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1、 HEBEI UNIVERSITY 密密 级:级: 分分 类类 号:号: 学校代码:学校代码:10075 学学 号:号:20090869 硕士学位论文 硕士学位论文 三指标阵空间的代数结构 学位申请人: 张蒙 指 导 教 师 : 白瑞蒲 教授 学 位 类 别 : 理学硕士 学 科 专 业 : 基础数学 授 予 单 位 : 河北大学 答 辩 日 期 : 二一二年五月 Classihed Index:CODE: 10075 U. D. C. :NO.: 20090869 A Dissertation for the Degree of Master Science Algebraic Struct
2、ures of 3-Indices Matrix Space Candidate : Zhang Meng Supervisor : Prof. Bai Ruipu Academic Degree Applied for : Master of Science Speciality : rundamental Mathematics University : Hebei University Date of Oral Examination : May, 2012 ? ? ?3-?3-? ?15?TM(r.s),? ?N-?A,?3?As, s1, 2, 3?4?t, t 1,2,3,4. ?
3、TM(r.s),?Lrs?3-?Jrs,? ? ?F? ? ?15? ? ? ?3-? ?3-? I AbStract Abstract The paper mainly concerns multiplicative structures of 3-indices matrix (cubic ma- trix) and algebraic structures of 3-indices matrix space . rifteen kinds of multiplications TM(r.s) for arbitrary two cubic matrices which satisfyin
4、g the associative law are pro- vided, and the relationship between diferent multiplications discussed. ror an N-order cubic matrix A, three kinds of determinant As, s 1,2,3 and four kinds of ”trace” t, t 1,2,3,4 are dehnied. According to these multiplications TM(r.s), the Lie algebra Lrsand 3 - Lie
5、algebra Jrsare constructed, respectively, and the structures are studied. Throughout the paper, the characteristic of a held F is assumed to be zero. The organization of the paper is as follows: Section 1 introduces the background and the development of matrix. Section 2 dehnes 15 kinds of multiplic
6、ations which satisfy the associative law. Section 3 dehnes the determinants and traces of the cubic matrix. Section 4 constructs of Lie algebras by cubic matrix. Section 5 constructs of 3-Lie algebras by cubic matrix. KeywordsLie algebra3-Lie algebracubic matricesdeterminanttrace II ? ? 1.?.1 2.? 2
7、3.? ?. 10 4.?. 14 5.?3-? 20 6.?. 28 ? 29 ? 31 ? 32 III ContentS Contents 1.Introduction . 1 2.Multiple-index matrix and basic multiplications 2 3.Determinant and traces of the cubic matrices .10 4.Structures of Lie algebras constructed by cubic matrices 14 5.Structures of 3-Lie algebras constructed
8、by cubic matrices 20 6.Conclusions 28 References 29 Papers written during postgraduate period 31 Achnowledgements 32 IV ?1? ?1? 1850?1858? ? ? ?3?n-?4? ? ?3?V.Abramov,R.Kerner,O.Liivapuu?KMN-?(? ?)?3-?2 ?ternary j commutator?4?H.Awata,M.Li,D.Minic ?QuantumNambu ?4?V.T.Filippov ?n-?Sh.M.Kasymov?E.N.K
9、uzlmin?V.T.Filippov? ?1?n-? ?Cartan?Engle?(?1,2).?n- ?n-?n?2? ?n-?(?9,12,13,20).? ? ? ?3-? ? 1)? ? 2)? 3)? 4)?3-?3-?3- ? 1 ? ?2? ?r? ?2.1? A (ai1.im),1 ik nk,1 k m,ai1.ime F(2.1) ?n1” ” nm?.?ai1.im?A?,?0. ?m 1?A?F?m 2?A?n1”n2 ?m 3?A?k-?m 3?A? ?2.2? A (ai1.im), ai1.ime F,1 ik nk,1 k m? ?r?n1” ” nm?A,
10、B e , 入 e F,? A B e , 入A e ,? A B (ai1.im bi1.im),(2.2) 入A (入ai1.im), 入 e F.(2.3) ?r?n1” ” nm? ?Eh1h2.hm (ej1j2.jm),?ej1j2.jm Jh1j1Jh2j2 Jhmjm,?hk jk? Jhkjk 1,?hk jk?Jhkjk 0, 1 hk,jk N.?A (ai1i2.im) e ,?(2.1),(2.2),(2.3)?A Z ai1i2.imEi1i2.im.?入i1i2.ime F, Z 入i1i2.imEi1i2.im 0 入i1i2.im 0.?r?n1” ” nm
11、? ? ?A (aijk), B (bijk) e , ?A “stB e , 1 s 3,1 t 9?A?B? ?2.3?A (aijk), B (bijk) e ,? (A “11B)ijk N l P=1 aijPbiPk,1 i,j,k N.TM(1.1) 2 ?2? (A “12B)ijk N l P=1 aPjkbijP,1 i,j,k N.TM(1.2) (A “13B)ijk N l P=1 aiPkbPjk,1 i,j,k N.TM(1.3) (A “21B)ijk N l P,q=1 aqjPbiPk,1 i,j,k N.TM(2.1) (A “22B)ijk N l P,
12、q=1 aPjkbiqP,1 i,j,k N.TM(2.2) (A “23B)ijk N l P,q=1 aiPqbPjk,1 i,j,k N.TM(2.3) (A “24B)ijk N l P,q=1 aijPbqPk,1 i,j,k N.TM(2.4) (A “25B)ijk N l P,q=1 aPqkbijP,1 i,j,k N.TM(2.5) (A “26B)ijk N l P,q=1 aiPkbPjq,1 i,j,k N.TM(2.6) (A “27B)ijk N l P,q=1 aqjkbiPk,1 i,j,k N.TM(2.7) (A “28B)ijk N l P,q=1 aiqkbijP,1 i,j,k N.TM(2.8) (A “29B)ijk N l P,q=1 aijqbPjk,1 i,j,k N.TM(2.9) (A “31B)ijk N l P,q,r=1 aiPqbrjk,1 i,j,k N.TM(3.1) (A “32B)ijk