Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

September 7, 2007

Lie n-Algebra Cohomology

Posted by Urs Schreiber

This here is supposed to move the discussion triggered by John’s “nice problem” in Higher Gauge Theory and Elliptic Cohomology into a new thread. If we already had that Wiki I’d develop everything there and just drop you all a link here. Since we are not at this point yet, here is another entry.


i) What is a characteristic class of an nn-bundle with structure Lie nn-algebra g (n)g_{(n)}?

ii) How are these characteristic classes described in terms of deRham cohomology, starting from an arbitrary connection on the nn-bundle?

ii) In particular, how do the characteristic classes of 2-bundles with g (n):=g ,[,]g_{(n)} := g_{\langle \cdot, [\cdot, \cdot\rangle]} the String group Lie 2-algebra relate to those of ordinary gg-bundles?

Here I’ll describe what looks like a nice answer to this nice problem. It is obtained by combining the nn-groupoid realization of universal nn-bundles in terms of tangent categories and inner automorphisms with the Lie nn-algebra technoloy described in Chern Lie (2n+1)(2n+1) and String and Chern-Simons Lie 3-algebras.

More details on the Lie nn-algebraic aspects are in

Lie nn-algebra cohomology

The tangent-categorical background useful to put this into perspective is discussed in

Tangent Categories (html).

The higher morphisms of morphisms of Lie nn-algebras which play a role are discussed in

Higher morphisms of Lie nn-algebras (html)

Lie nn-algebras in terms of differential algebra

Lie nn-groups are hard to understand, beyond low nn and without restrictive assumptions on their structure.

Lie nn-algebras are easier to understand:

a (semistrict) Lie nn-algebra is an nn-category g (n)g_{(n)} internal to Vect\mathrm{Vect} equipped with an antisymmetric bracket functor [,]:g (n)×g (n)g (n) [\cdot, \cdot ] : g_{(n)} \times g_{(n)} \to g_{(n)} which satisfies the Jacobi identitiy up to coherent equivalence.

What makes this easier to handle is that this entire structure turns out to arrange itself into one single ordinary differential algebra. Or coalgebra, rather.

First, take VV to be the vector space of nn-morphisms of g (n)g_{(n)}. This is naturally \mathbb{Z}-graded: V= k=0 n1V k. V = \oplus_{k = 0}^{n-1} V_{k} \,. The subspaces V 0V k1 V_0 \oplus \cdots \oplus V_{k-1} are the space of kk-morphisms of g (n)g_{(n)}. Alternatively, we may think of V kV_k alone as the space of kk-morphisms that start at the vanishing (k1)(k-1)-morphism.

The funny shift I have inserted (n1n-1 instead of nn) has very important meaning: we should really think of a Lie nn-algebra as a one-object Lie nn-algebroid. This means that what looks like an object in a Lie nn-algebra is best thought of as a 1-morphism, instead.

To reflect this, we pass from V V to its suspension sV. s V \,. This is the same vector space, but with the grading shifted by one sV= k=1 b(sV) k. s V = \oplus_{k=1}^b (sV)_k \,.

So sVsV is the space of morphisms of our Lie nn-algebra g (n)g_{(n)}. In addition to that information, g (n)g_{(n)} carries all the remaining information that make it a monoidal nn-category:

- source and target maps for all kk-morphisms

- identity-assigning maps for all kk-morphisms

- nn different composition laws of nn-morphisms - one of them the bracket functor [,][\cdot, \cdot] itself.

Remarkably, all this information is equivalently encoded in a single operator: a codifferential operator D:S c(sV)S c(sV) D : S^c( s V ) \to S^c(s V) of degree -1 on the free graded-co-commutative coalgebra S c(sV)S^c( s V) over sVsV. All the coherence conditions on the above structures are entirely encoded in the condition that this codifferential squares to 0 D 2=0. D ^2 = 0 \,.


(Semistrict) Lie nn-algebras are the same as free graded-co-commutative coalgebras on generators in degree 1kn1 \leq k \leq n equipped with a nilpotent degree -1 codifferential.

Moreover, free graded-cocommutative coalgras with nilpotent odd codifferential are, in turn, the same as what are called L L_{\infty}-algebras. So we may equivalently say

(Semistrict) Lie nn-algebras are the same as L L_{\infty}-algebras concentrated in degree 0kkeq(n1)0 \leq k \keq (n-1).

