The main result of the paper is the characterization of those locally compact quantum groups with projection, i.e. non-commutative analogs of semidirect products, which are extensions as defined by L. Vainerman and S. Vaes. It turns out that quantum groups with projection are usually not extensions.We discuss several examples including the quantum $\mathrm{U}_q(2)$. The major tool used to obtain these results is the co-duality for coideals in algebras of functions on locally compact quantum groups and the concept of a normal coideal.

中文翻译：

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具有局部紧凑量子群的投影和扩展的量子群

本文的主要结果是表征那些具有投影的局部紧凑量子群，即半直接乘积的非可交换类似物，这是由L. Vainerman和S. Vaes定义的扩展。事实证明，具有投影的量子组通常不是扩展。我们讨论几个示例，包括量子$ \ mathrm {U} _q（2）$。用于获得这些结果的主要工具是局部紧凑量子群上的函数代数中的同余体的对偶性和正常同体的概念。