History of Lie theory
imported>Pythagoras0님의 2013년 12월 10일 (화) 06:36 판
introduction
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
Coxeter-Dynkin diagrams
- 1934, Coxeter, 'Discrete groups generated by reflections'
- 1946, Dynkin, introduces 'simple roots' and the so-called Dynkin diagrams to classify simple Lie algebras
- 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
development of theory of linear algebraic groups
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 on linear algberaic groups
- 1962 Jacques Tits, introduced BN-pair, develops the theory of groups with a $(B,N)$ pair where $B$ is for Borel, $N$ is the normalizer of a maximal torus contained in $B$
- 1965 Tits, Bourbaki Seminar expose , introduced the Building
flag manifold and Borel subgroup
- general linear group G by the isotropy subgroup B of a standard flag (say, the group of upper triangular nonsingular matrices)
- connected maximal solvable subgroups became known as Borel subgroups, while the notion of flag manifold came to mean the quotient $G/B$
for an arbitrary connected reductive Lie group G and a Borel subgroup B
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
- Chevalley, establish a synthesis between the theory of Lie groups and the theory of finite groups
- classified the simple algebraic groups over an algebraically closed field
- proved the existence of analogous groups over any field
- Finite groups of Lie type
modern development
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
- 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.