nLab
quantum homogeneous space

Idea

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 $H$ and a right $H$-comodule $(E,\rho )$ (where $\rho :E\to E\otimes H$ is a right coaction), a subspace $A\subset E$ is coinvariant subspace foir coaction $\rho$ if $\rho (A)\subset A\otimes H$. If $(E,\rho )$ is a $H$-comodule algebra, viewed as a (formal dual to) noncommutative $\mathrm{Spec}H$-space, one can consider its coinvariant subalgebras. If $H=E$ and $\rho =\Delta$ then the coinvariant subalgebras represent (the formal duals of) quantum homogenous spaces of the (formal dual of) $H$.

For example, a quotient Hopf algebra $B$ of $H$, with quotient map $\pi :H\to B$, can be considered as representing a quantum subgroup. Then $B$ coacts on $H$ by map $(\mathrm{id}\circ \pi )\circ \Delta :H\to H\otimes B$. The space of $B$-coinvariants gives an example of a coinvariant subalgebra in $H$.

Literature

• Ulrich Krähmer, Dirac operators on quantum flag manifolds, Lett. Math. Phys. 67/1 (2004) 49-59, MR2005b:58009, doi, math.QA/0305071
• Zoran Škoda, Localizations for construction of quantum coset spaces, in “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications 61, pp. 265–298, Warszawa 2003, math.QA/0301090; Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–202, arXiv:0811.4770 MR2011b:14004
• A. U. Klimyk, K. Schmuedgen, Quantum groups and their representations, Springer 1997.
• H-J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990), no. 1-2, 167–195 MR92a:16047 doi
• K. De Commer, M. Yamashita, Tannaka-Krein duality for compact quantum homogeneous spaces. I. General theory, arxiv/1211.6552

category: noncommutative geometry