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

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

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.

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

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]]

made *curved L-infinity algebra* explicit

I am working on the next chapter in *geometry of physics*:

This is not done yet, but it should already be readable. To some extent I am taking the talk notes *Super Lie n-algebra of Super p-branes (schreiber)* and expand them into fully fledged lecture notes.

Since I am editing in a separate window, at this point I ask that everyone who feels like touching this page, even if it just concerns tiny changes (typos) to please alert me.

]]>quick entry *Haag–Lopuszanski–Sohnius theorem*

I have added explicit details to *super L-infinity algebra*.

Added a remark that the formal dual of this concept was introduced and made use of by supergravity theorists a full decade before plain $L_\infty$-algebras were considered by mathematicians.

(I am writing this material for inclusion in the more comprehensive lecture notes *geometry of physics – superalgebra* and *geometry of physics – fundamental super p-branes*.)

The Hinich-Pridham-Lurie theorem on formal moduli problems says that unbounded $L_\infty$-algebras over some field are equivalently the (“infinitesimally cohesive”) infinitesimal $\infty$-group objects over the derived site over that field.

May we say anything about an analogous statement for infinitesimal group objects in the tangent $\infty$-topos of the $\infty$-topos over the site of smooth manifolds?

There exists for instance the $L_\infty$-algebra whose CE-algebra is

$\{ d h_3 = 0, d \omega_{2p+2} = h_3 \wedge \omega_{2p} | p \in \mathbb{Z} \} \,.$This looks like it wants to correspond to the smooth parameterized spectrum whose base smooth stack is the smooth group 2-stack $\mathbf{B}^2 U(1)$, and whose smooth parameterized spectrum is the pullback along $\mathbf{B}^2 U(1) \to \mathbf{B}GL_1(\mathbf{KU})$ of the canonical smooth parameterized spectrum over $\mathbf{B}GL_1(\mathbf{KU})$, for $\mathbf{KU}$ a smooth sheaf of spectra representing multiplicative differential KU-theory.

So it looks like it wants to be this way. How much more can we say?

]]>I finally realized that this ought to exist. And sure enough, it had been constructed already: the *4d supergravity Lie 2-algebra*-extension of the 4d $N = 2$ super-Poincaré super-Lie algebra. I have added a minimum of an Idea-section and pointers to the references.

There is a stub adjoint representation which in my opinion should be the same entry as adjoint action, hence should be merged. Words representation and action are in general equivalent; to each action $G\times M\to M$ one assigns a representation $G\to End M$ and viceversa (up to nicetess of inner hom spaces etc.). True, the specialists in Lie theory like to prefer calling representation when they have a **linear representation** but their own textbooks start with nonlinear case. Thus action of a Lie group on a Lie group is nonlinear hance usually the action terminology used while on a Lie algebra more often the representation is used, but it is not a rule, and the distinction does not survive in generalizations (like the Hopf algebra); any sensible entry, as the main entry “adjoint action” should relate th nonlinear case and its linearization hence should not be in separate entries. I wanted to write some references for adjoint action for quantum groups but gave up as I do not know into which of the entries and expect a decision on fiture fate of the two entries first.

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

Chenchang Zhu had been running a course titled “higher bundle theory” in Göttingen last semester. It ended up being mostly about Lie groupoids and stacks. She and her students used the relevant $n$Lab pages as lectures notes, and they added more stuff to these $n$Lab pages as they saw the need.

I just learned of this from Chenchang.

She had created an $n$Lab page

which lists the $n$Lab entries that were used and edited.

For instance the first one is *Lie groupoid* and Chenchang Zhu as well as some of her students added some stuff to that entry, such as this section *Morphisms of Lie groupoids*. Below that they added a section on Morphisms of Lie algebroids. (Maybe some of this could be reorganized a little now.)

have started model structure for L-infinity algebras

]]>Hi, do someone have patience to explain how is the Lie differentiation of $\infty$-Lie groupoids? More precisely, I’m assuming that a $\infty$-Lie groupoid is a simplicial manifold satisfying the usual Kan fibrancy assumption (the restriction from n-simplices to the i-th horn is a surjective submersion) as in http://arxiv.org/abs/math/0603563 . However in nlab, as I understand, it’s assumed a more general case in the cohesive synthetic setting (which I know almost nothing about and apparently is not so explicit) using infinitesimally thickened simplexes. Is there a more explicit construction that differentiate a Lie groupoid (in the sense described above) to an $\infty$-Lie algebroid? Maybe an illustrative example would be how to make this construction using the nerve of an ordinary Lie groupoid to obtain the usual Lie algebroid (as a cochain complex concentrated in degreee 1).

Thanks in advance.

]]>The entry *Lie algebra extension* used to have only a discussion of the fairly exotic topic of classification in nonabelian Lie algebra cohomology. I have now added an Idea-section with some more introductory and more traditional remarks. This could well be expanded much further.