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On the adjoint representation of a hopf algebra
Proceedings of the Edinburgh Mathematical Society ( IF 0.7 ) Pub Date : 2020-11-11 , DOI: 10.1017/s0013091520000358
Stefan Kolb , Martin Lorenz , Bach Nguyen , Ramy Yammine

We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{{\textrm ad\,fin}}$, defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e., $H$ is finitely generated as module over a cocommutative Hopf subalgebra), we show that $H_{{\textrm ad\,fin}}$ is a Hopf subalgebra of $H$. This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any virtually pointed Hopf algebra. For general Hopf algebras, $H_{{\textrm ad\,fin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.

中文翻译:

关于hopf代数的伴随表示

我们考虑 Hopf 代数的伴随表示$H$专注于局部有限部分,$H_{{\textrm ad\,fin}}$,定义为所有有限维子表示的总和。对于虚拟共交换$H$(IE,$H$被有限地生成为协交换 Hopf 子代数上的模),我们证明$H_{{\textrm ad\,fin}}$是一个 Hopf 子代数$H$. 这是这里证明的事实的结果,即局部有限部分在任何虚拟指向的 Hopf 代数的模范畴上产生一个张量函子。对于一般 Hopf 代数,$H_{{\textrm ad\,fin}}$被证明是一个左共理想子代数。我们还从群论中为 Hopf 代数证明了 Dietzmann 引理的一个版本。
更新日期:2020-11-11
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