# nLab central extension

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

Given an object in algebra $A$ (such as an associative algebra, a group or a Lie algebra, etc.) then an extension $\widehat A \overset{p}{\longrightarrow} A$ (e.g. a group extension or Lie algebra extension etc.) is called a central extension if its kernel

(1)$1 \to ker(p) \overset{\phantom{A}\iota\phantom{A}}{\hookrightarrow} \widehat A \overset{\phantom{A}p\phantom{A}}{\longrightarrow} A \to 1$

is in the center $C(\widehat{A})$ of $\widehat A$

$ker(p) \subset C(\widehat{A}) \,.$

This means in particular that $ker(p)$ is “commutative” (e.g. a commutative algebra or abelian group or Lie algebra with vanishing Lie bracket etc.), but it means in addition that the elements of $ker(p)$ commute noit just among themselves, but also with all other elements of $\widehat A$.

Typically central extensions by some commutative algebraic object $ker(p)$ are classified by the suitable degree-2 cohomology group $H^2(A,ker(p))$ of $A$ with coefficients in $ker(p)$. In fact, typically there is an embedding of the situation into homotopical algebra/higher algebra such that this cohomology group is given by the homotopy classes of morphisms to a second delooping object $B ker(p)$ (in the context of groups: the delooping 2-group)

$H^2(A,ker(p)) \;\simeq\; \left\{ A \overset{\phi}{\longrightarrow} B ker(p) \right\}_{/homotopy}$

and under this identification the central extension is the homotopy fiber of the cocycle $\phi$ and the short exact sequence (1) is part of the long homotopy fiber sequence to the left induced by $\phi$:

$\array{ ker(p) &\overset{\iota}{\longrightarrow}& \widehat{A} \\ && \Big\downarrow \\ && A &\overset{ \phantom{AA}\phi\phantom{AA} }{\longrightarrow}& B ker(p) }$