History of Lie theory

19세기 프랑스 군론

• 갈루아
• Jordan
• 클라인과 리

리 군

• Sophus Lie—the precursor of the modern theory of Lie groups.
• Wilhelm Killing, who discovered almost all central concepts and theorems on the structure and classification of semisimple Lie algebras.
• Élie Cartan and is primarily concerned with developments that would now be interpreted as representations of Lie algebras, particularly simple and semisimple algebras.
• Hermann Weyl the development of representation theory of Lie groups and algebras.

development of representation theory of Lie groups

• 1913 Cartan spin representations
• 19?? Weyl unitarian trick : Complete reducibility
• Dynkin, The structure of semi-simple Lie algebras
  • amre,math.sco.transl.17

on fraktur

• http://mathoverflow.net/questions/87627/fraktur-symbols-for-lie-algebras-mathfrakg-etc
• The point is that group theory and Lie groups in particular were actively developed by German mathematicians in the nineteenth century; they were not inventing exotic notation when they used these particular letters as symbols

1950's

• 1954 Bruhat decomposition, Bruhat on the representation theory of complex Lie groups
• 1955 Chevalley [81,83] picked up on Bruhat decomposition immediately, and it became a basic tool in his work on the construction and classification of simple algebraic groups
• 1956 Borel,“Borel subgroup” of G as a result of the fundamental work
• 1962 Tits, introduced BN-pair
• 1965 Tits, Bourbaki Seminar expose , introduced the Building

refs

• Tits, Jacques. 1995. “Structures et Groupes de Weyl.” In Séminaire Bourbaki, Vol.\ 9, Exp.\ No.\ 288, 169–183. Paris: Soc. Math. France. http://www.ams.org/mathscinet-getitem?mr=1608796.
• Tits, Jacques. 1962. “Théorème de Bruhat et Sous-Groupes Paraboliques.” C. R. Acad. Sci. Paris 254: 2910–2912.

Tits, Jacques. 1995. “Structures et Groupes de Weyl.” In Séminaire Bourbaki, Vol.\ 9, Exp.\ No.\ 288, 169–183. Paris: Soc. Math. France. http://www.ams.org/mathscinet-getitem?mr=1608796.

• Borel, Armand. 1956. “Groupes Linéaires Algébriques.” Annals of Mathematics. Second Series 64: 20–82.
• Chevalley, C. 1955. “Sur Certains Groupes Simples.” The Tohoku Mathematical Journal. Second Series 7: 14–66.

리 타입의 유한군

• Finite groups of Lie type

modern development

• Kac-Moody algebras
• Quantum groups
• Canonical basis and dual canonical basis

memo

• history of theory of symmetric polynomials
• the role of invariant theory

articles

1. Elie Cartan Sur la structure des groupes de transformations finis et continus Cartan's famous 1894 thesis, cleaning up Killing's work on the classification Lie algebras.
2. Wilhelm Killing, "Die Zusammensetzung der stetigen endlichen Transformations-gruppen" 1888-1890 part 1part 2part 3part 4 Killing's classification of simple Lie complex Lie algebras.
3. S. Lie, F. Engel "Theorie der transformationsgruppen" 1888 Volume 1Volume 2Volume 3 Lie's monumental summary of his work on Lie groups and algebras.
4. Hermann Weyl, Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. 1925-1926 I, II, III. Weyl's paper on the representations of compact Lie groups, giving the Weyl character formula.
5. H. Weyl The classical groups ISBN 978-0-691-05756-9 A classic, describing the representation theory of lie groups and its relation to invariant theory
6. Chevalley, On certain simple groups

표준적인 교과서

• J.-P. Serre, Lie algebras and Lie groups ISBN 978-3540550082 Covers most of the basic theory of Lie algebras.
• J.-P. Serre, Complex semisimple Lie algebras ISBN 978-3-540-67827-4 Covers the classification and representation theory of complex Lie algebras.
• N. Jacobson, Lie algebras ISBN 978-0486638324 A good reference for all proofs about finite dimensional Lie algebras
• Claudio Procesi, Lie Groups: An Approach through Invariants and Representations, ISBN 978-0387260402. Similar to the course, with more emphasis on invariant theory.

expository

• Varadarajan, Historical review of Lie Theory
  • http://www.math.ucla.edu/~vsv/liegroups2007/liegroups2007.html
• Hawkins, Thomas. 1998. “From General Relativity to Group Representations: The Background to Weyl’s Papers of 1925–26.” In Matériaux Pour L’histoire Des Mathématiques Au XX\rm E$ Siècle (Nice, 1996), 3:69–100. Sémin. Congr. Paris: Soc. Math. France. http://www.ams.org/mathscinet-getitem?mr=1640256. http://www.emis.de/journals/SC/1998/3/pdf/smf_sem-cong_3_69-100.pdf
• T. Hawkins Emergence of the theory of Lie groups ISBN 978-0-387-98963-1 Covers the early history of the work by Lie, Killing, Cartan and Weyl, from 1868 to 1926.
• A. Borel Essays in the history of Lie groups and algebraic groups ISBN 978-0-8218-0288-5 Covers the history.
• "From Galois and Lie to Tits Buildings", The Coxeter Legacy: Reflections and Projections (ed. C. Davis and E.W. Ellers), Fields Inst. Publications volume 48, American Math. Soc. (2006), 45–62.
  
  • http://books.google.com/books?id=cKpBGcqpspIC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=twopage&q=chevalley&f=false
  • http://www.fields.utoronto.ca/programs/scientific/03-04/coxeterlegacy/abstracts.html