cotensor product

Given a monoidal category \mathcal{M} and a coalgebra CC in \mathcal{M} denote by C\mathcal{M}^{C} (resp. C resp. {}^{C}\mathcal{M}) the category of right (resp. left) C{C}-comodules; similarly for an algebra EE, denote by E{}_E\mathcal{M} (resp. E\mathcal{M}_E) the category of left EE-modules (right EE-modules); if the monoidal category is symmetric or there is instead an appropriate distributive law, then there are extensions of this notation to bimodules, bicomodules, relative Hopf modules, entwined modules etc. e.g. E B{}_E\mathcal{M}^B for left-right relative (E,B)(E,B)-Hopf modules where EE is a BB-comodule algebra over a bialgebra BB.

Let kk be a commutative unital ring, and let \mathcal{M} be kk-linear with (in particular it has zero morphisms). Given a coalgebra CC in \mathcal{M}, a left CC-comodule (N,ρ N:NNC)(N,\rho_N:N\to N\otimes C), a right CC-comodule (M,ρ M:MCM)(M,\rho_M:M\to C\otimes M), their cotensor product is an object in \mathcal{M} given by

NM:=ker(ρ Nid Mid Nρ M). N \Box M := \mathrm{ker} (\rho_N \otimes \mathrm{id}_M - \mathrm{id}_N \otimes \rho_M ).

If equalizers exist in \mathcal{M}, this formula extends to a bifunctor

= C: C× C.{}\Box = \Box^{C} : \mathcal{M}^{C} \times {}^{C}\mathcal{M} \rightarrow \mathcal{M}.

If BB is a bialgebra in \mathcal{M} and EE is a right BB-comodule algebra then the same formula defines a bifunctor

: E B× B E.\Box : {}_{E}\mathcal{M}^{B} \times {}^{B}\mathcal{M} \rightarrow {}_{E}\mathcal{M}.

Let now =( kMod, k)\mathcal{M}=({}_k\mathrm{Mod},\otimes_k) be the symmetric monoidal category of kk-modules.

Let DD be another kk-coalgebra, with coproduct Δ C\Delta_C. If DD is flat as a kk-module (e.g. kk is a field), and NN a left DD- right CC-bicomodule, then the cotensor product NMN \Box M is a DD-subcomodule of N kMN \otimes_k M. In particular, under the flatness assumption, if π:DC\pi : D \rightarrow C is a surjection of coalgebras then DD is a left DD- right CC-bicomodule via Δ D\Delta_D and (idπ)Δ D(\id \otimes \pi) \circ \Delta_D respectively, hence Ind C D:=D C\mathrm{Ind}^D_C := D \Box^C - is a functor from left CC- to left DD-comodules called the induction functor for left comodules from CC to DD.

Cotensor products in noncommutative geometry appear in the role of space of sections of a associated vector bundles of quantum principal bundles (which in affine case correspond to Hopf-Galois extensions). See e.g.

  • S. Majid, Foundations of quantum groups theory, 2nd extended edition, paperback, Cambridge Univ. Press 2000.

For a nonaffine extension of the sections of associated quantum vector bundle, using localization theory see

  • Z. Škoda, Coherent states for Hopf algebras, Lett. Math. Phys. 81 (2007), no. 1, 1–17. arXiv:math.QA//0303357

In Hopf algebra theory, cotensor products appear as early as in

  • John W. Milnor, John C. Moore, On the structure of Hopf algebras. Ann. of Math. (2) 81 1965 211–264.

The authors mention that they learned the notion from Mac Lane who knew it earlier in more general contexts. An important problem is that the cotensor product of bicomodules is in general (even for = kMod\mathcal{M}={}_k\mathrm{Mod}) not associative, even up to an isomorphism. Cotensor products play a prominent role in various treatments of Galois theory in noncommutative geometry; a particularly geometric approach is within a version of noncommutative algebraic geometry based on usage of monoidal categories, as sketched in

Revised on November 12, 2012 02:34:22 by Zoran Škoda (