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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
started Lie algebra cohomology,
(for the moment mainly to record that reference on super Lie algebra cocycles)
polished and expanded Lie algebra cohomology: added an Idea-section, collected the different definitions together, added explanations to the definition via oo-Lie algebra morphisms, expanded the section on Extension, started an Examples, section
The words "infinitesimal gauge transformation" in one entry point to gauge transformation while in gauge transformation to infinitesimal object. At both places allusion is just half-clear so far. Could you have exact statement ? Infinitesimal gauge transformations are infinitesimal object in which category/setup ? Can this explanation be more than allusive playing with words ?
I derive the precise formulas at infinity-Lie algebroid valued differential forms in the section integration of infinitesimal gauge transformations
added Whiteheadâ€™s lemma
I was being asked, and so I added a textbook reference to Chevalley-Eilenberg algebra, to Lie algebra cohomology and and a pointer to an article to nonabelian Lie algebra cohomology
I have recorded the following fact (here) form Solleveld 02, theorem 2.28:
Let
$(\mathfrak{g}, [-,-])$ be a Lie algebra of finite dimension;
$(V, \rho)$ a $\mathfrak{g}$-Lie algebra module of finite dimension, which is reducible;
$\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra which is reductive in $\mathfrak{g}$ in that its adjoint representation on $\mathfrak{g}$ is reducible
such that
$\mathfrak{g} = \mathfrak{h} \ltimes \mathfrak{a}$is a semidirect product Lie algebra (hence $\mathfrak{a}$ a Lie ideal).
Then the invariants in Lie algebra cohomology of $\mathfrak{a}$ (equivalently with respect to $\mathfrak{h}$ or all of $\mathfrak{g}$) coincide with the relative Lie algebra cohomology (using the invariant subcomplex!):
$H^\bullet(\mathfrak{a}; V)^{\mathfrak{h}} \;\simeq\; H^\bullet(\mathfrak{g}, \mathfrak{h}; V) \,.$1 to 7 of 7