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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 15th 2009
   • PermaLink
   Author: Urs
   Format: Markdownstarted [[Lie algebra cohomology]], (for the moment mainly to record that reference on super Lie algebra cocycles)

   started Lie algebra cohomology,

   (for the moment mainly to record that reference on super Lie algebra cocycles)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 21st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexpolished 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

   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

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeSep 21st 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownThe 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 ?

   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 ?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 21st 2010
   • (edited Sep 21st 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI derive the precise formulas at [[infinity-Lie algebroid valued differential forms]] in the section <a href="http://ncatlab.org/nlab/show/infinity-Lie+algebroid+valued+differential+forms#1morphisms_integration_of_infinitesimal_gauge_transformations_27">integration of infinitesimal gauge transformations</a>

   I derive the precise formulas at infinity-Lie algebroid valued differential forms in the section integration of infinitesimal gauge transformations

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeSep 21st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded Whitehead's lemma

   added Whitehead’s lemma

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeFeb 2nd 2011
   • (edited Feb 2nd 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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 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

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 26th 2018
   • (edited Feb 26th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have recorded the following fact ([here](https://ncatlab.org/nlab/show/Lie+algebra+cohomology#InvariantsInLieAlgebraCohomologyComputedByRelativeLieAlgebraCohomology)) form [Solleveld 02, theorem 2.28](https://ncatlab.org/nlab/show/Lie+algebra+cohomology#Solleveld02): Let 1. $(\mathfrak{g}, [-,-])$ be a [[Lie algebra]] of [[finite number|finite]] [[dimension]]; 1. $(V, \rho)$ a $\mathfrak{g}$-[[Lie algebra module]] of [[finite number|finite]] [[dimension]], which is [[reducible representation|reducible]]; 1. $\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra which is _[[reductive Lie algebra|reductive]] in $\mathfrak{g}$_ in that its [[adjoint representation]] on $\mathfrak{g}$ is [[reducible representation|reducible]] 1. 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!): $$ H^\bullet(\mathfrak{a}; V)^{\mathfrak{h}} \;\simeq\; H^\bullet(\mathfrak{g}, \mathfrak{h}; V) \,. $$

   I have recorded the following fact (here) form Solleveld 02, theorem 2.28:

   Let

   1. $(\mathfrak{g}, [-,-])$ be a Lie algebra of finite dimension;

   2. $(V, \rho)$ a $\mathfrak{g}$-Lie algebra module of finite dimension, which is reducible;

   3. $\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra which is reductive in $\mathfrak{g}$ in that its adjoint representation on $\mathfrak{g}$ is reducible

   4. 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!):

   $$H^\bullet(\mathfrak{a}; V)^{\mathfrak{h}} \;\simeq\; H^\bullet(\mathfrak{g}, \mathfrak{h}; V) \,.$$
8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJun 23rd 2021
   • (edited Jun 23rd 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to today's * [[C. A. Cremonini]], [[Pietro Antonio Grassi]], *Cohomology of Lie Superalgebras: Forms, Pseudoforms, and Integral Forms* ([arXiv:2106.11786](https://arxiv.org/abs/2106.11786)) <a href="https://ncatlab.org/nlab/revision/diff/Lie+algebra+cohomology/35">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie+algebra+cohomology/35">v35</a>, <a href="https://ncatlab.org/nlab/show/Lie+algebra+cohomology">current</a>

   added pointer to today’s

   • C. A. Cremonini, Pietro Antonio Grassi, Cohomology of Lie Superalgebras: Forms, Pseudoforms, and Integral Forms (arXiv:2106.11786)

   diff, v35, current