geometrically admissible action

It is often natural that a symmetry object acting on some space is in the category of spaces itself. For example, algebraic groups acts on varieties, topological groups on topological spaces, and group schemes over SS act on SS-schemes.

Suppose now we want to define an action of a Hopf kk-algebra HH on a noncommutative kk-scheme, where we view a Hopf algebra as a dual to a noncommutative analogue of an affine group object over Spec(k)Spec(k). In nonaffine situations noncommutative schemes are glued from algebras, but they are not described by a single algebra, but by a system of algebras; in fact a more canonical object is the category of quasicoherent sheaves which localizes to categories of modules over each of the algebras of the cover; these categories of modules are monadic over the category V= kModV = {}_{k}Mod which is the category of quasicoherent sheaves over Spec(k)Spec(k). The Hopf algebra, to be considered as a noncommutative space, has to be replace by the monoidal category = HMod\mathcal{M} = {}_{H}Mod which comes equipped with a canonical action on VV: tensor and forget the additional structure. The Hopf algebra should be reconstructed from the forgetful functor to VV; it is itself a comonoid in MM. What does it mean that HH acts on some noncommutative space represented by a category CC ? Well it has no meaning unless CC has a forgetful functor to VV as well (like in the case of possibly noncommutative kk-schemes). Now the action of Spec(H)Spec(H) corresponds to the action of the monodial category MM on CC. But not any action: the action which is compatible with the action of MM on VV. Such actions are called geometrically admissible actions of monoidal category. Admisibility is a notion relative to the action of the same monoidal category on the base category. In other words admissible actions are the actions of monoidal categories on categories which lift the defining action on the base. In special cases, the liftings correspond to the distributive laws. This is the origin of appearance of coalgebra bundles in noncommutative geometry: entwining structures induce lifts to admissible actions; however there are other cases of distributive laws and more general lifts which are not coming from entwinings between coalgebras and algebras.

Another example is given by the action of the monoidal category of sheaves on a topological group GG on a topological GG-space XX. The action Sh(G)Sh(G) on Sh(X)Sh(X) is given by the pushforward along the action G×XXG\times X\to X of the external tensor product of sheaves Sh(G)×Sh(X)Sh(G×X)Sh(G)\times Sh(X)\to Sh(G\times X).

As a general definition, we start with a fixed notion of a symmetry object as a monoidal category MM with distinguished action \lozenge on the base category VV. A geometrically admissible action ˜\tilde{\lozenge} in wider sense is any action of MM on some category CC over VV such that

M×C ˜ C M×V V\array{ M\times C&\stackrel{\tilde{\lozenge}}\to& C\\ \downarrow && \downarrow\\ M\times V&\stackrel{\lozenge}\to& V }


Remark: For admissible actions in much more narrow sense one also has a forgetful functor MVM\to V and one requires that the action below is in the fact factoring through that functor. In other words, the tensor product of VV lifts to an action of MM in Cat/VCat/V.

  • Z. Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–202, arXiv:0811.4770.

Created on February 11, 2011 at 18:34:19. See the history of this page for a list of all contributions to it.