History of Lie theory

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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


Killing

  • calculation of the characteristic polynomial of linear transformation given by adjoint representations
  • Killing form
  • found big commuting subalgebra, which later named as Cartan subalgebra
  • definition of rank
  • concept of root = appears in the factorization of the characteristic polynomial
  • semisimple = sum of a few simple groups
  • \(a_{ij}\) in terms of root strings

Cartan

  • rigorous characterization of maximal commuting subalgebra
  • semisimple = groups not possessing a solvable invariant subgroup
  • criterion for semisimplicity in terms of Killing form
  • After the development of the theory of algebraic groups, it was realized that Cartan's results on the structure of real semi-simple groups were really of two kinds
  1. some could be viewed as special cases of theorems on algebraic groups after a suitable reformulation
  2. the others, such as the conjugacy of maximal compact subgroups and the properties of symmetric spaces, were topological or differential geometric in nature.
  • In other words, some depended on the Zariski topology and the others on the ordinary topology and the \(C^{\infty}\)-structures associated to \(\mathbb{R}\)


development of representation theory of Lie groups

Weyl

  • Particularly striking was the fact that a topological result, the finiteness of the fundamental group of a compact semi-simple group, played a key role in the proof of an algebraic theorem, namely the full reducibility of the finite dimensional representations of the complex semi-simple Lie algebras.
  • This work led a bit later to the Peter-Weyl theorem, and also paved the way for the applications of Lie group representations to physics, which have steadily gained in importance since then.


on fraktur


Weyl group

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',introduced the Dynkin diagrams with arrows 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
  • Coleman, A. J. “Killing and the Coxeter Transformation of Kac-Moody Algebras.” Inventiones Mathematicae 95, no. 3 (October 1, 1989): 447–77. doi:10.1007/BF01393885.
  • Coxeter, H. S. M. ‘The Evolution of Coxeter-Dynkin Diagrams’. In Polytopes: Abstract, Convex and Computational, edited by T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivić Weiss, 21–42. NATO ASI Series 440. Springer Netherlands, 1994. http://link.springer.com/chapter/10.1007/978-94-011-0924-6_2.
  • Maximal subalgebras of Lie algebras

on universal enveloping algebras


Chevalley

  • integral basis of \(\mathfrak{g}\)

The theory of integral forms for finite-dimensional simple Lie algebras was first studied by Chevalley in 1955. His work led to the construction of Chevalley groups (of adjoint type). The representation theory of Chevalley groups relies on the existence of integral forms for the universal enveloping algebras associated to these simple finite-dimensional Lie algebras.


Kostant

  • integral basis of \(U(\mathfrak{g})\)

In 1966, suitable integral forms were discovered by Cartier and Kostant independently. They obtained precise information about these integral forms through integral bases (bases whose \(\mathbb{Z}\)-span is the integral form). The construction of such bases relies heavily on straightening identities in the universal enveloping algebra, which allow one to write certain elements in Poincare-Birkhoff-Witt (PBW) order. Cartier and Kostant's \(\mathbb{Z}\)-form led to the construction of Lie groups and Lie algebras over a field of positive characteristic, generalizing Chevalley's construction. This in turn led to the development of representation theory over a field of positive characteristic, [H].


Serre

Also in 1966, Serre showed that a finite dimensional Lie algebra can be presented by generators and relations determined solely by the Cartan matrix. With a generalized Cartan matrix one can use the Serre presentation to define the class of Kac-Moody Lie algebras.

development of theory of linear algebraic groups

Bruhat and subsequent works

  • see Bruhat decomposition also
  • 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

Underline (red), Dec 12, 2013, 2:49 AM: Apart from flat factors, these spaces are homo- geneous spaces of semi-simple groups, and are products of quotients G/K, where either G is compact semi-simple, K the fixed point set of an involution of G and G/K has positive curvature, or G is simple non-compact with finite center, K is a maximal compact subgroup, and G/K has negative curvature. A beautiful illustration of the inter- play between groups and differential geometry is the proof of the conjugacy of the maximal compact subgroups of a semi-simple group via a fixed point theorem asserting that any compact group of isometries of a complete simply connected Riemannian manifold with negative curvature has a fixed point. For about thirty years, this was the only one.

Underline (red), Dec 12, 2013, 2:49 AM: This, and the classification of the bounded symmetric domains, carried out a bit later by E. Cartan, made it clear that semi-simple groups and symmetric spaces offered a natural framework to study reduction theory with respect to arithmetic groups and automorphic forms in several vari- ables, developed notably by C. L. Siegel in the thirties and the forties.


--- Page 3 ---

Underline (red), Dec 12, 2013, 2:49 AM: However, in the early fifties, the growing importance of certain algebraic homogeneous spaces and the development of abstract algebraic geometry made the need of a more general theory of linear algebraic groups

Underline (red), Dec 12, 2013, 2:49 AM: I hope this gives some idea of the global arguments which replaced Lie algebra considerations. The theory was rather quickly developed in this framework. A major achievement was the classification of simple algebraic groups by C. Chevalley, which he proved to be indepen- dent of K: The simple groups over K are classified by the root systems and lattices, in the same way as over C.


--- Page 4 ---

Underline (red), Dec 12, 2013, 2:49 AM: The general study of the groups G(k) was developed from about 1957 on. It led notably to a structure theory of semi-simple, (or, slightly more generally, reductive) groups over arbitrary fields by J. Tits and myself in the early sixties. The notions of Cartan subgroups, roots, Weyl groups, Bruhat decomposition of the classical theory have suitable analogues


--- Page 5 ---

Underline (red), Dec 12, 2013, 2:49 AM: After the develop- ment of the theory of algebraic groups, it was realized that Cartan's results on the structure of real semi-simple groups were really of two kinds: some, suitably reformu- lated, could be viewed as special cases of theorems on alge- braic groups while the others, such as the conjugacy of maximal compact subgroups and the properties of sym- metric spaces, were topological or differential geometric in nature. In other words, some depended on the Zariski topology and the others on the ordinary topology and the C~-structures associated to ~ . The former were now available over non-archimedean fields, and there remained to be seen whether the others also had some counterpart.

Underline (red), Dec 12, 2013, 2:49 AM: The theory of Lie groups and algebraic groups has now attained a considerable degree of completeness and has found many applications. Its usefulness is nowhere more in evidence than in the present study of automorphic forms and of their connections with arithmetic and alge- braic geometry.


--- Page 6 ---

Underline (red), Dec 12, 2013, 2:49 AM: In fact, I would rather schema- tize the structure of mathematics by a complicated graph, where the vertices are the various parts of mathematics and the edges describe the connections between them. These connections sometimes go one way, sometimes both ways, and the vertices can act both as sources and sinks. The development of the individual topics is of course the life and blood of mathematics, but, in the same way as a graph is more than the union of its vertices, mathematics is much more than the sum of its parts. It is the presence of those numerous, sometimes unexpected edges, which makes mathematics a coherent body of knowledge, and testifies to its fundamental unity, in spite of its being too vast to be comprehended by one single mind.


related items


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.


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  • [{'LOWER': 'borel'}, {'LEMMA': 'subgroup'}]
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