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November 23, 2006

A Third Model of the String Lie 2-Algebra

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 𝔤\mathfrak{g} has a god-given one-parameter family of categorifications 𝔤 k\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), 1699-1722.

    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 𝔰𝔩 2()\mathfrak{sl}_2(\mathbb{R}), where the generator is usually called Godbillon-Vey class.

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 𝔤\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 𝔤 k\mathfrak{g}_k we still have [x,[y,z]]=[[x,y],z]+[x,[y,z]] [x,[y,z]] = [[x,y],z] + [x,[y,z]] for any x,y,z𝔤x,y,z \in \mathfrak{g}, but we define the Jacobiator in a nontrivial way: it’s given by J x,y,z=k[x,y],z J_{x,y,z} = k \langle [x,y], z \rangle where ,\langle \cdot, \cdot \rangle is the Killing form and kk is any complex number. We get a one-parameter family of Lie 2-algebras 𝔤 k\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. 𝔤 k\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’, string G\mathrm{string}_G. The reason for these terms is that it’s constructed using paths in GG together with the central extension of the loop group of GG - a key player in string theory.

Both From Loop Groups to 2-Groups, and Andre Henriques’ paper Integrating L 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:BC t : B \to C together with a representation of CC on BB satisfying two equations.

Any differential crossed module gives a 4-term exact sequence of Lie algebras 0ABtCD0 0 \to A \to B \stackrel{t}{\to} C \to D \to 0 where A=ker(t) A = ker (t) and D=coker(t) 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 AA and DD correspond to elements of the Lie algebra cohomology group H 3(D,A). H^3(D,A). If 𝔤\mathfrak{g} is a complex simple Lie algebra, H 3(𝔤,)=. 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 𝔤 k\mathfrak{g}_k.

What Wagemann does is explicitly construct a 4-term exact sequence 0ABtCD0 0 \to A \to B \stackrel{t}{\to} C \to D \to 0 that does the job when k=1k = 1. It looks like this: 0(U𝔤) #(U𝔤 +)× α𝔤𝔤0 0 \to \mathbb{C} \to (U\mathfrak{g})^# \to (U\mathfrak{g}^+) \times_\alpha \mathfrak{g} \to \mathfrak{g} \to 0 Note that 𝔤\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 𝔤 k\mathfrak{g}_k. But what’s the stuff in the middle?

U𝔤U\mathfrak{g} is the universal enveloping algebra of 𝔤\mathfrak{g}. By the Poincaré-Birkhoff-Witt theorem, as a vector space we have U𝔤𝔤S 2𝔤 U\mathfrak{g} \cong \mathbb{C} \oplus \mathfrak{g} \oplus S^2 \mathfrak{g} \oplus \cdots where S n𝔤S^n \mathfrak{g} is the space of nnth-degree polynomials in 𝔤\mathfrak{g}. So, (U𝔤) * *𝔤 *S 2𝔤 * (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 (U𝔤) #(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: (U𝔤) #= *𝔤 *S 2𝔤 * (U\mathfrak{g})^# = \mathbb{C}^* \oplus \mathfrak{g}^* \oplus S^2 \mathfrak{g}^* \oplus \cdots

U𝔤 +U\mathfrak{g}^+ is the augmentation ideal of the universal enveloping algebra - that is, the subspace like this: U𝔤 +𝔤S 2𝔤 U\mathfrak{g}^+ \cong \mathfrak{g} \oplus S^2 \mathfrak{g} \oplus \cdots (U𝔤 +) #(U\mathfrak{g}^+)^# is the restricted dual of the augmentation ideal: (U𝔤 +) #=𝔤 *S 2𝔤 * (U\mathfrak{g}^+)^# = \mathfrak{g}^* \oplus S^2 \mathfrak{g}^* \oplus \cdots

Finally, (U𝔤 +)× α𝔤(U\mathfrak{g}^+) \times_\alpha \mathfrak{g} is some central extension of the Lie algebra 𝔤\mathfrak{g} by the vector space U𝔤 +U\mathfrak{g}^+, defined by some 2-cocycle α\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:

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!

Posted at November 23, 2006 4:29 AM UTC

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Re: A Third Model of the String Lie 2-Algebra

moths to a candle flame

I challenge all potential moths to fly to the next candle flame: what is the familiy of Lie 3-algebras canonically associated with any Lie algebra?

My proposal: the Chern-Simons Lie-3-algebra #.

Posted by: urs on November 23, 2006 9:42 AM | Permalink | Reply to this

Re: A Third Model of the String Lie 2-Algebra

For a different formulation of the final theorem 9 in Wagemann’s paper, see subsection 3.5 of my math-ph/0002016.

Posted by: Thomas Larsson on November 23, 2006 10:04 AM | Permalink | Reply to this

Re: A Third Model of the String Lie 2-Algebra

taken to calling it the ‘string Lie 2-algebra’, string G\mathrm{string}_G. The reason for these terms is that it’s constructed using paths in GG together with the central extension of the loop group of GG - a key player in string theory.

Just for the record, the reason for calling the string Lie 2-algebra the string Lie 2-algebra is somewhat stronger than this statement might suggest.

There is a topological (1-)group that is called the “String group” because it plays for superstrings the role that the Spin group plays for spinning particles.

I once tried to review that here.

And the 2-group that integrates what I call the string Lie 2-algebra is the smooth version of this topological group, in the sense that the String group is the realization of the nerve of that 2-group.

This is reviewed here.

Posted by: urs on November 23, 2006 12:00 PM | Permalink | Reply to this

Re: A Third Model of the String Lie 2-Algebra

Behrang Noohi has recently published in HHA
Notes on 2-groupoids, 2-groups and crossed modules. It is certianly well written.
He writes: Some of these results are presumably folklore and some appear to be new.

The arXiv version is

math.CT/0512106 [abs, ps, pdf, other] :
Title: Notes on 2-groupoids, 2-groups and crossed-modules
Authors: Behrang Noohi
Comments: 27 pages
Subj-class: Category Theory

see also
2. math.CT/0506313 [abs, ps, pdf, other] :
Title: On weak maps between 2-groups
Authors: Behrang Noohi

Would the n-cat cafe bloggers care to comment?

Posted by: jim stasheff on December 2, 2006 1:40 AM | Permalink | Reply to this

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