Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science connection constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 15th 2009

    started Lie algebra cohomology,

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

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2010

    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

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeSep 21st 2010

    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 ?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2010
    • (edited Sep 21st 2010)
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2010

    added Whitehead’s lemma

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2011
    • (edited Feb 2nd 2011)

    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

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2018
    • (edited Feb 26th 2018)

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


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

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

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

    4. such that

      𝔤=𝔥𝔞 \mathfrak{g} = \mathfrak{h} \ltimes \mathfrak{a}

      is a semidirect product Lie algebra (hence 𝔞\mathfrak{a} a Lie ideal).

    Then the invariants in Lie algebra cohomology of 𝔞\mathfrak{a} (equivalently with respect to 𝔥\mathfrak{h} or all of 𝔤\mathfrak{g}) coincide with the relative Lie algebra cohomology (using the invariant subcomplex!):

    H (𝔞;V) 𝔥H (𝔤,𝔥;V). H^\bullet(\mathfrak{a}; V)^{\mathfrak{h}} \;\simeq\; H^\bullet(\mathfrak{g}, \mathfrak{h}; V) \,.
Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)