On angular momentum Helmholtz theorems and cohomology of Lie algebras

Stetkaer, Henrik

doi:10.1090/S0002-9947-1975-0410775-3

On angular momentum Helmholtz theorems and cohomology of Lie algebras
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by Henrik Stetkaer

Trans. Amer. Math. Soc. 214 (1975), 349-374

DOI: https://doi.org/10.1090/S0002-9947-1975-0410775-3

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Abstract:

Helmholtz’ 2nd theorem (that every vector field on ${{\mathbf {R}}^3}$ with vanishing curl is gradient of a function) can be viewed as a statement about the group of translations of ${{\mathbf {R}}^3}$. We prove similar theorems for other Lie transformation groups, in particular for semidirect products of abelian and compact semisimple groups. Using Hodge theory we also obtain results analogous to the 1st Helmholtz theorem, but only for compact Lie transformation groups.

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Geomerty of quantum theory