Heisenberg代数的范畴化和MacMahon函数  

Categorification of Heisenberg algebra and macMahon function

作　　者：王娜[1] 王志玺[2] 吴可[2] 杨洁[2] 杨紫峰[2] WANG Na WANG ZhiXi WU Ke YANG Jie YANG ZiFeng(School of Mathematics and Statistics, Henan University, Kaifeng 475001, China School of Mathematical Science, Capital Normal University, Beijing 100048, China)

机构地区：[1]河南大学数学与统计学院,开封475001 [2]首都师范大学数学科学学院,北京100048

出　　处：《中国科学：物理学、力学、天文学》2017年第2期29-44,共16页Scientia Sinica Physica,Mechanica & Astronomica

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

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.As an elementary infinite dimensional Lie algebra structure,Heisenberg algebra plays an important role in field theory.In classical theory,it is generated by free oscillators which can also be viewed as ascending operators and descending operators in representation theory.From a point view of the theory of categories,these are functors between categories which satisfy some special properties and thus behave like the corresponding algebraic operators.Starting from a one dimensional vector space,this paper constructs a categorification ＇H of a deformed Heiserberg algebra ＇H_（Z（[t,t^（-1）]）） by Cautis and Licata＇s method.In this construction,＇H is a 2-category,where the set of 1-morphisms contains the categorified versions of the free oscillators in the Heisenberg algebra,and the 2-morphisms constitute a graded vector space.The 2-morphisms determine the isomorphism classes of 1-morphisms,this is,the Grothendieck ring of the category.The grading on the 2-morphisms gives rise to a deformation parameter on the Heisenberg algebra,and also enables the authors to show that the Grothendieck ring of ＇H is ＇H_（Z（[t,t^（-1）]））.The authors also give a Fock representation of the category ＇H.From the Fock representation of the category ＇H,we can see that such a grading can be realized by cohomological shiftings on derived categories which are related to representations of symmetric groups.As an application,the authors also discuss some partition functions related to the MacMahon function of 3D Young diagram.They expect further applications of the results of this paper.

关 键 词：HEISENBERG代数 MacMahon函数 范畴化 Grothendieck环 

分 类 号：O152.5[理学—数学]