ガロア第一論文及びその関連の資料スレ

[過去ﾛｸﾞ] ガロア第一論文及びその関連の資料スレ (1002ﾚｽ)
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355: １３２人目の素数さん [] 2023/02/12(日) 09:51:32.23 ID:t5GdbcIg

>>354
つづき

A systematic study of Kac-Moody algebras was started independently by V.G. Kac [Ka] and R.V. Moody [Mo], and subsequently many results of the theory of finite-dimensional semi-simple Lie algebras have been carried over to Kac-Moody algebras. The main technical tool of the theory is the generalized Casimir operator (cf. Casimir element), which can be constructed provided that the matrix A
is symmetrizable, i.e. A=DB
for some invertible diagonal matrix D
and symmetric matrix B
[Ka2]. In the non-symmetrizable case more sophisticated geometric methods are required [Ku], [Ma].

One of the most important ingredients of the theory of Kac-Moody algebras are integrable highest-weight representations (cf. also Representation with a highest weight vector).

The numerous applications of Kac-Moody algebras are mainly related to the fact that the Kac-Moody algebras associated to positive semi-definite indecomposable Cartan matrices (called affine matrices) admit a very explicit construction. (A matrix is called indecomposable if it does not become block-diagonal after arbitrary permutation of the index set.)
These Kac-Moody algebras are called affine algebras.

This observation leads to geometric applications of affine algebras and the corresponding groups, called the loop groups (see [PrSe]).

つづく

http://rio2016.5ch.net/test/read.cgi/math/1615510393/355

356: １３２人目の素数さん [] 2023/02/12(日) 09:51:54.00 ID:t5GdbcIg

>>355
つづき

The basic representation of g(A(1))
is then defined on V
by the following formulas [FrKa]:

π(u(n))=u(n),u∈h
π(E(n)α)=Xn(α)cα,π(k)=1;
This is called the homogeneous vertex operator construction of the basic representation.

The vertex operators were introduced in string theory around 1969, but the vertex operator construction entered string theory only at its revival in the mid 1980s. Thus, the representation theory of affine algebras became an important ingredient of string theory (see [GrScWi]).

The vertex operators turned out to be useful even in the theory of finite simple groups. Namely, a twist of the homogeneous vertex operator construction based on the Leech lattice produced the 196883-dimensional Griess algebra and its automorphism group, the famous finite simple Monster group (see Sporadic simple group) [FrLeMe].

The vertex operator constructions were, quite unexpectedly, applied to the theory of soliton equations. This was based on the observation (see [DaJiKaMi]) that the orbit of the vector vΛ0
of the basic representation under the loop group satisfies an infinite hierarchy of partial differential equations, the simplest of them being classical soliton equations, like the Korteweg-de Vries equation.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1615510393/356

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