nLab cleft extension

Let HH be a Hopf algebra, a right HH-comodule algebra EE is an HH-extension of a subalgebra UEU\subset E if U=E coHU=E^{co H} is precisely the subalgebra of HH-coinvariants. The HH-extension UEU\subset E is cleft if there exist a convolution-invertible HH-comodule map γ:HE\gamma:H\to E.

If UEU\hookrightarrow E is a cleft HH-extension, then the cleavage γ\gamma can always be chosen normalized in the sense that γ(1)=1\gamma(1)=1; because if it is not normalized we can rescale γ\gamma to form a normalized cleavage γ=γ 1(1)γ\gamma'=\gamma^{-1}(1)\gamma (indeed, 11 is group-like, hence γ(1)\gamma(1) is invertible with inverse (γ(1)) 1=γ 1(1)(\gamma(1))^{-1}=\gamma^{-1}(1)).

It is easy to show that the rule

hu:=γ(h (1))uγ 1(h (2))h\triangleright u := \sum \gamma(h_{(1)})u\gamma^{-1}(h_{(2)})

defines a measuring :HUU\triangleright:H\otimes U\to U i.e. h(uv)=(h (1)u)(h (2)v)h\triangleright(uv)=\sum (h_{(1)}\triangleright u)(h_{(2)}\triangleright v) and if γ\gamma is chosen normalized, then h1=ϵ(h)1h\triangleright 1 = \epsilon(h) 1. Define a convolution invertible map σHom k(HH,U)\sigma\in Hom_k(H\otimes H,U) by

σ(h,k)=γ(h (1))γ(k (1))γ 1(h (2)k (2)),h,kH.\sigma(h,k) = \sum \gamma(h_{(1)})\gamma(k_{(1)})\gamma^{-1}(h_{(2)}k_{(2)}),\,\,\,\,\,h,k\in H.

Then the pair (,σ)(\triangleright,\sigma) defines the data for the cocycled crossed product algebra U σHU\sharp_\sigma H which is canonically isomorphic to BB as an HH-extension of UU1U σHU\cong U\otimes 1\hookrightarrow U\sharp_\sigma H, and i.e. as a right HH-comodule algebra with the isomorphism fixing UU as given.

Conversely, every cocycled product U σHU\sharp_\sigma H is cleft via γ:h1h\gamma: h\mapsto 1\sharp h and the cocycle σ\sigma built out of γ\gamma is the same one, which helped build the cocycled crossed product.

Every cleft extension is a particular case of a Hopf-Galois extension.

  • Y. Doi, Mitsuhiro Takeuchi, Cleft comodule algebras for a bialgebra, Comm. Alg. 14 (1986) 801–818 doi

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

  • S. Montgomery, Hopf algebras and their actions on rings, CBMS 82, AMS 1993.

There are generalizations for Hopf algebroids:

There are some globalizations of cleft extensions. For the smash product case of the globalization some details are written in

Last revised on December 8, 2019 at 15:13:41. See the history of this page for a list of all contributions to it.