Furthermore, it is often useful to pass to the dual description: the quasi-free (since the algebra is free, but not the differential) codifferential coalgebra (S c(sV),D)(S^c(s V), D) gives rise to a quasi-free differential algebra ( (sV *),d) (\wedge^\bullet (s V^*), d) simply by setting dω:=ω(D()) d \omega := - \omega(D(\cdot)) for all dual elements ωsV *\omega \in sV^*.

Hence finally we get

(Semistrict) Lie nn-algebras are the same as free graded-commutative algebras on generators in degree 1kn1 \leq k \leq n equipped with a nilpotent degree +1 codifferential.

This bridge

categorified Lie algebra \leftrightarrow ordinary differential algebra

connects two huge continents. Hence crossing this bridge back and forth is very fruitful. On the left we have conceptual understanding that helps to understand the large-scale properties of our constructions. On the right we have computational control, which allows us to handle the small-scale properties of our constructions.

Universal nn-Bundles in terms of groupoids

It turns out that for G (n)G_{(n)} an nn-group, one obtains a sequence G (n) INN 0(G (n)) ΣG (n) \array{ G_{(n)} \\ \downarrow \\ \mathrm{INN}_0(G_{(n)}) \\ \downarrow \\ \Sigma G_{(n)} } of nn-groupoids (discussed up to n=2n=2 here) which plays the role of the universal G (n)G_{(n)}-nn-bundle in the following sense:

For XX any space and YXY \to X a good cover of it, morphisms of nn-groupoids Y [2]ΣG (n) Y^{[2]} \to \Sigma G_{(n)} classify G (n)G_{(n)}-nn-bundles over XX. The total “space” of such an nn-bundle, in it’s groupoid incarnation is the pullback Y [2]× ΣG (n)INN 0(G (n)) INN 0(G (n)) Y [2] ΣG (n). \array{ Y^{[2]} \times_{\Sigma G_{(n)}} \mathrm{INN}_0(G_{(n)}) &\to&\mathrm{INN}_0(G_{(n)}) \\ \downarrow &&\downarrow \\ Y^{[2]}&\to&\Sigma G_{(n)} } \,. But more its true. As described in more detail (though still not in full detail) in Tangent Categories, the fact that INN 0(G (n))\mathrm{INN}_0(G_{(n)}) is itself an (n+1)(n+1)-group lets us iterate this construction and form G (n) INN 0(G (n)) ΣG (n) INN 0(G (n)) INN 0(INN 0(G (n))) ΣINN 0(G (n)) ΣG (n) ΣINN 0(G (n)). \array{ G_{(n)} &\to& \mathrm{INN}_0(G_{(n)}) &\to& \Sigma G_{(n)} \\ \downarrow && \downarrow &\Downarrow^\simeq& \downarrow \\ \mathrm{INN}_0(G_{(n)}) &\to& \mathrm{INN}_0(\mathrm{INN}_0(G_{(n)})) &\to& \Sigma \mathrm{INN}_0(G_{(n)}) \\ \downarrow &\Downarrow^{\simeq}& \downarrow \\ \Sigma G_{(n)} &\to& \Sigma \mathrm{INN}_0(G_{(n)}) } \,. Here INN 0(INN 0(G (n)))\mathrm{INN}_0(\mathrm{INN}_0(G_{(n)})) is the total space (in terms of n+1n+1-groupoids) of the universal INN 0(G (n))\mathrm{INN}_0(G_{(n)})-(n+1)(n+1)-bundle. Since INN 0(G (n))\mathrm{INN}_0(G_{(n)}) is trivializable, so is this (n+1)(n+1)-bundle. We may think of this trivializable (n+1)(n+1)-bundle as the universal trivializable (n+1)(n+1)-bundle which is trivialized by the universal G (n)G_{(n)}-nn-bundle.

