# nLab universal central extension

Contents

### Context

#### Algebra

higher algebra

universal algebra

group theory

# Contents

## Definition

For any algebraic object, such as an associative algebra, group, Lie algebra, etc. such that there is a notion of central extension, the universal central extension is, if it exists, the initial object in the category of central extensions of that object.

## Examples

For $G$ a discrete group which is perfect, the universal central extension of $G$ is the essentially unique group $K$ that is a Schur covering group? of $G$.

## Properties

• The universal central extension of a perfect group is also perfect.

• The universal central extension of a perfect group is a Schur-trivial group, and hence a superperfect group (superperfect means that it’s perfect and Schur-trivial).

• The universal central extension operator is idempotent, i.e., the universal central extension of the universal central extension is the universal central extension. This follows directly from the universal central extension being a Schur-trivial group.

## References

### General

Discussion in semi-abelian categories:

• Jose Casas, Tim Van der Linden, A relative theory of universal central extensions (arXiv:0908.3762)

### For super Lie algebras

• Erhard Neher, An introduction to universal central extensions of Lie superalgebras (pdf)

Last revised on July 31, 2018 at 04:58:25. See the history of this page for a list of all contributions to it.