Let $H$ be a bialgebra, a right $H$-comodule algebra $E$ is an **$H$-extension** of a subalgebra $U\subset E$ if $U=E^{co H}$ is precisely the subalgebra of $H$-coinvariants. The $H$-extension $U\subset E$ is **cleft** if there exist a **convolution-invertible $H$-comodule map** $\gamma:H\to E$. In other words, there is also a map $\gamma^{-1}:H\to E$ (the convolution inverse of $\gamma$) such that for all $h\in H$, $\gamma(h_{(1)})\gamma^{-1}(h_{(2)}) = \epsilon(h) = \gamma^{-1}(h_{(1)})\gamma(h_{(2)})$.

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 $\gamma'=\gamma^{-1}(1)\gamma$ (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

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

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

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

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

Conversely, if $H$ is a Hopf algebra, every cocycled product $U\sharp_\sigma H$ with invertible cocycle $\sigma$ is cleft via $\gamma: h\mapsto 1\sharp h$ with convolution inverse $\gamma^{-1}(h) = \sigma^{-1}(Sh_{(2)},h_{(3)})\otimes h_{(1)}$ 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. Any Hopf–Galois extension with the normal basis property is necessarily a cleft extension (Doi Takeuchi 1986, Theorem 9).

- Yukio Doi, Mitsuhiro Takeuchi,
*Cleft comodule algebras for a bialgebra*, Comm. Alg.**14**(1986) 801–818 doi - Shahn Majid,
*Foundations of quantum group theory*, Cambridge University Press 1995, 2000. - Yukio Doi,
*Equivalent crossed products for a Hopf algebra*, Commun. Alg. 17 (1989) 12 doi - R. J. Blattner, Miriam Cohen, Susan Montgomery,
*Crossed products and inner actions of Hopf algebras*, Trans. Amer. Math. Soc. 298 (1986) 671–711

Chapter 7, Crossed products, in

- Susan 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

Last revised on April 25, 2024 at 12:38:43. See the history of this page for a list of all contributions to it.