Bornological quantum groups
Bornological quantum groups
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DOI：
10.2140/pjm.2008.235.93
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发表时间：
2005-11
影响因子：
0.6
通讯作者：
Christian Voigt
中科院分区：
文献类型：
--
作者：
Christian Voigt
我们引入并研究了出生量子群的概念。这将van Daele意义下的代数量子群理论从代数环境推广到了边界向量空间的框架。利用出生向量空间，后一种理论的范围可以大大扩展。特别地，该理论涵盖了任意局部紧群及其对偶的光滑卷积代数。此外，幂零李群的Schwartz代数自然地是出生量子群，类似地，人们可以考虑由各种衰减条件定义的有限生成离散群上的函数的代数。另一个例子来源于Rieffel意义下的形变量子化。除了描述这些例子外，我们还得到了一些关于边界量子群的一般性结果。特别地，我们构造了一个出生量子群的对偶，并证明了Pontrjagin对偶定理。
We introduce and study the concept of a bornological quantum group. This generalizes the theory of algebraic quantum groups in the sense of van Daele from the algebraic setting to the framework of bornological vector spaces. Working with bornological vector spaces, the scope of the latter theory can be extended considerably. In particular, the bornological theory covers smooth convolution algebras of arbitrary locally compact groups and their duals. Moreover Schwartz algebras of nilpotent Lie groups are bornological quantum groups in a natural way, and similarly one may consider algebras of functions on finitely generated discrete groups defined by various decay conditions. Another source of examples arises from deformation quantization in the sense of Rieffel. Apart from describing these examples we obtain some general results on bornological quantum groups. In particular, we construct the dual of a bornological quantum group and prove the Pontrjagin duality theorem.