A Third Model of the String Lie 2Algebra
Posted by John Baez
One of the main themes of this blog is categorification: taking mathematical structures that are sets with extra structure, and replacing equations by isomorphisms to make them into categories. A wonderful fact is that any Lie algebra $\mathfrak{g}$ has a godgiven oneparameter family of categorifications $\mathfrak{g}_k$. We already have two ways to construct this gadget. Now this paper gives a third:

Friederich Wagemann, On Lie algebra crossed modules,
Communications in Algebra 34 (2006), 16991722.
Abstract: This article constructs a crossed module corresponding to the generator of the third cohomology group with trivial coefficients of a complex simple Lie algebra. This generator reads as $\langle [,],\rangle$, constructed from the Lie bracket $[,]$ and the Killing form $\langle ,  \rangle$. The construction is inspired by the corresponding construction for the Lie algebra of formal vector fields in one formal variable on $\mathbb{R}$, and its subalgebra $\mathfrak{sl}_2(\mathbb{R})$, where the generator is usually called GodbillonVey class.
The first approach to categorifying Lie algebras was developed in Alissa Crans’ thesis and published in HDA6. This construction gives a ‘semistrict Lie 2algebra’  a category that’s like a Lie algebra, but where the Jacobi identity holds only up to a specified isomorphism, called the Jaocbiator. The idea is simple. We start with a complex Lie algebra $\mathfrak{g}$ and take this as our space of objects. For each object, we then put in a 1dimensional space $\mathbb{C}$ of endomorphisms. In $\mathfrak{g}_k$ we still have $[x,[y,z]] = [[x,y],z] + [x,[y,z]]$ for any $x,y,z \in \mathfrak{g}$, but we define the Jacobiator in a nontrivial way: it’s given by $J_{x,y,z} = k \langle [x,y], z \rangle$ where $\langle \cdot, \cdot \rangle$ is the Killing form and $k$ is any complex number. We get a oneparameter family of Lie 2algebras $\mathfrak{g}_k$.
The second approach was developed in a paper by Alissa Crans, Urs Schreiber, Danny Stevenson and myself, called From Loop Groups to 2Groups. Here we did a wellknown sort of tradeoff. A Lie 2algebra is skeletal if isomorphic objects are equal. It’s strict if the Jacobiator is the identity. Any Lie 2algebra is equivalent to a strict one, and to a skeletal one, but not usually one that’s simultaneously strict and skeletal. $\mathfrak{g}_k$ is skeletal but not strict  so we constructed a new, nonisomorphic but equivalent Lie 2algebra which is strict but not skeletal. We called it the ‘path Lie 2algebra’, but Urs has taken to calling it the ‘string Lie 2algebra’, $\mathrm{string}_G$. The reason for these terms is that it’s constructed using paths in $G$ together with the central extension of the loop group of $G$  a key player in string theory.
Both From Loop Groups to 2Groups, and Andre Henriques’ paper Integrating $L_\infty$Algebras, discuss the problem of constructing a Lie 2group associated to this Lie 2algebra.
Wagemann’s new paper describes yet another nonisomorphic but equivalent Lie 2algebra. Like the string Lie 2algebra, it is strict but not skeletal. However, it’s described in a purely algebraic way!
As shown in my paper Higher YangMills Theory, a strict Lie 2algebra is the same thing as a ‘differential crossed module’  the Lie algebra analogue of a crossed module. Wagemann calls such a thing a ‘Lie algebra crossed module’. It consists of a Lie algebra homomorphism $t : B \to C$ together with a representation of $C$ on $B$ satisfying two equations.
Any differential crossed module gives a 4term exact sequence of Lie algebras $0 \to A \to B \stackrel{t}{\to} C \to D \to 0$ where $A = ker (t)$ and $D = coker(t)$ Conversely, given such an exact sequence, we get a differential crossed module!
Since exact sequences can be classified using cohomology, Gerstenhaber was able to show that equivalence classes of differential crossed modules with fixed $A$ and $D$ correspond to elements of the Lie algebra cohomology group $H^3(D,A).$ If $\mathfrak{g}$ is a complex simple Lie algebra, $H^3(\mathfrak{g},\mathbb{C}) = \mathbb{C}.$ So, we get a 1parameter family of differential crossed modules  and thus Lie 2algebras  from any simple Lie algebra! These are equivalent to the Lie 2algebras $\mathfrak{g}_k$.
What Wagemann does is explicitly construct a 4term exact sequence $0 \to A \to B \stackrel{t}{\to} C \to D \to 0$ that does the job when $k = 1$. It looks like this: $0 \to \mathbb{C} \to (U\mathfrak{g})^# \to (U\mathfrak{g}^+) \times_\alpha \mathfrak{g} \to \mathfrak{g} \to 0$ Note that $\mathfrak{g}$ and $\mathbb{C}$ show up at the ends here  these are secretly the objects and endomorphisms for our good old Lie 2algebra $\mathfrak{g}_k$. But what’s the stuff in the middle?
$U\mathfrak{g}$ is the universal enveloping algebra of $\mathfrak{g}$. By the PoincaréBirkhoffWitt theorem, as a vector space we have $U\mathfrak{g} \cong \mathbb{C} \oplus \mathfrak{g} \oplus S^2 \mathfrak{g} \oplus \cdots$ where $S^n \mathfrak{g}$ is the space of $n$thdegree polynomials in $\mathfrak{g}$. So, $(U\mathfrak{g})^* \supset \mathbb{C}^* \oplus \mathfrak{g}^* \oplus S^2 \mathfrak{g}^* \oplus \cdots$ The thing on the right is called the restricted dual, denoted $(U\mathfrak{g})^#$. It’s a bit smaller than the actual dual, since if we have an element of an infinite direct sum, only finitely many terms can be nonzero. So: $(U\mathfrak{g})^# = \mathbb{C}^* \oplus \mathfrak{g}^* \oplus S^2 \mathfrak{g}^* \oplus \cdots$
$U\mathfrak{g}^+$ is the augmentation ideal of the universal enveloping algebra  that is, the subspace like this: $U\mathfrak{g}^+ \cong \mathfrak{g} \oplus S^2 \mathfrak{g} \oplus \cdots$ $(U\mathfrak{g}^+)^#$ is the restricted dual of the augmentation ideal: $(U\mathfrak{g}^+)^# = \mathfrak{g}^* \oplus S^2 \mathfrak{g}^* \oplus \cdots$
Finally, $(U\mathfrak{g}^+) \times_\alpha \mathfrak{g}$ is some central extension of the Lie algebra $\mathfrak{g}$ by the vector space $U\mathfrak{g}^+$, defined by some 2cocycle $\alpha$.
Wagemann’s new paper does not refer to any of the previous work just described. Unfortunately it was published before being put on the arXiv, so nobody could tell him in time. On the other hand, I wish I’d been aware of his previous paper:
 Friedrich Wagemann, A crossed module giving the GodbillonVey cocycle.
and Gerstenhaber’s work on Lie algebra crossed modules and 3cocycles, in this classic paper… which, err… umm… I’ve never read:
 Murray Gerstenhaber, On deformations of rings and algebras, II, Ann. Math. 84 (1966), 119.
It all goes to show that intrinsically interesting objects attract mathematicians like moths to a candle flame. Their discovery, and rediscovery, is almost inevitable!
Re: A Third Model of the String Lie 2Algebra
I challenge all potential moths to fly to the next candle flame: what is the familiy of Lie 3algebras canonically associated with any Lie algebra?
My proposal: the ChernSimons Lie3algebra #.