nLab
central extension

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Idea

Given an object in algebra AA (such as an associative algebra, a group or a Lie algebra, etc.) then an extension A^pA\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)1ker(p)AιAA^ApAA1 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(A^)C(\widehat{A}) of A^\widehat A

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

This means in particular that ker(p)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)ker(p) commute noit just among themselves, but also with all other elements of A^\widehat A.

Typically central extensions by some commutative algebraic object ker(p)ker(p) are classified by the suitable degree-2 cohomology group H 2(A,ker(p))H^2(A,ker(p)) of AA with coefficients in ker(p)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 Bker(p)B ker(p) (in the context of groups: the delooping 2-group)

H 2(A,ker(p)){AϕBker(p)} /homotopy 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 ϕ\phi and the short exact sequence (1) is part of the long homotopy fiber sequence to the left induced by ϕ\phi:

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

Examples

References

See also

Discussion of application in physics:

  • G. M. Tuynman, W. A. Wiegenbrinck, Central extensions and physics (pdf)

Last revised on August 2, 2018 at 09:04:07. See the history of this page for a list of all contributions to it.