Zoran Skoda
cleft extension of a space cover


Recall the category of noncommutative space covers in the sense of Kontsevich-Rosenberg. The objects are called here simply space covers; an object is given by a ring UU, a UU-coring CC over UU and a structure map s:UUCs:U\otimes U\to C which is a map of corings and epimorphism of UU-bimodules. (see also: noncommutative space as a cover).

Let HH be a Hopf algebra. We are going to generalize the cleft HH-extensions of rings to cleft HH-extensions of space covers.


Let (U,C,s)(U,C,s) be a space cover with structure map s=s C:UUCs=s_C:U\otimes U \to C and HH a Hopf algebra. Then a HH-cleft extension of (U,C,s)(U,C,s) consists of the following data:

  • a right HH-comodule algebra EE which is an HH-cleft extension of U=E coHU = E^{co H}

  • an EE-coring DD in the category of right HH-comodules equipped with a monomorphism CDC\hookrightarrow D of UU-bimodules such that the image is precisely the subbimodule of HH-coinvariants in DD; we identify CC with its image and require that DD is generated as an EE-bimodule with CC.

  • a structure map s D:EEDs_D:E\otimes E\to D such that (E,D,s D)(E,D,s_D) is a space cover, and s D:EEDs_D:E\otimes E\to D is a map in the monoidal category of right HH-comodules

  • extension of UU-bimodules CDC\hookrightarrow D is cleft compatibly with cleavage of UEU\hookrightarrow E in the following sense. Let γ:HE\gamma:H\to E be the cleavage of EE. Then it induces an measuring (in smash case an action) :HUU\triangleright:H\otimes U\to U as well known, i.e. hu=γ(h (1))uγ 1(h (2))h\triangleright u = \gamma(h_{(1)})u \gamma^{-1}(h_{(2)}). Now there should be left and right cleavage on DD in the sense that there are two “extensions” of this measuring to a semimeasuring of CC (we will take these semimeasurings as a part of the structure). This means that for all uUu\in U, cCc\in C and k=1,2k = 1,2 k:HCC\triangleright_k:H\otimes C\to C is a map such that

    h 1(s C(u1))=s C((hu)1) h\triangleright_1 (s_C(u\otimes 1)) = s_C((h\triangleright u)\otimes 1)
    h 2(s C(1u))=s C(1(hu)) h\triangleright_2 (s_C(1\otimes u)) = s_C(1\otimes (h\triangleright u))
    h k(s C(11))=ϵ(h)s C(11) h\triangleright_k (s_C(1\otimes 1)) = \epsilon(h) s_C(1\otimes 1)
    h 1(uc)=(h (1)u)(h (2) 1c) h\triangleright_1 (u c) = \sum (h_{(1)}\triangleright u)(h_{(2)}\triangleright_1 c)
    h 2(cu)=(h (1) 2c)(h (2)u) h\triangleright_2 (c u) = \sum (h_{(1)}\triangleright_2 c) (h_{(2)}\triangleright u)

    In fact, regarding that s Cs_C is a map of UU-bimodules, the first two properties follow from the other three. Now 1\triangleright_1 will be called the left and 2\triangleright_2 the right semimeasuring of HH on CC.

Constructing smash product case from cocycles

There is a construction which produces some examples from local data. Here I sketch this construction for the simpler case of a smash product rather than a general case of cleft product.

Namely, we start with (U,C,s)(U,C,s), Hopf action \triangleright of HH on UU with semimeasuring extensions k\triangleright_k as above, and a transition cocycle y:HCy:H\to C in the following sense:

  • y:HCy:H\to C is convolution invertible in the sense that there is y 1:HCy^{-1}:H\to C such that y 1(h (1)) Uy(h (2))=y(h (1)) Uy 1(h (2))=ϵ(h)1 U1C UC\sum y^{-1}(h_{(1)})\otimes_U y(h_{(2)}) = \sum y(h_{(1)})\otimes_U y^{-1}(h_{(2)}) = \epsilon(h) 1\otimes_U 1\in C\otimes_U C
  • yy satisfies the cocycle condition:
    y(h (1)) Uy(h (2))=Δ C(y(h)) \sum y(h_{(1)})\otimes_U y(h_{(2)}) = \Delta_C(y(h))

    Notice that this equation on the left involves comultiplication in the Hopf algebra, while on the right hand side the comultiplication of the coring.

  • yy convolution intertwines the action \triangleright of HH on UU in the sense that for all uUu\in U and for all hHh\in H the two conditions
    Δ C((h (1)u)y(h (2)))=y(h (1)) U(h (2) 1s C(1u)) \Delta_C(\sum (h_{(1)}\triangleright u) y(h_{(2)})) = \sum y(h_{(1)}) \otimes_U (h_{(2)}\triangleright_1 s_C(1\otimes u))
    (h (1) 2s C(u1)) Uy(h (2))=Δ C(y(h (1))(h (2)u)) \sum (h_{(1)}\triangleright_2 s_C(u\otimes 1))\otimes_U y(h_{(2)}) = \Delta_C(\sum y(h_{(1)}) (h_{(2)}\triangleright u))

    hold as equalities in C UCC\otimes_U C.

