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

Dynkin diagrams

• 1946, Dynkin, introduces 'simple roots' and the so-called Dynkin diagrams
• 1955, Tits, 'on certain classes of homogeneous spaces of Lie groups', Dynkin diagrams used today.

refs

• Tits, J. 1955. “Sur Certaines Classes D’espaces Homogènes de Groupes de Lie.” Acad. Roy. Belg. Cl. Sci. Mém. Coll. in $8^\circ$ 29 (3): 268.
• Dynkin, Evgeniĭ Borisovich. Classification of simple Lie groups, 2000. Selected Papers of E.B. Dynkin with Commentary. American Mathematical Soc.
• Dynkin, 1947 , Structure of semisimple Lie algebras

Bruhat and subsequence works

• 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

• 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.
• 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.
• S. Lie, F. Engel "Theorie der transformationsgruppen" 1888 Volume 1Volume 2Volume 3 Lie's monumental summary of his work on Lie groups and algebras.
• 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.
• 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
• Chevalley, C. 1955. “Sur Certains Groupes Simples.” The Tohoku Mathematical Journal. Second Series 7: 14–66.

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.
• Borel, A. 1980. “On the Development of Lie Group Theory.” The Mathematical Intelligencer 2 (2) (June 1): 67–72. doi:10.1007/BF03023375. http://link.springer.com/article/10.1007%2FBF03023375
• Borel, Armand. 2001. Essays in the History of Lie Groups and Algebraic Groups. American Mathematical Society. Covers the history. book review
• "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