[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-316601-105":59,"doc-detail-316601-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","finely-homogeneous-computations-in-free-lie-algebras","Finely homogeneous computations in free Lie algebras","","The paper presents algorithms for free Lie algebras based on Lyndon words and finely homogeneous multidegrees. It first gives a fast method to compute, for every prescribed multidegree, the maximal Lyndon word under lexicographic order, and then provides a procedure to enumerate all Lyndon words associated with a given finely homogeneous class. It also derives a simple algorithm, using Duval’s Lyndon-word successor method, to compute the next standard bracketing of a Lyndon word by lexicographic order by length. An application is discussed in control theory, where letters represent vector fields within the command Lie algebra.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/finely-homogeneous-computations-in-free-lie-algebras/316601/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/finely-homogeneous-computations-in-free-lie-algebras/316601.png","ImageObject",300,407,{"name":92,"@type":93},"Sage","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-09-20",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the main goal regarding Lyndon words in the paper?","Question",{"text":112,"@type":113},"To compute maximal Lyndon words under lexicographic order for each given multidegree, and to generate all Lyndon words belonging to a specified finely homogeneous class.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the paper relate its algorithms to Duval’s method?",{"text":117,"@type":113},"It uses Duval’s algorithm for generating Lyndon word successors to build a simple procedure that computes the next standard bracketing of a Lyndon word by lexicographic order by length.",{"name":119,"@type":110,"acceptedAnswer":120},"What application area is mentioned for the standard bracketing algorithm?",{"text":121,"@type":113},"Control theory, where one computes within the command Lie algebra of a dynamical system and the letters are interpreted as vector fields.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},316601,1791040320,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},687197207057,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Discrete Mathematics and Theoretical Computer Science 1, 1997, 101–114  \nFinely homogeneous computations in free Lie algebras  \nPhilippe Andary  \nUniversitde Rouen, Facult´e des Sciences, Laboratoire d’Informatique de Rouen, F-76821 Mont-Saint-Aignan Cdex, France  \nE-Mail: [andary@dir.univ-rouen. fr](andary@dir.univ-rouen. fr)  \nWe first give a fast algorithm to compute the maximal Lyndon word (with respect to lexicographic order) ofLy􀀋 (A) for every given multidegree 􀀋 in Nk . We then give an algorithm to compute all the words living in Ly􀀋 (A) for any given 􀀋 in Nk . The best known method for generating Lyndon words is that of Duval [1], which gives a way to go from every Lyndon word of length n to its successor (with respect to lexicographic order by length), in space and worst case time complexity O (n) . Finally, we give a simple algorithm which uses Duval’s method (the one above) to compute the next standard bracketing ofa Lyndon word for lexicographic order by length. We can find an interesting application of this algorithm in control theory, where one wants to compute within the command Lie algebra of a dynamical system (letters are actually vector fields) .  \nKeywords: Lie algebras, finely homogeneous computations  \n1 Introduction  \nLet A = fa1 ; a2 ; : : : ; akg be a set with k elements, and QhAi the associative (non-commutative) algebra on A. Defining a Lie bracket ([x; y ] = xy 􀀀 y x) on this Q-module turns it into a Lie algebra, and we will denote by L (A) its Lie subalgebra generated by A (i.e. L (A) is the free Lie algebra on A and QhAi its enveloping algebra) . A will now be called an alphabet, whose elements are the letters, and A􀀃 is the free monoid (the set of all words) over A.  \nWe know that L(A) is a graded Q-module  \nL (A) = M Ln (A) (1)  \nn􀀕 1  \nwhere Ln (A) is the submodule of L (A) generated by homogeneous components with degree n. Furthermore, Ln (A) is the direct sum of finely homogeneous submodules L􀀋 (A) with multidegree 􀀋 2 Nk such  \nthat j􀀋 = n. Hence we have  \nL (A) = M L􀀋 (A) (2)  \n􀀋 2Nk  \n1365–8050 􀀍c 1997 Chapman & Hall  \n102 Philippe Andary  \nAll the classical monomial basis of L(A) are finely homogeneous (see Reutenauer [2], for example, or Viennot [3]), i.e. their components are finely homogeneous Lie monomials, thus every computation on Lie polynomials (the elements of L(A)) is nothing but a computation in some fine homogeneity class.  \nWe want to emphasize the use of a particular basis of L(A), the Lyndon basis, introduced by Chen et al. in the late 1950s [4] (the interested reader can find all the details of its construction in Lothaire [5]) . For this purpose, we choose a total order a1 \u003C a2 \u003C : : : \u003C ak on A, which induces a lexicographic order on A􀀃 . A word ` is a Lyndon word if it is primitive and minimal in its conjuguacy class. This means that ` cannot be written as un for given u in A􀀃 and n 􀀕 2, and ` 􀀔 vu whenever ` = uv . We denote by Ly(A)  \nthe set of Lyndon words over A.  \nFor every Lyndon word ` we define its (right) standard factorization (`0 ; `00) as the unique couple of Lyndon words such that ` = `0`00 and `00 is of maximal length. Moreover, the recursive Lie bracketing of the standard factorization will be called the (right) standard bracketing of`, denoted by [`] .  \nIt would be convenient to define the left standard factorization ( u; v) of ` as the unique couple of Lyndon words such that ` = uv and u is of maximal length. Then the recursive Lie bracketing of the left standard factorization will be called the left standard bracketing of`, denoted by (`) .  \nNote that A is included in Ly (A) and that [a] = a = (a) for every letter a. As an exemple, let A = fa; bg and ` = aaabab, then ` is Lyndon because it is primitive and ` \u003C s for each of its proper suffix s. The standard bracketings of ` are  \n[`]  \n(`)  \n=  \n=  \n[a; [[a; [a; b]] ; [a; b]]]  \n[[a; [a; [a; b]]] ; [a; b]]  \nIt is well known that Ly(A) is a factorization of A􀀃 and that [Ly(A) ] is a basis of L (A) (called the L","cbCaisUHDf059MWw","https://ap.wps.com/l/cbCaisUHDf059MWw","pdf",202142,14,"English","# Introduction\n# Maximal Finely Homogeneous Lyndon Words\n## Introduction","[{\"question\":\"What is the main goal regarding Lyndon words in the paper?\",\"answer\":\"To compute maximal Lyndon words under lexicographic order for each given multidegree, and to generate all Lyndon words belonging to a specified finely homogeneous class.\"},{\"question\":\"How does the paper relate its algorithms to Duval’s method?\",\"answer\":\"It uses Duval’s algorithm for generating Lyndon word successors to build a simple procedure that computes the next standard bracketing of a Lyndon word by lexicographic order by length.\"},{\"question\":\"What application area is mentioned for the standard bracketing algorithm?\",\"answer\":\"Control theory, where one computes within the command Lie algebra of a dynamical system and the letters are interpreted as vector fields.\"}]","Finely homogeneous computations in free Lie algebras | PDF",1789919513,35]