One should have a theorem of the form that that U,C,s C,,yU, C, s_C, \triangleright,y suffice to construct D,s DD, s_D etc. such that the data form a cleft HH-extension of (U,C,s C)(U,C,s_C) and that every cleft HH-extension, up to a natural isomorphism comes from a cocycle of that form.

Then there is an idea how to compare different cocycles and finally the Čech cohomology related to such cocycles.

Now the main examples should come from covers by localizations and globalized torsors over them (there is a different theory of those). We will here call \mathcal{E} instead of previous EE as there is EE for another object in this example. If E λE_\lambda is a family of localizations of a right HH-comodule algebra EE, then we say that the family is coaction compatible if in each localization there is a (automatically unique) extension of the coaction ρ\rho to the localization ρ λ:E λE λH\rho_\lambda :E_\lambda\to E_\lambda\otimes H as an algebra map, and then it is automatically a comodule algebra. This means that ρ λi λ=(i λH)ρ\rho_\lambda\circ i_\lambda = (i_\lambda\otimes H)\otimes \rho, where i λ:EE λi_\lambda:E\to E_\lambda is the localization map (the component of the unit of the adjunction between the localization functor and its adjoint, taken at the object EE). Then U λ=(E λ) coHU_\lambda = (E_\lambda)^{co H} for ρ λ\rho_\lambda coaction). Define E λμ=Q μ(E λ)=E μ EE λE_{\lambda\mu} = Q_\mu(E_\lambda) = E_\mu\otimes_E E_\lambda, U λμ=(E λμ) coHU_{\lambda\mu} = (E_{\lambda\mu})^{co H}, U= λU λU = \prod_\lambda U_\lambda, C= λμU λμC = \prod_{\lambda\mu}U_{\lambda\mu}, = λE λ\mathcal{E} = \prod_\lambda E_\lambda and D= λμE λμD = \prod_{\lambda\mu} E_{\lambda\mu}. The action 1\triangleright_1 is then estricting to 1 λu μ1_\lambda\otimes u_\mu as an extension of action λ\triangleright_\lambda on u μu_\mu understood as an element in U μλU_{\mu\lambda} (along canonical map from U μU_\mu), and similarly for the other leg. Thus we have, for uU μu\in U_\mu,

(h (1) μu)y μλ(h (2))=y μλ(h (1))(h (2) λu) (h_{(1)}\triangleright_\mu u) y_{\mu\lambda}(h_{(2)}) = y_{\mu\lambda}(h_{(1)})(h_{(2)}\triangleright_\lambda u)

as an equality in U λμλC UCU_{\lambda\mu\lambda}\subset C\otimes_U C,
where y μλy_{\mu\lambda} is the projection of yy to U λμ=s C(U μU λ)U_{\lambda\mu} = s_C(U_\mu\otimes U_\lambda). One denotes the restriction s μλ=s C|U λU μs_{\mu\lambda} = s_C|U_\lambda\otimes U_\mu. Notice that λ\triangleright_\lambda is here extended from U λU_\lambda to U μλU_{\mu\lambda} (one could also extend to U λμU_{\lambda\mu} in another leg). The cocycle condition is for every λ,μ,ν,h\lambda,\mu,\nu,h we have y λμ(h (1))y μν(h (2))=y λν(h)y_{\lambda\mu}(h_{(1)})y_{\mu\nu}(h_{(2)}) = y_{\lambda\nu}(h) in U νμλU_{\nu\mu\lambda}. The convolution invertibility is for every μ\mu, y μν(h (1))y νμ(h (2))=ϵ(h)1y_{\mu\nu}(h_{(1)})y_{\nu\mu}(h_{(2)}) = \epsilon(h)1 in U μU_{\mu}. Though one should check if this fits with the coring version (it looks like there is U μλU_{\mu\lambda}-condition there as well, what is strange, but one can not isolate.

Finally at least in a smash product case one expects some sort of DCHD\cong C\otimes H coming from the data. For example, by that isomorphism of HH-comodules one should have (ch)(uh)=c(h (1)u)h (2)h(c\otimes h)(u\otimes h')= c(h_{(1)}\triangleright u)\otimes h_{(2)} h' but (uh)(ch)=u(h (1) 1c)h (2)h(u\otimes h)(c\otimes h') = u(h_{(1)}\triangleright_1 c) \otimes h_{(2)}h'; here 1\triangleright_1 is used and not version 2, probably because it is right smash product and left actions. This looks a bit nonsymmetric.

Last revised on April 19, 2012 at 00:01:31. See the history of this page for a list of all contributions to it.