---
tag: XJS07AJ50Y
kind: concept
name: Semisimple Lie group
msc: ["22E46", "22E15", "22E10"]
specializes: [42ZCE14AD2]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_Lie_group
---

## Summary

A semisimple Lie group is a connected [[Lie group|7JBRGCVJRB]] whose Lie algebra is a semisimple Lie algebra, that is, has no nonzero solvable ideals.
Equivalently, its Lie algebra is a direct sum of simple Lie algebras.
So, up to a covering and a central quotient, the group is a product of simple Lie groups.
Semisimple Lie groups have a rich structure theory, with Cartan, Iwasawa and Bruhat decompositions, and a deep representation theory.
By the Levi decomposition, a finite-dimensional real or complex Lie algebra is a semisimple subalgebra plus its largest solvable ideal.
So semisimple and solvable groups are the two basic kinds.

## Explanations

### L1 human

A semisimple Lie group is a Lie group built from simple Lie groups, with no commutative or "almost commutative" continuous part.
The group $SO(4)$ is a typical example.
Its Lie algebra splits into two copies of the Lie algebra of $SO(3)$, one for each of two independent kinds of rotation of four-dimensional space.
The unitary group $U(n)$ is not semisimple, because the scalar matrices $e^{i\theta} I$ form a commutative continuous part.
Removing that part by fixing the determinant gives the semisimple group $SU(n)$.