Not signed in

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

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry 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 homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics 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 superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 18th 2010
   • (edited Aug 18th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHave been polishing and expanding the first part of [[invariant polynomial]].

   Have been polishing and expanding the first part of invariant polynomial.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 19th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexspelled out in hopefully pedagogical detail at [[invariant polynomial]] in the section <a href="http://ncatlab.org/nlab/show/invariant+polynomial#OnLieAlgebras">On Lie algebras</a> the proof that the definition given there (closed elements in shifted generators inthe Weil algebra) is equivalent to the traditional one

   spelled out in hopefully pedagogical detail at invariant polynomial in the section On Lie algebras the proof that the definition given there (closed elements in shifted generators inthe Weil algebra) is equivalent to the traditional one

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 26th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexAt [[invariant polynomial]] had been missing a definition of the notion of equivalence of invariant polynomials on $L_\infty$-algebras that are not Lie algebras (where the notion does not play much of a role). I have added that into the Definition-section and then started adding to the Examples- and Properties-section a discusison on how this produces the expected behaviour that for $$ b^{n-1} \mathbb{R} \to \mathfrak{g}_\mu \to \mathfrak{g} $$ a shifted central extension of $L_\infty$-algebras induced from a transgressive cocycle $\mu : \mathfrak{g} \to b^n \mathbb{R}$ transgressing to $\langle-\rangle_\mu$, the invariant polynomials of $\mathfrak{g}_\mu$ are generated from the generators of those of $\mathfrak{g}$ except $\langle - \rangle_\mu$.

   At invariant polynomial had been missing a definition of the notion of equivalence of invariant polynomials on $L_\infty$-algebras that are not Lie algebras (where the notion does not play much of a role). I have added that into the Definition-section and then started adding to the Examples- and Properties-section a discusison on how this produces the expected behaviour that for

   $$b^{n-1} \mathbb{R} \to \mathfrak{g}_\mu \to \mathfrak{g}$$

   a shifted central extension of $L_\infty$-algebras induced from a transgressive cocycle $\mu : \mathfrak{g} \to b^n \mathbb{R}$ transgressing to $\langle-\rangle_\mu$, the invariant polynomials of $\mathfrak{g}_\mu$ are generated from the generators of those of $\mathfrak{g}$ except $\langle - \rangle_\mu$.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 30th 2011
   • (edited Jun 30th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded in a new subsection [on reductive Lie algebras](http://ncatlab.org/nlab/show/invariant+polynomial#OnReductiveLieAlg) the statement that the invariant polynomials there form a free graded algebra on the indecomposables

   added in a new subsection on reductive Lie algebras the statement that the invariant polynomials there form a free graded algebra on the indecomposables

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeAug 28th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to the original references by Weil, Cartan and Chern (same original references that are now at _[[Chern-Weil homomorphism]]_) <a href="https://ncatlab.org/nlab/revision/diff/invariant+polynomial/37">diff</a>, <a href="https://ncatlab.org/nlab/revision/invariant+polynomial/37">v37</a>, <a href="https://ncatlab.org/nlab/show/invariant+polynomial">current</a>

   added pointer to the original references by Weil, Cartan and Chern (same original references that are now at Chern-Weil homomorphism)

   diff, v37, current