Heisenberg代数的范畴化和MacMahon函数

发布时间：2018-03-21 12:53

本文选题：Heisenberg代数　切入点：MacMahon函数　出处：《中国科学:物理学力学天文学》2017年02期 　论文类型：期刊论文

【摘要】：作为一类基本的无限维李代数结构,Heisenberg代数在场论中扮演了很重要的角色.在经典理论中,它是利用自由谐振子生成的.这样的自由谐振子在表示论中可以看作是升箅子和降箅子.在范畴论中,它们是范畴之间的函子,满足一些特珠的性质,因此看起来像相对应的代数箅子.本文从一维向量空间出发,把Cautis和Licata的方法推广到单个形变Heisenberg代数,'H_(Z([t,t~(-1)]))的情况,给出了它的范畴化'H.在这样的构造中,'H为一个2-范畴,它的1-态射构成的集合包含了Heisenberg代数中自由谐振子的范畴化,它的所有2-态射组成了一个分次向量空间.在这个范畴中,2-态射决定了1-态射的同构类,即范畴的Grothendieck环.2-态射上的分次导致了Heisenberg代数的一个形变参数,并且也因此使本文证明了,'H的Grothendieck环为,'H_(Z([t,t~(-1)])).本文同时给出了范畴,'H的一个Fock表示.从'H的Fock表示中可以看到,2-态射上的分次可以由与对称群相关的表示导出范畴中的上同调次数平移来实现.作为Heisenberg代数范畴化的应用,本文还讨论了与三维Young图的MacMahon函数相关的配分函数.这篇文章的结果期望有更进一步的应用.
[Abstract]:As a basic infinite dimensional lie algebra structure, Heisenberg algebra plays an important role in field theory. It is generated by using free harmonic oscillators. Such free harmonic oscillators can be regarded in representation theory as ascending and descending gratings. In category theory, they are functors between categories and satisfy the properties of some special beads. In this paper, the method of Cautis and Licata is extended to the case of a single deformed Heisenberg algebra H _ S ([t ~ t ~ (1) ~ (-1)]), and its categorization is given. In this kind of structure, H is a 2-category. The set of its 1-morphisms contains the categorization of free harmonic oscillators in Heisenberg algebras, and all its 2-morphisms form a graded vector space. In this category, 2-morphisms determine the isomorphism of 1-morphisms. That is, the degree on the Grothendieck ring of category. 2-morphism leads to a deformed parameter of Heisenberg algebra. It is also proved in this paper that the Grothendieck ring of H is a Fock representation of the category H, and that the graded order over a morphism can be derived from the representation of a symmetric group from the Fock representation of H, and a Fock representation of the category H is also given in this paper. In this paper, it is also proved that the Grothendieck ring of H is a Grothendieck ring of H ([t t t + 1]). From the Fock representation of H, it can be seen that the graded order over a morphism can be derived from the representation of symmetric groups. As an application of categorization of Heisenberg algebras, This paper also discusses the partition functions related to the MacMahon functions of 3D Young graphs. The results of this paper are expected to be further applied.
【作者单位】：河南大学数学与统计学院;首都师范大学数学科学学院;
【基金】：国家自然科学基金(编号:11031005,11426089,11447146,11401400) 北京市教育委员会重点项目(编号:KZ201210028032)资助
【分类号】：O152.5

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