nLab equivariant noncommutative algebraic geometry

This entry is intended to be about symmetry objects acting on noncommutative spaces and objects/constructions which are invariant or equivariant under their action.

Motivation

One would like to have symmetry objects like algebraic groups and Lie algebras in noncommutative geometry (including algebraic flavour). The group-like objects should be noncommutative spaces themselves, they should have representation theory, they should act on other noncommutative spaces, define quotient spaces and so on. One also expects to have a equivalence between the category of $G$-equivariant “sheaves” on a total noncommutative principal $G$-bundle (where $G$ is an appropriate symmetry object) and usual “sheaves” on the quotient. A first massive appearance were quantum groups, and one should be warned that quantum groups are not cogroup objects in the category of noncommutative rings, because they are Hopf algebras with respect to the tensor product rather than categorical coproduct of algebras.

Main characters

Here one should write about quantum groups (Drinfeld, Manin, Woronowicz, Jimbo, Lusztig, Faddeev-Reshetikin-Tahtajan, Majid), Hopf algebras, Hopf algebroids (quantum groupoids), quantum Lie algebras, entwinings/distributive laws, quantum flag varieties, (co)module (co)algebras, quantum principal bundles, associated bundles, Drinfel’d center, equivariant cyclic homology etc.

Literature

• V. G. Drinfel’d, Quantum groups, Proc. Int. Cong. Math. 1986, Vol. 1, 2 798–820, AMS 1987.

• S. Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.

• B. Parshall, J.Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.

• Zoran Škoda, Some equivariant constructions in noncommutative geometry, Georgian Math. J. 16 (2009) 1; 183–202 arXiv:0811.4770

• Yuri Manin, Quantum groups and noncommutative geometry, CRM, Montreal (1988)

• Ludwig Faddeev, N. Reshetikin, L. Tahtajan, Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 (1989) 178 (transl. Leningrad Math. J. 1 (1990), 193-225)

• Susan Montgomery, Hopf algebras and their action on rings, AMS 1994, 240p.

• Yan Soibelman, On the quantum flag manifold, Funk. analiz i ego pril. 26, 3, 90–92, 1992

• Z. Škoda, Localizations for construction of quantum coset spaces, math.QA/0301090, Banach Center Publ. vol.61, pp. 265–298, Warszawa 2003.

• Nikolai Reshetikhin, A. A. Voronov, Alan Weinstein, Semiquantum geometry, math.q-alg/9606007

• Tomasz Brzeziński, Shahn Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), no. 2, 467–492 (arXiv version).

• Brian Day, Ross Street, Monoidal bicategories and Hopf algebroids, Adv. Math. 129, (1997) 99-157 doi

category: algebraic geometry, noncommutative geometry