A Note on the Antipode for Algebraic Quantum Groups
A Note on the Antipode for Algebraic Quantum Groups
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DOI：
10.4153/cmb-2011-079-4
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发表时间：
2012-06
期刊：
影响因子：
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通讯作者：
L. Delvaux;A. Van Daele;Shuanhong Wang
中科院分区：
文献类型：
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作者：
L. Delvaux;A. Van Daele;Shuanhong Wang
最近，Beattie，Bulacu和Torrecillas证明了余Frobenius Hopf代数的对极的四次方的拉德福公式。在本文中，我们证明了这个公式可以证明任何正则乘子Hopf代数积分（代数量子群）。当然，这不仅包括有限维Hopf代数的情况，也包括任何具有积分的Hopf代数（co-Frobenius Hopf代数）。此外，事实证明，在这种更一般的情况下的证明，事实上，遵循几行从著名的公式获得较早的理论，正规乘子Hopf代数与积分。我们讨论这些公式和它们在这个理论中的重要性。我们也提到他们的推广，特别是（在某种意义上）更一般的局部紧量子群理论。这样做，也因为证明的主要结果本身是非常短的，目前的说明成为主要是一个短暂的性质。
Abstract Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra. In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals. We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.