双环群的Gerbal表示
Gerbal Representations of Double Loop Groups
摘要：
在约化李群及其李代数的循环群的表示论中，它们的非平凡二阶上同调类起着至关重要的作用，这些上同调类产生了它们的中心扩张（仿射Kac-Moody群和李代数）。循环群嵌入到C（（t））的连续自同构群GL_ \infty中，这些类来自GL_ \infty的第二上同调类.以类似的方式，双环群嵌入到C（（t））（（s））的自同构群中，记为GL_{\infty,\infty}，它具有非平凡的第三上同调。在本文中，我们解释了如何实现第三个上同调类的表示论的一组：它自然出现时，我们考虑表示范畴，而不是向量空间。我们称之为“格巴尔表象。“然后，我们构造了GL_{\infty,\infty}（以及双环群）的Gerbal表示，在无限维Clifford代数上的模范畴上实现了它的非平凡第三上同调类。这是一个二维模拟的费米福克表示的普通圈群。
英文摘要：
A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their non-trivial second cohomology classes which give rise to their central extensions (the affine Kac-Moody groups and Lie algebras). Loop groups embed into the group GL_\infty of continuous automorphisms of C((t)), and these classes come from a second cohomology class of GL_\infty. In a similar way, double loop groups embed into a group of automorphisms of C((t))((s)), denoted by GL_{\infty,\infty}, which has a non-trivial third cohomology. In this paper we explain how to realize a third cohomology class in representation theory of a group: it naturally arises when we consider representations on categories rather than vector spaces. We call them "gerbal representations." We then construct a gerbal representation of GL_{\infty,\infty} (and hence of double loop groups), realizing its non-trivial third cohomology class, on a category of modules over an infinite-dimensional Clifford algebra. This is a two-dimensional analogue of the fermionic Fock representations of the ordinary loop groups.
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