• 
    

    
    

      99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

      A Class of Simple Modules for Electrical Lie Algebra of Type D5

      2023-05-18 14:37:08CHENYulu陳雨露SHENRanZHANGJiangang張建剛
      關鍵詞:雨露

      CHEN Yulu(陳雨露), SHEN Ran(申 冉)*, ZHANG Jiangang(張建剛)

      1 College of Science, Donghua University, Shanghai 201620, China

      2 College of Science &Mathematics, Shanghai Normal University, Shanghai 200233, China

      Abstract:The simple modules for electrical Lie algebra of type D5 were investigated. The sufficient and necessary criteria of the simple Z-graded highest weight modules were established by means of determining the singular vectors of the Verma modules. The simple highest weight module is isomorphic to either that for the symplectic Lie algebra sp4 or Verma module.

      Key words:electrical Lie algebra; simple module; highest weight module

      Introduction

      We denote the sets of all integers, nonnegative integers, and complex numbers byZ,Z+, andC, respectively. And we denote the universal enveloping algebra of Lie algebraεoverCbyU(ε).All vector spaces and algebras are overCthroughout this paper.

      The study of electrical networks could trace back more than a century when Ohm and Kirchhoff researched electrical circuits. It remains a classic object in many branches of mathematics such as graph theory[1], as well as a wide range of applications in materials science and medical imaging[2]. Electrical Lie algebras, originating from the study of two operations, namely, adjoining a boundary spike and adjoining a boundary edge to a circular planar electrical network[3], were introduced by Lametal.[4]who gave the definition of electrical Lie algebras of finite Dynkin type and the electrical Serre relations. They mainly studied the electrical Lie groups and the structure of electrical Lie algebras of typeA2n.And the simple modules ofεA2n?sp2nhave been studied in Ref. [5]. They also showed that the electrical Lie algebraεG2was six-dimensional and conjectured that the dimension of the electrical Lie algebra of typeXcoincided with the number of positive roots |Φ(X)+|, whereΦ(X)+was the set of positive roots ofX.

      Su[6]proved the dimension conjecture of electrical Lie algebras for classical types and studied their structures. He showed thatεA2n+1?sp2n+1,εBn?εAn⊕εAn-1andεC2n?εA2n(Vλ⊕V0), whereV0was the trivial representation andVλwas the simple highest weight representation for the symplectic Lie algebrasp2n.Therefore, the structure ofεA2n+1was obtained since that ofsp2n+1had been studied[7-8]. The universal enveloping algebra ofεA2n+1, also known as a symplectic oscillator Lie algebra or Jacobi Lie algebra, is an infinitesimal Hecke algebra[9-10], and the categories of weight modules forεA2n+1were classified in Ref. [11].

      Recently, Gaoetal.[12]studied the structure and representations forεD4.They showedεD4?sl2was a 3-step nilpotent Lie algebra constructed from the colored complete bipartite graphK2,2with coloring mapc, and classified all the simple highest weight modules. Besides, they provedεG2?sch⊕Cz, whereschis the centerless Schr?dinger algebra. Thus the representations forεG2were known as those for Schr?dinger algebra, which were well studied in Refs.[13-17].

      The electrical Lie algebra of typeD5is isomorphic to the semidirect product ofsp4with a 2-step nilpotent Lie algebra constructed from the colored 4-order tournamentT4.

      In this paper, the electrical Lie algebraεD5is viewed as aZ-graded Lie algebra since it has more practical applications. And we study a class of simple highest weight modules for it. The paper is organized as follows. In section 1, some basic definitions and results are collected. In section 2, we give the classification of a class of simple highest weight modules forεD5.And in section 3, we get some conclusions.

      1 Preliminaries

      In this section, the definition of electrical Lie algebra for typeXand a spanning set of it were given in Ref. [4] as follows.

      Definition1LetXbe a Dynkin diagram of finite type, andA=(aij) be the corresponding Cartan matrix. Define the electrical Lie algebraεXof typeXto be the Lie algebra generated by {ei}, whereiruns over the vertex indices inX, modulo the relations

      We studied the structure ofεD5and gave the following structure theorem.

      Theorem1We claim that

      εD5?sp4g(T4,c)

      with basis

      wheresp4is the symplectic Lie algebra andg(T4,c) is a 2-step nilpotent Lie algebra constructed from the colored 4-order tournamentT4.The brackets ofεD5are defined in Table 1.

      2 A Class of Highest Weight Modules for εD5

      The electrical Lie algebraε=εD5is a direct sum of the subalgebra isomorphic tosp4and an ideal. It isZ-graded with respect to the adjoint action ofh.Fori∈Zwe set

      εi:={x∈ε|[h,x]=ix},

      ε0:=spanC{h,p1,p2,p3,v1,v2,w1,z},

      Anε-moduleMis called the highest weight module if there is a vector1λsuch thatM=U(ε)1λ,ε+1λ=0, andh1λ=λ1λ, whereλ∈C.Thus in other wordsM=U(ε≤0)1λ.Then

      is a basis ofMdue to the Poincaré-Birkhoff-Witt Theorem.

