Álgebra de Kac-Moody

Nota: Para a álgebra de Kac-Moody generalizada, veja Álgebra de Lie.

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v d e

A álgebra de Kac-Moody, nomeada em honra de Victor Kac e Robert Moody, (também conhecida como álgebra de Kac-Moody Lie) é definida da seguinte forma.

Dado,

1) Uma n×n matriz generalizada de Cartan C = (c_ij) de classificação r.

2) Um vetor de espaço

{\mathfrak {h}}

sobre os números complexos de dimensão 2n − r

3) Um conjunto de n elementos linearmente independentes

\alpha _{i}^{\vee }\

{\mathfrak {h}}

e um conjunto de n elementos linearmente independentes

\alpha _{i}

do espaço dual

{\mathfrak {h}}^{*}

, de tal modo que

\alpha _{i}(\alpha _{j}^{\vee })=c_{ji}

. Os

\alpha _{i}

são analógicos para as raízes simples^[1] de uma semi-simples álgebra de Lie, e os

\alpha _{i}^{\vee }

para as co-raízes simples.

A álgebra de Kac-Moody é a álgebra de Lie ${\mathfrak {g}}$ definida por geradores $e_{i}$ e $f_{i}$ ( $i\in \{1,\ldots ,n\}$ ) e os elementos de ${\mathfrak {h}}$ e as relações.

$[h,h']=0\$ para $h,h'\in {\mathfrak {h}}$ ;
$[h,e_{i}]=\alpha _{i}(h)e_{i}$ , para $h\in {\mathfrak {h}}$ ;
$[h,f_{i}]=-\alpha _{i}(h)f_{i}$ , para $h\in {\mathfrak {h}}$ ;
$[e_{i},f_{j}]=\delta _{ij}\alpha _{i}^{\vee }$ , onde $\delta _{ij}$ é o delta de Kronecker
${\textrm {ad}}(e_{i})^{1-c_{ij}}(e_{j})=0$ e ${\textrm {ad}}(f_{i})^{1-c_{ij}}(f_{j})=0$ , onde ${\textrm {ad}}:{\mathfrak {g}}\to {\textrm {End}}({\mathfrak {g}}),{\textrm {ad}}(x)(y)=[x,y],$ é a representação adjunta^[2] de ${\mathfrak {g}}$ .

A álgebra de Lie real (possivelmente de dimensão infinita) é também considerada uma álgebra de Kac-Moody, se a sua complexificação é uma álgebra de Kac-Moody^[3]^[4] ^[5].

Referências

↑ Structure of the Root Spaces for Simple Lie Algebras - [https://web.archive.org/web/20100710004059/http://panda.unm.edu/Courses/Finley/p500Fall05/rootss.pdf Arquivado em 10 de julho de 2010, no Wayback Machine.]
↑ Adjoint Representation por Rowland, Todd -
↑ Conrado Damato de Lacerda (11 de maio de 2012). «Introdução às Álgebras de Kac-Moody» (PDF). Universidade Estadual de Campinas
↑ Anjos, R.C.; Ferreira, L.A. (24 de novembro de 2008). «Cargas conservadas em teorias de sólitons» (PDF). XII Workshop da Pós-Graduação do IFSC
↑ Rita de C dos Anjos, Luiz Agostinho Ferreira (26 de setembro de 2008). «Método de obtenção de cargas conservadas em teorias de sólitons». Instituto de Física de São Carlos, Universidade de São Paulo

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Conceitos básicos	Escalar Vetor Espaço vetorial Projeção de um vetor Espaço vetorial gerado Transformação linear Projeção Independência linear Combinação linear Base Espaço coluna Espaço linha Espaço dual Ortogonalidade Núcleo Valor próprio Método dos mínimos quadrados Produto diádico Espaço com produto interno Produto escalar Transposição Processo de Gram-Schmidt Sistema de equações lineares
Matrizes	Matriz Produto de matrizes Decomposição LU Menor Posto matricial Regra de Cramer Matriz inversa Eliminação de Gauss Matriz de transformação Matriz em bloco Matriz unimodular
Álgebra linear numérica	Vírgula flutuante Estabilidade numérica BLAS Matriz esparsa Comparação de bibliotecas de álgebra linear Comparação de softwares de análise numérica

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