高维代数VII：群化
Higher-Dimensional Algebra VII: Groupoidification
摘要：
群胚化是一种范畴化的形式，其中向量空间被群胚取代，线性算子被群胚的跨度取代。我们介绍了这个想法，详细阐述了“degroupoidification”：一个系统的过程，把groupoid和跨度到向量空间和线性算子。然后我们提出了群化的三个应用。第一个是费曼图。量子谐振子的希尔伯特空间自然地产生于有限集和双射的广群化。这允许一个纯粹的组合解释创造和湮灭运营商，他们的交换关系，场运营商，他们的正常有序的权力，最后费曼图。第二个应用是Hecke代数。我们解释如何groupoidify的Hecke代数与Dynkin图的变形参数q是一个素数的权力。我们用最简单的例子来说明这一点，来自A2 Dynkin图。在这个例子中，我们表明，解决方案的杨-巴克斯特方程内置到A2 Hecke代数自然产生的公理适用于有限域上的射影平面与q元素的射影几何。第三个应用是霍尔代数。我们解释了如何从一个简单的花边类的表示霍尔代数的标准建设可以被看作是一个例子degroupoidification。这反过来又提供了一种新的方法来分类--或者更准确地说，群化--量子群中与量子相关的正部分。
英文摘要：
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of "degroupoidification": a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang-Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field with q elements. The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of representations of a simply-laced quiver can be seen as an example of degroupoidification. This in turn provides a new way to categorify - or more precisely, groupoidify - the positive part of the quantum group associated to the quiver.
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