      LetMbe a simple highest weightε-module with the highest weight vector1λ.Verma module overεis the1-dimensionalε+⊕ε0-module defined by

      ε+1λ=0,h1λ=λ1λ,p11λ=λ11λ,p21λ=λ21λ,p31λ=λ31λ,

      v11λ=μ11λ,v21λ=μ21λ,w11λ=μ1λ,z1λ=c1λ,

      where (λ,λ1,λ2,λ3,μ1,μ2,μ,c)∈C8.

      From Table1, we obtain the following lemma.

      Lemma1Claim that

      λ1=λ2=λ3=μ1=μ2=0,

      and

      μ=c.

      ProofFrom Table 1, we know that

      Hence

      which impliesλ2=0.Similarly,λ1=λ3=μ1=μ2=0.

      Then from

      w1-z=[v1,v2]=v1v2-v2v1,

      we have

      (w1-z)1λ=(μ-c)1λ=(v1v2-v2v1)1λ=

      (μ2μ1-μ1μ2)1λ=0,

      and

      μ=c.

      Hence the Verma moduleV=V(λ,μ), where (λ,μ)∈C2.

      Any simple highest weight module with the highest weight vector1λis a simple quotient ofV(λ,μ).Therefore, to classify the simple highest weightε-modules, it suffices to classify all simple quotients ofV(λ,μ).Thus we determine the maximal proper submodules forV(λ,μ).

      The following identities can be proved through direct computations.

      Lemma2Fork∈Z+, we have

      We discuss the simple highest weight modulesMin two situations:μ=0 andμ≠0 as follows.

      Proposition1Mis isomorphic to a simple highest weightsp4-module forμ=0.

      Table 1 Brackets in εD5

      which is a nonzero proper submodule. It forces

      wherea(r,k,s,t)∈C{0}=C*.Similarly

      Therefore,Mis isomorphic to a simple highest weightsp4-module, which was studied in Ref. [18] with two fundamental weights,L1andL2.

      Proposition2Verma moduleV(λ,μ) is irreducible ifμ≠0.

      ProofRecall that a vectoruis called a singular vector in a moduleV(λ,μ)ifε+u=0. It may be written in the form

      0=

      1λ,

      1λ,

      1λ,

      we know thatn=0.Thus,u∈C1λ, and the theorem follows.

      To summarize, we classify a class of the simple highest weight modules forεD5as follows.

      Theorem2LetMbe a simple highest weightε-module with the highest weight vector1λ.For (λ,μ)∈C2,V(λ,μ) is the Verma module overε.

      (1) Ifμ=0, thenMis isomorphic to a simple highest weightsp4-module.

      (2) Ifμ≠0, thenM?V(λ,μ) sinceV(λ,μ) is irreducible.

      3 Conclusions

      In this paper, we classify the simpleZ-graded highest weight modulesMfor electrical Lie algebra of typeD5.We seek singular vectors to determine the maximal proper submodule of Verma moduleV(λ,μ).Then the sufficient and necessary criteria for simple modules are obtained.

      Ifμ=0,Mis isomorphic to a simple highest weightsp4-module, which has two fundamental weights,L1andL2.The highest weight module of the symplectic Lie algebrasp4is irreducible if it is the module with the highest weightL1,L1+L2, 2L1or 2L1+L2.Ifμ≠0, we haveM?V(λ,μ), whereV(λ,μ) is irreducible.

      猜你喜歡
      雨露
      許雨露美術作品
      電影文學(2022年21期)2022-12-31 09:59:10
      《花紅承雨露》
      散文詩(2022年13期)2022-07-29 09:06:16
      雨露
      “稀奇古怪”的牛老師
      好狗偏擋道
      都市人(2021年3期)2021-08-11 06:49:56
      紅色雨露滋養(yǎng)下的鄉(xiāng)村沃土
      金橋(2021年7期)2021-07-22 01:55:42
      陽光 雨露 夢想——凝心聚力、砥礪前行的敦煌路小學
      甘肅教育(2020年4期)2020-09-11 07:41:00
      墻頭草
      雨露
      學生天地(2019年33期)2019-08-25 08:56:16
      襄城县| 邵东县| 灌云县| 海丰县| 长子县| 唐山市| 广安市| 揭阳市| 奈曼旗| 兴宁市| 三江| 兴安县| 沙河市| 泉州市| 封开县| 西安市| 紫云| 阿尔山市| 滕州市| 江津市| 柞水县| 收藏| 西藏| 宁海县| 靖安县| 修水县| 安福县| 长沙县| 西峡县| 扬中市| 吉林市| 南宫市| 金坛市| 和龙市| 渑池县| 筠连县| 治多县| 贵阳市| 马山县| 东乡县| 武乡县|