[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-665654-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-665654-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","dynkin-diagrams-chapter-8-classification-of-simple-lie-algebras","Dynkin Diagrams - Chapter 8 - Classification of Simple Lie Algebras","","Chapter 8 explains how to construct Dynkin diagrams for semisimple compact finite-dimensional Lie groups from constraints on how simple roots interact. It defines bonds using the integers p and q derived from the Cartan matrix of simple roots, and gives graphical rules for directed and undirected connections. It then classifies simple Lie algebras by analyzing which diagram configurations are impossible, using arguments about disconnected diagrams, planar angle constraints, and linear dependence.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/dynkin-diagrams-chapter-8-classification-of-simple-lie-algebras/665654/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/dynkin-diagrams-chapter-8-classification-of-simple-lie-algebras/665654.png","ImageObject",300,407,{"name":42,"@type":43},"Terk","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":52,"interactionType":53,"userInteractionCount":22},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"How are bonds and arrows determined in a Dynkin diagram for simple roots?","Question",{"text":62,"@type":63},"The chapter uses the values derived from the Cartan matrix of simple roots, with p and q specifying connections. It gives explicit diagram rules: number of lines depends on -α·2β (or the equivalent -2α·β), arrows point from the shorter root to the longer, and roots are not connected when the corresponding value is 0.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"Why must the diagram of a simple Lie algebra be connected?",{"text":67,"@type":63},"If simple roots split into two disconnected sets A and B, the chapter shows the commutation relations imply the corresponding structure decomposes as a direct product, producing a semisimple but not simple group. Therefore disconnected diagrams cannot represent simple algebras.",{"name":69,"@type":60,"acceptedAnswer":70},"What key constraints eliminate impossible Dynkin diagram configurations?",{"text":71,"@type":63},"The text rules out configurations that force geometric or algebraic contradictions. It uses angle-sum/vector-plane arguments to exclude certain three-root patterns, and it proves linear dependence for several forbidden multi-connection cases (e.g., branch-point and double-bond combinations, or a root linked to four others).","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},665654,1793305504,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":92,"show_sort_weight":93,"slug":94},"Exam",70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":22,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":30,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":79,"read_time":111},1099525198933,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","100. Last Latexed: April 25, 2017 at 9:45 Joel A. Shapiro  \nChapter 8  \nDynkin Diagrams  \nWe now describe how to use the constraints  \n~  \nα·2β = q~~ ~~−2p  \nwhere p is the number of times E can hit 􀀌E without annihilating, and q is the number of times E −  can. This is encoded in the Cartan matrix for the simple roots,  \nAji = 2jα~~ ~~i = q − p.  \nWe will draw a Dynkin diagram for each semisimple compact 􀀌nite-dimensional Lie group in which each simple root will be represented by a dot or small circle, and they will be connected by bonds according to the p values. As we saw, these p values are quite limited, but we will also 􀀌nd constraints on the way even these limited number of bonds can be put together. We shall see that many conceivable diagrams are forbidden, and therefore that the set of all such groups is quite limited.  \nFor each simple root draw a small open cicle.  \n · β  \nIf α and β are two simple roots, and −2 = 2 or 3, draw 2 or 3 lines,  \nα2  \nrespectively, between their corresponding circles, ith~an arrow fom α to β .  \nThat is, from the shorter root to the longer. If −2α · β = −2 · β = 1, draw  \nα2 2  \na single line without an arrow between the two roots. If  · β = 0, do not connect the two circles corresponding to α and β .  \nWith these rules we can draw diagrams such as  G 2  \nthese, and investigate if there is a group corresponding  \nto that diagram. We will see, however, that there are  A  \nrestrictions on what is possible. It will turn out that  \nthe diagrams on the right do correspond to Lie groups,  B 3  \nwhile those below do not.  \nC  \nD 4  \n8.1 Classi􀀌cation of Simple Lie Algebras  \nWe have seen that given any algebra, there exist m simple roots, where m is the rank of the algebra, with relations that can be represented by a Dynkin diagram.  \nLet us 􀀌rst show any algebra with a disconnected diagrams cannot be simple. Suppose that the simple roots can be divided into two sets, A and B, with the vertices of A disconnected from B. Then, for α ∈ A and β ∈ B , E−α |Eβ i = 0, which is alwas true for simple roots, so q = 0 . As there is no connection of α and β ,  · β = 0, and p = 0, so Eα |Eβ i = 0 . Each simple root in A commutes with each in B and with its conjugate. The other roots in A are commutators of simple ones, and similarly for B, so all roots in A commute with all roots in B. The group is therefore the direct product of that generated by A and that generated by B. Each of these is a nontrivial invariant subalgebra, so the whole group is not simple. Of course the direct product of simple groups is semisimple, but not simple.  \nNext, we rule out diagrams with three simple roots connected as shown, whether or not there are additional connections to other roots.  \nγ  \nα β  \n1  \nα γ  \nβ  \n2  \nα γ  \nβ  \n3  \n618: Last Latexed: April 25, 2017 at 9:45 101  \nThe angles between these roots are as given in the table, with the angles even bigger if any additional lines are added. Each is a vector with its tail at the origin. As the sum of three angles between three such vectors must be ≤ 360◦ , and equal to 360◦ only if the three vectors lie in a plane, we see the three roots must lie in a plane, and therefore cannot be linearly independent, which is impossible.  \n\n|  | (1) | (2) | (3) |\n| --- | --- | --- | --- |\n| θαβ | 120◦ | 120◦ | 135◦ |\n| θβγ | 120◦ | 150◦ | 135◦ |\n| θγα | 120◦ | 90◦ | 90◦ |\n| P θ | 360◦ | 360◦ | 360◦ |\n\nThus the only diagram with a “triple bond” is the diagram for G2 .  \nA set of vectors which corresponds to the Dynkin diagram rules, having the magnitudes and angles as indicated by the diagram, is called a 􀀅 system, without worrying about whether each 􀀅 system corresponds to an algebra.  \nNow consider a 􀀅 system containing two simple roots α and β connected to each other with a single line, with unspeci􀀌ed connections to the rest of the system. Then the 􀀅 system which results from contracting the lie, collapsing α and β toa single root ~σ =  + β . is also a 􀀅 system. For σ 2 = α2 = β2 ","cbCailz0DuqVcB27","https://ap.wps.com/l/cbCailz0DuqVcB27","pdf",84321,"English","# Chapter 8 Dynkin Diagrams\n## Classification of Simple Lie Algebras","[{\"question\":\"How are bonds and arrows determined in a Dynkin diagram for simple roots?\",\"answer\":\"The chapter uses the values derived from the Cartan matrix of simple roots, with p and q specifying connections. It gives explicit diagram rules: number of lines depends on -α·2β (or the equivalent -2α·β), arrows point from the shorter root to the longer, and roots are not connected when the corresponding value is 0.\"},{\"question\":\"Why must the diagram of a simple Lie algebra be connected?\",\"answer\":\"If simple roots split into two disconnected sets A and B, the chapter shows the commutation relations imply the corresponding structure decomposes as a direct product, producing a semisimple but not simple group. Therefore disconnected diagrams cannot represent simple algebras.\"},{\"question\":\"What key constraints eliminate impossible Dynkin diagram configurations?\",\"answer\":\"The text rules out configurations that force geometric or algebraic contradictions. It uses angle-sum/vector-plane arguments to exclude certain three-root patterns, and it proves linear dependence for several forbidden multi-connection cases (e.g., branch-point and double-bond combinations, or a root linked to four others).\"}]","Dynkin Diagrams - Chapter 8 - Classification of Simple Lie Algebras | PDF"]