nLab cleft extension

Definition

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 $h\in H$, $\gamma(h_{(1)})\gamma^{-1}(h_{(2)}) = \epsilon(h) = \gamma^{-1}(h_{(1)})\gamma(h_{(2)})$.

Normalization of $\gamma$

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

Correspondence between cleft extensions and cocycled crossed products

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.

Proposition

Every cleft $H$-extension $U\hookrightarrow E$ is a Hopf-Galois extension, namely the inverse of the canonical map $E\otimes_U E\to E\otimes H$ is given by $e\otimes h\mapsto e\gamma^{-1}(h_{(1)})\otimes_U\gamma(h_{(2)})$, where $\gamma:H\to E$ is the cleaving map.

Any Hopf–Galois extension with the normal basis property is necessarily a cleft extension (Doi Takeuchi 1986, Theorem 9).

Literature

• 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