the coincidence lefschetz number for self-maps of lie groups
Let/. It :0 ---0 G be any two self maps of a compact connected oriented Lie group G. In this paper, for each positive integer k , we associate an integer with fk,hi . We relate this number with Lefschetz coincidence number. We deduce that for any two differentiable maps f, there exists a positive in...
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University of Baghdad, College of Science for Women
2006-09-01
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| description | Let/. It :0 ---0 G be any two self maps of a compact connected oriented Lie group G. In this paper, for each positive integer k , we associate an integer with fk,hi . We relate this number with Lefschetz coincidence number. We deduce that for any two differentiable maps f, there exists a positive integer k such that k 5.2+1 , and there is a point x C G such that ft (x) = (x) , where A is the rank of G . Introduction Let G be an n-dimensional com -pact connected Lie group with multip-lication p ( .e 44:0 xG--+G such that p ( x , y) = x.y ) and unit e . Let [G, G] be the set of homotopy classes of maps G G . Given two maps f , f G ---• Jollowing [3], we write f. f 'to denote the map G-.Gdefined by 01.11® =A/WO= fiat® ,sea Given a point g EC and a differ-entiable map F: G G , write GA to denote the tangent space of G at g [4,p.10] , and denote by d x F the linear map rig F :Tx0 T, (x)G induced by F , it is called the differential of Fat g [4,p.22]. Let LA, Rx :0 G be respec-tively the left translation Lx(i)=4..(g,e) , and the right translation Rx(1)./..(gcg). Then there is a natural homomorphism Ad ,the adjoin representation, from G to GL(G•), (the group of nonsingular linear transformations of Qdefined as follows:- Ad(g)= deRe, od,Lx. Note that d xRc, ad.; =d(4,( Lx(e)))0 de; =d.(4, 04)=4(40 Re) = d(4(4, (e)))0 (44, =d ar, o (44, . Since G is connected , the image of Ad belongs to the connected component of G(G)containing the identity,i.e. for each g E 0, detAd(g) > 0 . By Exercise Al • Dr.-Prof.-Department of Mathematics- College of Science- University of Baghdad. •• Dr.-Department of Mathematics- College of Science for Woman- University of Baghdad. |
| format | Article |
| id | doaj-art-6ce74c93ba514646b699cd374e564f5e |
| institution | DOAJ |
| issn | 2078-8665 2411-7986 |
| language | English |
| publishDate | 2006-09-01 |
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| spelling | doaj-art-6ce74c93ba514646b699cd374e564f5e2025-08-20T02:52:20ZengUniversity of Baghdad, College of Science for Womenمجلة بغداد للعلوم2078-86652411-79862006-09-013310.21123/bsj.3.3.465-469the coincidence lefschetz number for self-maps of lie groupsBaghdad Science JournalLet/. It :0 ---0 G be any two self maps of a compact connected oriented Lie group G. In this paper, for each positive integer k , we associate an integer with fk,hi . We relate this number with Lefschetz coincidence number. We deduce that for any two differentiable maps f, there exists a positive integer k such that k 5.2+1 , and there is a point x C G such that ft (x) = (x) , where A is the rank of G . Introduction Let G be an n-dimensional com -pact connected Lie group with multip-lication p ( .e 44:0 xG--+G such that p ( x , y) = x.y ) and unit e . Let [G, G] be the set of homotopy classes of maps G G . Given two maps f , f G ---• Jollowing [3], we write f. f 'to denote the map G-.Gdefined by 01.11® =A/WO= fiat® ,sea Given a point g EC and a differ-entiable map F: G G , write GA to denote the tangent space of G at g [4,p.10] , and denote by d x F the linear map rig F :Tx0 T, (x)G induced by F , it is called the differential of Fat g [4,p.22]. Let LA, Rx :0 G be respec-tively the left translation Lx(i)=4..(g,e) , and the right translation Rx(1)./..(gcg). Then there is a natural homomorphism Ad ,the adjoin representation, from G to GL(G•), (the group of nonsingular linear transformations of Qdefined as follows:- Ad(g)= deRe, od,Lx. Note that d xRc, ad.; =d(4,( Lx(e)))0 de; =d.(4, 04)=4(40 Re) = d(4(4, (e)))0 (44, =d ar, o (44, . Since G is connected , the image of Ad belongs to the connected component of G(G)containing the identity,i.e. for each g E 0, detAd(g) > 0 . By Exercise Al • Dr.-Prof.-Department of Mathematics- College of Science- University of Baghdad. •• Dr.-Department of Mathematics- College of Science for Woman- University of Baghdad.http://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/727 |
| spellingShingle | Baghdad Science Journal the coincidence lefschetz number for self-maps of lie groups مجلة بغداد للعلوم |
| title | the coincidence lefschetz number for self-maps of lie groups |
| title_full | the coincidence lefschetz number for self-maps of lie groups |
| title_fullStr | the coincidence lefschetz number for self-maps of lie groups |
| title_full_unstemmed | the coincidence lefschetz number for self-maps of lie groups |
| title_short | the coincidence lefschetz number for self-maps of lie groups |
| title_sort | coincidence lefschetz number for self maps of lie groups |
| url | http://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/727 |
| work_keys_str_mv | AT baghdadsciencejournal thecoincidencelefschetznumberforselfmapsofliegroups AT baghdadsciencejournal coincidencelefschetznumberforselfmapsofliegroups |