代数量子群II。结构和示例
Algebraic quantum groups II. Constructions and examples
摘要：
设G是群，A是G上有限支撑的复变函数代数. G中的乘积在A上产生一个余积，使A成为一个乘子霍普夫代数。实际上，由于存在积分，我们得到一个代数量子群。设H是G的有限子群，考虑A中在H的双陪集上为常数的函数的子代数。余积一般不会使这个代数保持不变，但我们可以修改它，使它使子代数保持不变（在这个意义上，图像是在这个子代数与自身的张量积的乘子代数中）。然而，子代数上的修改的余积将不再是代数映射。因此，一般来说我们没有代数量子群，而是一个所谓的代数量子超群。*-代数量子群A中的类群投影以一种自然的方式产生 *-代数量子超群，非常类似于子群对与群相关联的 *-代数量子群所做的那样。在本文中，我们进一步推动这一结果。一方面，我们不再假设 *-结构，另一方面，我们允许类群投影属于A的乘子代数M（A），而不仅仅是A本身。这样做，我们不仅得到了一些著名的早期代数量子超群的例子，但也有一些有趣的新的。
英文摘要：
Let G be a group and let A be the algebra of complex functions on G with finite support. The product in G gives rise to a coproduct on A making it a multiplier Hopf algebra. In fact, because there exist integrals, we get an algebraic quantum group. Now let H be a finite subgroup of G and consider the subalgebra of functions in A that are constant on double cosets of H. The coproduct in general will not leave this algebra invariant but we can modify it so that it will leave the subalgebra invariant (in the sense that the image is in the multiplier algebra of the tensor product of this subalgebra with itself). However, the modified coproduct on the subalgebra will no longer be an algebra map. So, in general we do not have an algebraic quantum group but a so-called algebraic quantum hypergroup. Group-like projections in a *-algebraic quantum group A give rise, in a natural way, to *-algebraic quantum hypergroups, very much like subgroups do as above for a *-algebraic quantum group associated to a group. In this paper we push this result further. On the one hand, we no longer assume the *-structure while on the other hand, we allow the group-like projection to belong to the multiplier algebra M(A) of A and not only to A itself. Doing so, we not only get some well-known earlier examples of algebraic quantum hypergroups but also some interesting new ones.
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