nLab
cleft extension

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

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

It is easy to show that the rule

$$h▹u:=\sum \gamma (h_{(1)})u{\gamma }^{-1}(h_{(2)})$$h\triangleright u := \sum \gamma(h_{(1)})u\gamma^{-1}(h_{(2)})

defines a measuring $▹:H\otimes U\to U$ i.e. $h▹(\mathrm{uv})=\sum (h_{(1)}▹u)(h_{(2)}▹v)$ and if $\gamma$ is chosen normalized, then $h▹1=ϵ(h)1$. Define a convolution invertible map $\sigma \in {\mathrm{Hom}}_k(H\otimes H,U)$ by

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

Then the pair $(▹,\sigma )$ defines the data for the cocycled crossed product algebra $U{♯}_{\sigma }H$ which is canonically isomorphic to $B$ as an $H$-extension of $U\cong U\otimes 1\hookrightarrow U{♯}_{\sigma }H$, and i.e. as a right $H$-comodule algebra with the isomorphism fixing $U$ as given.

Conversely, every cocycled product $U{♯}_{\sigma }H$ is cleft via $\gamma :h\mapsto 1♯h$ and the cocycle $\sigma$ built out of $\gamma$ 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, M. Takeuchi, Cleft comodule algebras for a bialgebra, Comm. Alg. 14 (1986) 801–818

• 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:

• Gabriella Böhm, Tomasz Brzeziński, Cleft extensions of Hopf algebroids, Appl. Cat. Str. 14 (2006) 431-469; math.QA/0510253

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

• Zoran Škoda, Čech cocycles for quantum principal bundles, arxiv/1111.5316
• Z. Škoda, cleft extension of a space cover