Forma di Maurer-Cartan

In matematica, la forma di Maurer–Cartan associata ad ogni gruppo di Lie $G$ è una particolare 1-forma differenziale su $G$ che codifica l'informazione a livello infinitesimo circa la struttura del gruppo $G$ . Fu usata dal matematico Élie Cartan come ingrediente fondamentale del suo metodo dei riferimenti mobili e porta il suo nome accanto a quello di Ludwig Maurer.

Definizione

Sia $G$ un gruppo di Lie, ${\mathfrak {g}}=T_{e}G$ la sua algebra di Lie.

Ogni elemento $g\in G$ induce la seguente moltiplicazione (che risulta essere un diffeomorfismo)

L_{g^{-1}}:G\rightarrow G

L_{g^{-1}}(h):=g^{-1}h

e la mappa tangente (detta anche differenziale)

(TL_{g^{-1}})_{g}:T_{g}G\rightarrow T_{e}G={\mathfrak {g}}

La 1-forma di Maurer-Cartan $\omega \in \Omega ^{1}(G,{\mathfrak {g}})$ è definita da:

\omega (v):=(TL_{g^{-1}})_{g}(v)\,,

per ogni vettore tangente $v\in T_{g}G,g\in G$ ^[1].

Note

^ Jeffrey M. Lee, Chapter: 5.6 The Maurer Cartan Form, in Manifolds and differential geometry, Providence, R.I., American Mathematical Society, 2009, ISBN 0-8218-4815-1.

Bibliografia

(EN) Nicolas Bourbaki (1989): Elements of Mathematics. Lie groups and Lie algebras, Springer, ISBN 3-540-50218-1.

Voci correlate

Gruppo di Lie

Portale Matematica: accedi alle voci di Wikipedia che trattano di matematica

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