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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

530: １３２人目の素数さん [] 2023/02/17(金) 16:33:02.23 ID:PDN8ps3Q

>>529
つづき

Connection with cyclotomic fields
The early proofs of quadratic reciprocity are relatively unilluminating. The situation changed when Gauss used Gauss sums to show that quadratic fields are subfields of cyclotomic fields, and implicitly deduced quadratic reciprocity from a reciprocity theorem for cyclotomic fields. His proof was cast in modern form by later algebraic number theorists. This proof served as a template for class field theory, which can be viewed as a vast generalization of quadratic reciprocity.

Robert Langlands formulated the Langlands program, which gives a conjectural vast generalization of class field theory. He wrote:[27]

I confess that, as a student unaware of the history of the subject and unaware of the connection with cyclotomy, I did not find the law or its so-called elementary proofs appealing. I suppose, although I would not have (and could not have) expressed myself in this way that I saw it as little more than a mathematical curiosity, fit more for amateurs than for the attention of the serious mathematician that I then hoped to become. It was only in Hermann Weyl's book on the algebraic theory of numbers[28] that I appreciated it as anything more.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1615510393/530

567: １３２人目の素数さん [] 2023/02/18(土) 11:11:49.23 ID:dtkuCIRJ

>>541
>reciprocity：相互律（相互法則）
>という用語は、ガウスは使っていない
>reciprocityは、現代風のルジャンドルの記号で書いたときに
>pとqとの入れ替えで不変になっていることから来る

reciprocity：相互律（相互法則）について

https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%89%E3%83%AA%E3%82%A2%E3%83%B3%EF%BC%9D%E3%83%9E%E3%83%AA%E3%83%BB%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB#CITEREF%E3%83%AB%E3%82%B8%E3%83%A3%E3%83%B3%E3%83%89%E3%83%AB2007
アドリアン＝マリ・ルジャンドル（仏: Adrien-Marie Legendre、1752年9月18日 - 1833年1月10日）
1798年の著書『数の理論に関する試作（Essai sur la Theorie des Nombres）』は、ドイツの天文学者、数学者、物理学者であるカール・フリードリヒ・ガウスの1801年の著書『整数論（Disquisitiones Arithmeticae）』の登場により、影に埋もれることとなった[2]。
(引用終り)

これ、下記PDFで
ルジャンドル記号の導入があって
”§VI Theoreme contenant une loi reciprocite qui exite entre deux nombres premiers quelconques”
とあるから
”reciprocite”（相互律）の用語は、ルジャンドルからだね

(参考)
https://archive.org/details/essaisurlathor00lege/page/n1/mode/2up
Essai sur la theorie des nombres
by Legendre, A. M. (Adrien Marie), 1752-1833
Publication date 1798
Topics Number theory
Publisher Paris, Duprat
https://ia804700.us.archive.org/24/items/essaisurlathor00lege/essaisurlathor00lege.pdf

ここの
§VI Theoreme contenant une loi reciprocite qui exite entre deux nombres premiers quelconques  ・・・214
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1615510393/567

736: １３２人目の素数さん [] 2023/02/23(木) 19:31:08.23 ID:03KDcN8J

>>735
つづき

https://ja.wikipedia.org/wiki/%E3%83%A9%E3%83%B3%E3%82%B9%E3%83%AD%E3%83%83%E3%83%88%E3%83%BB%E3%83%9B%E3%82%B0%E3%83%99%E3%83%B3
ランスロット・ホグベン
ランスロット・トマス・ホグベン （1895年12月9日 - 1975年8月22日）は、イギリスの動物学者、遺伝学者。
『百万人の数学』『市民の科学』をはじめ、科学・数学・言語の啓蒙書の執筆者としてよく知られる。マルクス主義者でもあり独立労働党でも活動、人工言語・インターグロッサ（英語版）の考案者である。妻は、数学者・統計学者でフェミニストのエニッド・チャールズ（英語版）。
ホグベンは『百万人の数学』(1936年)、『市民の科学』(1938年)と一般向けの科学のベストセラーを2冊出版した。これらはとても野心的な書籍であった。

https://en.wikipedia.org/wiki/Lancelot_Hogben
Lancelot Thomas Hogben FRS[1] FRSE (9 December 1895 ? 22 August 1975)
British experimental zoologist and medical statistician.

Popular science writing
Hogben produced two best-selling works of popular science, Mathematics for the Million (1936) and Science for the Citizen (1938). Mathematics for the Million received widespread praise, with H. G. Wells saying that "Mathematics for the Million is a great book, a book of first-class importance".[20] The book was also lauded by Albert Einstein, Bertrand Russell and Julian Huxley.[20][21]Mathematics for the Million was reprinted after Hogben's death.[21]

References
21  "Mathematics for the Million...praised by Einstein, H. G. Wells and others, it was reprinted in paperback in 1993." De Smith, Michael John, Maths for the Mystified : An Exploration of the History of Mathematics and Its Relationship to Modern-Day Science and Computing.Leicester : Matador, 2006. (p.192)
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1615510393/736

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