While important in general, this is not the aspect of the above diagram which shall concern us here. Rather, we note that by pushing our pullback Y [2]× ΣG (n)INN 0(G (n)) INN 0(G (n)) Y [2] ΣG (n) \array{ Y^{[2]} \times_{\Sigma G_{(n)}} \mathrm{INN}_0(G_{(n)}) &\to&\mathrm{INN}_0(G_{(n)}) \\ \downarrow &&\downarrow \\ Y^{[2]}&\to&\Sigma G_{(n)} } sideways to the right over the above tableau, we choose a connection (in terms of parallel nn-transport) on our G (n)G_{(n)}-bundle. The morphism

Y [2] ΣG (n) \array{ Y^{[2]} \\ \downarrow \\ \Sigma G_{(n)} } which encodes the transition function/descent data of the nn-bundle may be extended to a morphism Y [2] C 2(Y) (g,tra,curv) ΣG (n) ΣINN 0(G (n)) \array{ Y^{[2]} &\to& C_2(Y) \\ \downarrow && \downarrow^{(g,\mathrm{tra},\mathrm{curv})} \\ \Sigma G_{(n)} &\to& \Sigma \mathrm{INN}_0(G_{(n)}) } which encodes

- the nonabelian G (n)G_{(n)}-cocycle data gg

- a compatible G (n)G_{(n)}-connection tra\mathrm{tra} (hence a nonabelian differential G (n)G_{(n)}-cocycle)

- and the corresponding curvature curv\mathrm{curv}.

This is the situation in the world of Lie nn-groupoids. It is conceptually powerful, but hard to deal with in detail. So pass it over our bridge: by looking at this very situation in the world of Lie nn-algebras and their differential algebra description, we shall arrive at an understanding of characteristic classes of these nn-bundles.

The general strategy is to some extent summarized in this figure:

Universal nn-bundles in terms of Lie nn-algebras

The Lie nn-group G (n)G_{(n)} turns into the Lie nn-algebra g (n)g_{(n)}. The Lie (n+1)(n+1)-group INN 0(G (n))\mathrm{INN}_0(G_{(n)}) turns into the Lie (n+1)(n+1)-algebra inn(g (n))\mathrm{inn}(g_{(n)}).

The Lie nn-algebra invariant polynomials give rise to an abelian Lie rr-algebra, bg (n) b g_{(n)} where rr is the degree of the maximal degree invariant polynomial

Notice that bg (n)b g_{(n)} is not the Lie nn-algebra of ΣG (n)\Sigma G_{(n)}.

In this business here it is very important to distinguish between nn-groups regarded as monoidal (n1)(n-1)-groupoids and the corresponding one-object nn-groupoids:

- G (n)G_{(n)} is an nn-group. Its Lie nn-algebra is g (n)g_{(n)}

- ΣG (n)\Sigma G_{(n)} is then, in general, not monoidal any more. Hence it has, in general, no Lie (n+1)(n+1)-algebra associated with it.

But the Lie rr-algebra bg (n)b g_{(n)} is something like the “best approximation” to this non-existent Lie nn-algebra:

the graded-commutative algebra underlying ΣG (n)\Sigma G_{(n)} is (ssg (n) *). \wedge^\bullet( s s g_{(n)}^* ) \,. That there is, in general, no (n+1)(n+1)-group structure on ΣG (n)\Sigma G_{(n)} is reflected in the fact that the differential d g (n)d_{g_{(n)}} coming from G (n)G_{(n)} does not extend to the above algebra. On the other hand, the differential d inn(g (n))d_{\mathrm{inn}(g_{(n)})} can be restricted to this algebra. But it doesn’t close, in general, on that algebra. bg (n)b g_{(n)} corresponds to the maximal subalgebra on which d inn(g)d_{\mathrm{inn}(g)} does vanish. And this turns out to be nothing but the algebra of invariant polynomials on g (n)g_{(n)} bg (n) *:=inv(g (n)) b g_{(n)}^* := \mathrm{inv}(g_{(n)})

(Compare slide 135-140.)

There is surely a more abstract way to understand what bg (n)b g_{(n)} is. But for the moment let’s use this pedestrian description.

Then g (n) * inn(g (n)) * bg (n) * \array{ g_{(n)}^* \\ \uparrow \\ \mathrm{inn}(g_{(n)})^* \\ \uparrow \\ b g_{(n)}^* } is supposed to play the role of the universal g (n)g_{(n)}-nn-bundle in the world of Lie nn-algebras.

For XX a smooth space, we address a morphism of differential graded algebras

Ω (X)bg (n) * \Omega^\bullet( X ) \leftarrow b g_{(n)}^*

as a classifying map of g (n)g_{(n)}-bundles in the world of Lie nn-algebras. A choice of such morphism is precisely a choice of a closed rr-form K iΩ (X) K_i \in \Omega^\bullet(X) on XX for each degree rr invariant polynomial k i k_i on g (n)g_{(n)}.

