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 quotients and so on. 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.
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.
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