 .

We require that $𝜑 0 = 1 𝐺$ and $𝜑 𝑎 𝜑 𝑏 = 𝜑 𝑎 + 𝑏$ for all $𝑎, 𝑏 \in (𝜖 -, 𝜖 +)$ such that $𝑎 + 𝑏 \in (𝜖 -, 𝜖 +)$ .