Hence we write

Ω (X){K i}bg (n) *. \Omega^\bullet( X ) \stackrel{\{K_i\}}{\leftarrow} b g_{(n)}^* \,.

Completing the cone

inn(g (n)) * Ω (X) {K i} bg (n) * \array{ && \mathrm{inn}(g_{(n)})^* \\ && \uparrow \\ \Omega^\bullet( X ) &\stackrel{\{K_i\}}{\leftarrow}& b g_{(n)}^* }

to the left amounts to choosing a bundle

p:PX p : P \to X

over XX together with a g (n)g_{(n)}-valued connection AA on its total space

Ω (P) (A,F A) inn(g (n)) * p * Ω (X) {K i} bg (n) * \array{ \Omega^\bullet(P) &\stackrel{(A,F_A)}{\leftarrow}& \mathrm{inn}(g_{(n)})^* \\ \uparrow^{p^*} && \uparrow \\ \Omega^\bullet( X ) &\stackrel{\{K_i\}}{\leftarrow}& b g_{(n)}^* }

which is such that feeding its nn-curvatures F AF_A into all invariant g (n)g_{(n)} polynomials k ik_i produces the previously fixed characteristic classes

K i=k i(F A)Ω (X). K_i = k_i(F_A) \in \Omega^\bullet(X) \,.

Demanding furthermore that the pushout of

g (n) * Ω (P) (A,F A) inn(g (n)) * p * Ω (X) {K i=k i(F A)} bg (n) * \array{ && g_{(n)}^* \\ && \uparrow \\ \Omega^\bullet(P) &\stackrel{(A,F_A)}{\leftarrow}& \mathrm{inn}(g_{(n)})^* \\ \uparrow^{p^*} && \uparrow \\ \Omega^\bullet( X ) &\stackrel{\{K_i = k_i(F_A)\}}{\leftarrow}& b g_{(n)}^* }


Ω li (|G (n)|) g (n) * i * Ω (P) (A,F A) inn(g (n)) * p * Ω (X) {K i=k i(F A)} bg (n) * \array{ \Omega^\bullet_{\mathrm{li}}(|G_{(n)}|) &\stackrel{\simeq}{\leftarrow}& g_{(n)}^* \\ \uparrow^{i^*} && \uparrow \\ \Omega^\bullet(P) &\stackrel{(A,F_A)}{\leftarrow}& \mathrm{inn}(g_{(n)})^* \\ \uparrow^{p^*} && \uparrow \\ \Omega^\bullet( X ) &\stackrel{\{K_i = k_i(F_A)\}}{\leftarrow}& b g_{(n)}^* }

amounts to imposing the first Cartan condition: that the connection AA restricted to the fibers of PP mimics the Lie algebra cohomology of g (n)g_{(n)}.

(I expect, judging from Jim Stasheff’s hint that requiring the lower of the two squares to also be suitably universal, probably a pushout, will amount to the second Cartan condition, namely the equivariance of AA. But I am not sure yet how to see this.)

This, then, should be the description of g (n)g_{(n)}-nn-bundles in terms of Lie nn-algebras and their duals.

Characteristic classes of String and g μg_{\mu}-nn-bundles

It is then an not hard to see that

Proposition 6 For gg an ordinary Lie algebra with (n+1)(n+1)-cocycle μ k\mu_k which is in transgression with an invariant polynomial kk on gg, and for g μ k g_{\mu_k} the corresponding Baez-Crans type Lie nn-algebra, we have inv(g μ k)inv(g)/k. \mathrm{inv}(g_{\mu_k}) \simeq \mathrm{inv}(g)/k \,. Hence the characteristic classes of g μ kg_{\mu_k}-nn-bundles PXP \to X in the above sense are those of the corresponding gg-1-bundles, modulo the class coming represented by K=k(F A)K = k(F_A) for the given invariant polynomial kk.

For gg simple and k=,k = \langle \cdot, \cdot\rangle this implies the answer to the nice problem™ that John Baez mentioned in Higher Gauge Theory and Elliptic Cohomology.

There would be more to say. And I intended to say more here. But I am running out of time. So for the moment I end here by relegating all further details to the discussion that can be found in Lie nn-algebra cohomology.

