nLab central extension

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 not 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 $\phi$:

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

Examples

• A Poisson bracket Lie algebra is a central extension of Lie algebras of a Lie algebra of Hamiltonian vector fields,

• the corresponding quantomorphism group is a central extension of groups of the diffeological group of Hamiltonian symplectomorphism.

• A Heisenberg group is a sub-central extension such a quantomorphism group-extension, over Hamiltonian symplectomorphisms on a Lie group that act by left multiplication.

• A spin group is a central extension of a special orthogonal group by the group of order 2 $\mathbb{Z}/2$.

• …

Related concepts

• universal central extension, maximal central extension

• higher central extension

• universal higher central extension

• nilpotent group

References

See also

• Wikipedia, Central extension

Discussion of application in physics:

• G. M. Tuynman, W. A. Wiegenbrinck, Central extensions and physics, J. Geom. Physics 4:2(1987), 207–258.(doi:10.1016/0393-0440(87)90027-1)