added a table with some homotopy groups in the unstable range to *orthogonal group – Homotopy groups*

started a bare minimum at *Poisson-Lie T-duality*, for the moment just so as to have a place to record the two original references

created *equivariant de Rham cohomology* with a brief note on the Cartan model.

(I seem to remember that we had discussion of this in the general context of Lie algebroids elsewhere already, several years back. But now I cannot find it….)

]]>Today I was asked for what I know about the development of the theory of Kan-fibrant simplicial manifolds. I realized that the nLab does not discuss this, so I have started a page now with the facts that come to mind right away. (Likely I forgot various things that should still be added.)

]]>expanded *E6* a bit.

have split-off a stub *positive energy representation* from *loop group*

have started model structure for L-infinity algebras

]]>*unitary matrix*. just for completeness

at *Atiyah Lie groupoid* was this old query box discussion, which hereby I am moving from there to here:

+– {: .query} What is all of this $diag$ stuff? I don't understand either $(P \times P)/_{diag} G$ or $(P_x \times P_x)_{diag} G$. —Toby

David Roberts: It’s to do with the diagonal action of $G$ on $P\times P$ as opposed to the antidiagonal (if $G$ is abelian) or the action on only one factor. I agree that it’s a bad notation.

*Toby*: How well do you think it works now, with the notation suppressed and a note added in words? (For what it's worth, the diagonal action seems to me the only obvious thing to do here, although admittedly the others that you mention do exist.)

*Todd*: I personally believe it works well. A small note is that this construction can also be regarded as a tensor product, regarding the first factor $P$ as a right $G$-module and the second a left module, where the actions are related by $g p = p g^{-1}$.

*Toby*: H'm, maybe we should write diagonal action if there's something interesting to say about it.
=–

added some very basic facts on $SU(2)$ here to *special unitary group*. Just so as to be able to link to them.

I tried to polish the "Idea" and the "References" section at [[Courant algebroid]] to something more comprehensive.

]]>added a little bit to *foliation*: a brief list of equivalent alternative definitions and and Idea-section with some general remarks.

added to *G2* the definition of $G_2$ as the subgroup of $GL(7)$ that preserves the associative 3-form.

started something at *ADE classification*, but am out of steam (and time) now.

have started something at *orthosymplectic super Lie algebra* and have added little bits and pieces to various related entries, such as first sketchy notes at *super Lie algebra – classification* and at *supersymmetry – Classification – superconformal symmetry*.

Nothing of this is done yet, but I need to call it quits now.

]]>New entry PBW theorem and stub primitive element. Related new stubs filtered ring and associated graded ring with redirects filtered algebra, associated graded algebra.

]]>started a minimum at *Loday-Quillen-Tsygan theorem*. Added the brief statement also to the Properties-section of related entries: *general linear Lie algebra*, *Lie algebra homology* and *cyclic homology*

I gave *Dickey bracket* its own entry (just a brief Idea-section and references)

(the term “Dickey bracket” used to redirect to *conserved current*, where however it was mentioned only in the references. Now it should be easier to discern what the pointer is pointing to. Of course the entry remains a stub nonetheless.)

I noticed that *exceptional Lie algebra* was still a missing entry. Just in order to make links work, I created a stub for it. No time for more at the moment.

I had need to point specifically to ideals in Lie algebras, so I gave them a little entry *Lie ideal*.

In the nLab article on the universal enveloping algebra, the section describing the Hopf algebra structure originally stated that “the coproduct $\Delta: U L \to U(L \coprod L)\cong U L\otimes UL$ is induced by the diagonal map $L \to L \coprod L$.”

I assume that this is a mistake, and I have since changed the coproduct $\coprod$ to a product $\times$. However, I don’t know a great deal about Hopf algebras, so please correct me if I’ve made a mistake here.

]]>The stub entry *model structure on simplicial Lie algebras* used to point to *model structure on simplicial algebras*. But is it really a special case of the discussion there?

Quillen 69 leaves the definition of the model structure to the reader. Is it with weak equivalences and fibrations those on the underlying simplicial sets? Is this a simplicially enriched model category?

]]>am starting [[model structure on dg-coalgebras]].

In the process I

created a stub for [[dg-coalgebra]]

and linked to it from [[L-infinity algebra]]