Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
Given a group and two subgroups and , then the set of double cosets, , is composed of equivalence classes of elements of , where two elements are regarded as equivalent if they differ by left multiplication with an element in and right multiplication with an element of .
This construction generalises to topological groups and Lie groups, where it is called a double coset space. The double coset space for a compact Lie group and and closed connected subgroups, with acting freely on the singe coset space , is also known as a biquotient space.
From the nPOV, a weak form of the double coset construction is often more natural, , and can be defined as the homotopy pullback of maps of delooped groups, and . This holds more generally for any three ∞-groups, with any homomorphisms and .
the following needs to state the assumption on the kind of group, e.g. finite group, compact Lie groups, …
Double coset decompositions are useful in representation theory, for example in George Mackey‘s formula for the restriction back to of a module induced from . Let be a linear representation of . Then
where , and is the representation of which is as a vector space, but with action where now . This formula, sometimes called Mackey’s decomposition theorem, amounts to the Beck-Chevalley condition for the homotopy pullback square. It appears, as the Mackey axiom, in one of the original definitions of Mackey functor (Bouc 02).
The Gromoll-Meyer sphere – an exotic 7-sphere – arises as the biquotient of Sp(2) by two copies of Sp(1).
Serge Bouc, Mackey functors, (pdf)
Burt Totaro, Cheeger manifolds and the classification of biquotients (arXiv:math/0210247)
Last revised on August 1, 2019 at 07:48:40. See the history of this page for a list of all contributions to it.