nLab quantum homogeneous space


There are many variants of a generalization of the concept of homogeneous space to noncommutative geometry. Some of them are sometimes referred to as quantum homogeneous spaces. Variants of quantum flag manifolds, are here the main examples in several different frameworks.

Homogeneous spaces for coactions of Hopf algebras

Most often here one considers the homogeneous spaces for Hopf algebras (viewed as quantum groups).

Given a Hopf algebra HH and a right HH-comodule (E,ρ)(E,\rho) (where ρ:EEH\rho:E\to E\otimes H is a right coaction), a subspace AEA\subset E is coinvariant subspace for coaction ρ\rho if ρ(A)AH\rho(A)\subset A\otimes H. If (E,ρ)(E,\rho) is a HH-comodule algebra, viewed as a (formal dual to) noncommutative SpecHSpec H-space, one can consider its coinvariant subalgebras. If H=EH = E and ρ=Δ\rho = \Delta then the coinvariant subalgebras represent (the formal duals of) quantum homogenous spaces of the (formal dual of) HH.

For example, a quotient Hopf algebra BB of HH, with quotient map π:HB\pi: H\to B, can be considered as representing a quantum subgroup. Then BB coacts on HH by map (idπ)Δ:HHB(id\circ\pi)\circ\Delta: H\to H\otimes B. The space of BB-coinvariants gives an example of a coinvariant subalgebra in HH.


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Last revised on April 13, 2023 at 14:32:38. See the history of this page for a list of all contributions to it.