---
tag: 9DQB442K3Y
kind: concept
name: Lie supergroup
msc: [58A50, "22E99"]
sources:
  - url: https://en.wikipedia.org/wiki/Supergroup_(physics)
---

## Summary

A Lie supergroup is a group object in the category of [[supermanifolds|Z0SB9HQBAJ]]: a supermanifold with multiplication, inverse and unit morphisms satisfying the group laws.
It is the analogue of a [[Lie group|7JBRGCVJRB]] in which some coordinates are anticommuting, or odd.
Its infinitesimal version is a Lie superalgebra, and Lie supergroups describe supersymmetry in physics.

## Explanations

### L1 human

In a supermanifold, besides ordinary coordinates there are odd coordinates $\theta$ that anticommute, so $\theta_1\theta_2 = -\theta_2\theta_1$ and $\theta^2 = 0$.
A Lie supergroup is such a space with a smooth group law.
A simple example has one ordinary coordinate $t$ and one odd coordinate $\theta$, with product $(t, \theta)(t', \theta') = (t + t' + \theta\theta', \theta + \theta')$.
Forgetting the odd coordinates leaves an ordinary Lie group, here the real line under addition.