The n-Category Café

The first approach to categorifying Lie algebras was developed in Alissa Crans’ thesis and published in HDA6. This construction gives a ‘semistrict Lie 2-algebra’ - 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 1-dimensional 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 one-parameter family of Lie 2-algebras $\mathfrak{g}_k$.

The second approach was developed in a paper by Alissa Crans, Urs Schreiber, Danny Stevenson and myself, called From Loop Groups to 2-Groups. Here we did a well-known sort of tradeoff. A Lie 2-algebra is skeletal if isomorphic objects are equal. It’s strict if the Jacobiator is the identity. Any Lie 2-algebra 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 2-algebra which is strict but not skeletal. We called it the ‘path Lie 2-algebra’, but Urs has taken to calling it the ‘string Lie 2-algebra’, $\mathrm{string}_G$. 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 2-Groups, and Andre Henriques’ paper Integrating $L_\infty$-Algebras, discuss the problem of constructing a Lie 2-group associated to this Lie 2-algebra.

Wagemann’s new paper describes yet another nonisomorphic but equivalent Lie 2-algebra. Like the string Lie 2-algebra, it is strict but not skeletal. However, it’s described in a purely algebraic way!

As shown in my paper Higher Yang-Mills Theory, a strict Lie 2-algebra 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 4-term 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 1-parameter family of differential crossed modules - and thus Lie 2-algebras - from any simple Lie algebra! These are equivalent to the Lie 2-algebras $\mathfrak{g}_k$.

What Wagemann does is explicitly construct a 4-term 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 2-algebra $\mathfrak{g}_k$. But what’s the stuff in the middle?

$U\mathfrak{g}$ is the universal enveloping algebra of $\mathfrak{g}$. By the Poincaré-Birkhoff-Witt 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 $n$th-degree 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 2-cocycle $\alpha$.

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 Godbillon-Vey cocycle.

and Gerstenhaber’s work on Lie algebra crossed modules and 3-cocycles, in this classic paper… which, err… umm… I’ve never read:      

• Murray Gerstenhaber, On deformations of rings and algebras, II, Ann. Math. 84 (1966), 1-19.

It all goes to show that intrinsically interesting objects attract mathematicians like moths to a candle flame. Their discovery, and rediscovery, is almost inevitable!