symmetric monoidal (∞,1)-category of spectra
Given an object in algebra (such as an associative algebra, a group or a Lie algebra, etc.) then an extension (e.g. a group extension or Lie algebra extension etc.) is called a central extension if its kernel
is in the center of
This means in particular that 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 commute not just among themselves, but also with all other elements of .
Typically central extensions by some commutative algebraic object are classified by the suitable degree-2 cohomology group of with coefficients in . 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 (in the context of groups: the delooping 2-group)
and under this identification the central extension is the homotopy fiber of the cocycle and the short exact sequence (1) is part of the long homotopy fiber sequence to the left induced by :
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 .
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See also
Discussion of application in physics:
Last revised on April 17, 2021 at 14:58:52. See the history of this page for a list of all contributions to it.