---
tag: SNA3DVPK0C
kind: concept
name: Left-invariant vector field
msc: ["22E60", "22E15"]
specializes: [0299VCW1TH]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_group
---

## Summary

A left-invariant vector field on a [[Lie group|7JBRGCVJRB]] is a [[vector field|0299VCW1TH]] that is unchanged by every left translation.
Such a field is fixed by its value at the identity, so left-invariant vector fields correspond to tangent vectors at the identity.
They are closed under the [[Lie bracket of vector fields|68FGQEBD6J]].
This makes the tangent space at the identity a Lie algebra, the Lie algebra of the group.

## Explanations

### L1 human

Pick an arrow at the identity element of a Lie group.
Carry it to every other point $g$ by multiplying on the left by $g$; you get an arrow at every point, all fitting together smoothly.
On the plane $\R^2$ as an additive group, this gives the constant vector fields: the same arrow attached at every point.
On the circle group, it gives a field of arrows of the same length all pointing anticlockwise, or all clockwise.