Lie brackets on Hopf algebra cohomology
Lie brackets on Hopf algebra cohomology
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DOI：
10.2140/pjm.2022.316.395
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发表时间：
2021-01
影响因子：
0.6
通讯作者：
Tek.in Karadaug;S. Witherspoon
中科院分区：
文献类型：
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作者：
Tek.in Karadaug;S. Witherspoon
根据 Farinati、Solotar 和 Taillefer 的工作，已知拟三角形 Hopf 代数的 Hopf 代数上同调（作为 Gerstenhaber 括号下的分级李代数）是阿贝尔的。出于这对于非准三角 Hopf 代数是否成立的问题，我们证明了 Hopf 代数上同调上的 Gerstenhaber 括号可以通过任意投影解析来表达，使用 Volkov 的同伦提升推广到一些精确的幺半群范畴。这是我们更一般结果的一个特例，即上同调的括号运算在精确幺半群函子下得以保留——这样的函子是将 Hopf 代数上同调嵌入到 Hochschild 上同调中。因此，我们证明了 Hopf 代数上同调上的李结构对于所有量子基本阿贝尔群来说都是正度阿贝尔的，其中大多数是非准三角形的。
By work of Farinati, Solotar, and Taillefer, it is known that the Hopf algebra cohomology of a quasi-triangular Hopf algebra, as a graded Lie algebra under the Gerstenhaber bracket, is abelian. Motivated by the question of whether this holds for nonquasi-triangular Hopf algebras, we show that Gerstenhaber brackets on Hopf algebra cohomology can be expressed via an arbitrary projective resolution using Volkov’s homotopy liftings as generalized to some exact monoidal categories. This is a special case of our more general result that a bracket operation on cohomology is preserved under exact monoidal functors—one such functor is an embedding of Hopf algebra cohomology into Hochschild cohomology. As a consequence, we show that this Lie structure on Hopf algebra cohomology is abelian in positive degrees for all quantum elementary abelian groups, most of which are nonquasi-triangular.