论文标题
Hopf-Galois扩展和扭曲的Hopf代数
Hopf-Galois extensions and twisted Hopf algebroids
论文作者
论文摘要
我们表明，带有结构量子组$ h $的量子束$ p $ $ p $ $ p $ $ p $的ehresmann-schauenburg bialgebroid实际上是左Hopf algebroid $ l（p，h）$。我们进一步表明，如果$ h $是coquasitriangular，则$ l（p，h）$具有符合某些最小公理的抗dopode map $ s $。众所周知，具有基本$ b $的琐碎量子主要捆绑包或cleft hopf galois扩展名是Cocycle Cross Products $ b \ b \#_σh$用于Cocycle-action对（$ \ vartriangleright $，$σ$），我们查看某些“ $ \ vartrianglerigh $ s eaction ocation”是一个实用的“ $ \ vartrian type”。在这种情况下，我们还表明，相关的左HOPF代数有一个抗底螺纹，遵守我们的最小公理。我们表明，如果$ l $是任何左HOPF代数，那么它的cotwist $ l^q $也是以前的Bialgebroid Drinfeld Cotwist理论的扩展。我们表明，对于$ l（b \#_σh，h）= l（b \＃h）^{\tildeσ} $，用于hopf algebroid cotwist $ q = \tildeσ$。因此，开启关联类型的$σ$作为Drinfeld Cotwist出现在Hopf代数级别。我们将Aggine量子组$ \ hat {u_q（sl_2）} $和$ u_q（sl_2）$的量子Weyl组作为关联类型的示例。
We show that the Ehresmann-Schauenburg bialgebroid of a quantum principal bundle $P$ or Hopf Galois extension with structure quantum group $H$ is in fact a left Hopf algebroid $L(P,H)$. We show further that if $H$ is coquasitriangular then $L(P,H)$ has an antipode map $S$ obeying certain minimal axioms. Trivial quantum principal bundles or cleft Hopf Galois extensions with base $B$ are known to be cocycle cross products $B\#_σH$ for a cocycle-action pair ($\vartriangleright$,$σ$) and we look at these of a certain `associative type' where $ \vartriangleright$ is an actual action. In this case also, we show that the associated left Hopf algebroid has an antipode obeying our minimal axioms. We show that if $L$ is any left Hopf algebroid then so is its cotwist $L^ς$ as an extension of the previous bialgebroid Drinfeld cotwist theory. We show that in the case of associative type, $L(B\#_σH,H)=L(B\# H)^{\tildeσ}$ for a Hopf algebroid cotwist $ς=\tildeσ$. Thus, switching on $σ$ of associative type appears at the Hopf algebroid level as a Drinfeld cotwist. We view the affine quantum group $\hat{U_q(sl_2)}$ and the quantum Weyl group of $u_q(sl_2)$ as examples of associative type.