Posted at September 7, 2007 2:08 PM UTC

TrackBack URL for this Entry:

8 Comments & 11 Trackbacks

Re: Lie n-Algebra Cohomology

I tried to help illuminate what’s going on by making the comparison between the integral picture and the differential picture.

But I realize that this may seem more confusing than enlightning, since there is a tricky issue involved which requires extra care:

It’s all about that point which I tried to emphasize in The second edge of the cube:

in order to understand how a (Cartan-)Ehresmann connection, namely

- a 1-form AA on the total space PP of some GG-bundle PXP \to X satisfying two conditions

fits into the big picture where connections are parallel transport functors, possibly expressed in terms of local data and descent data

it is crucial to realize (which is maybe clear to everybody but which I, personally, have never ever seen emphasized or even mentioned) that

the definition of a (Cartan-)Ehresmann connection follows as a special case of the general local-connection-with-descent/gluing-data definition when one chooses the cover

YXY \to X

of base space to coincide with the total space of the bundle itself, i.e. when one sets

P=YP = Y.

The experts will not be shocked by this statement. But it deserves emphasis nevertheless, I believe. This is very important for understanding what is going on above, and for making the connection between the Lie nn-groupoid and the Lie nn-algebra picture.

This I need to better describe.

While I am thinking about how to describe this better: can anyone help me see the answer to the open issue I mentioned:

I think I know that requiring

Ω li (G) g * Ω (P) (A,FA) inn(g) * \array{ \Omega^\bullet_{\mathrm{li}}(G) &\stackrel{\simeq}{\leftarrow}& g^* \\ \uparrow && \uparrow \\ \Omega^\bullet(P) &\stackrel{(A,FA)}{\leftarrow}& \mathrm{inn}(g)^* }

to be a pushout is the first condition on the Ehresmann connection AA: it says that AA pulled back to a fiber has to become the canonical gg-valued 1-form on GG.

I am suspecting that, similarly, requiring Ω (P) (A,FA) inn(g) * Ω (X) {K i=k i(F A)} bg * \array{ \Omega^\bullet(P) &\stackrel{(A,FA)}{\leftarrow}& \mathrm{inn}(g)^* \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\stackrel{\{K_i = k_i(F_A)\}}{\leftarrow}& b g^* } to be universal in one sense or other will be equivalent to the second condition on the Ehresmann connection AA: the equivariance under the gg-action.

That’s strongly suggested by Jim’s comment. But I am not exactly sure yet how to see it.

Posted by: Urs Schreiber on September 8, 2007 1:50 PM | Permalink | Reply to this

Current algebra cohomology

You post inspired me to work out the DGA corresponding to the Kac-Moody-like 2-cocycles of the current algebra in general dimension.

Consider the algebra of maps from n-dimensional space to a finite-dimensional Lie algebra g equipped with a Killing form < . , . >. Let J be a de Rham 0-form and a Lie 1-form valued in the adjoint g rep, and let S be a de Rham 1-form and a Lie 1-form valued in the trivial g rep. In components, J = Ja(x) ea and S = Sμ(x) e dxμ, where ea and e are bases for the adjoint and trivial modules.

There are two differentials, the de Rham differential d and the Lie differential δ, defined by

δ J = J ∧ J
δ S = < J, dJ >

where ∧ denotes the wedge product of Lie 1-forms. It turns out that d2 = d δ + δ d = 0, but δ2 != 0, because

δ2 S = d < J ∧ J, J >

Now I think that in this situation, you can always write down a new differential of the form

s = d + δ + more,

which is nilpotent in the proper sense.

In 1D, the only part of S that survives is the integral

c = ∫ S

We recover the usual affine cocycle,

δ c = ∫ < J, dJ >

and δ is properly nilpotent.

Posted by: Thomas Larsson on September 12, 2007 10:39 AM | Permalink | Reply to this

Re: Current algebra cohomology

Thanks, Thomas!

It’s good that you mention this again.

Maybe if we’d slightly modified this setup by letting SS be a mere scalar and setting δS=J,dJ \delta S = \int \langle J, d J \rangle in the first place?

At some point I need to sit down and figure this out systematically, because these affine algebras must be the transgression to parameter space of the corresponding String-type Lie nn-algebras, in a sense.

For instance the algebra you are talking about here is, for n=1n=1, part of (in that SS is like the central part of the full centrally extended loop group) the large-strict version of the small-weak String Lie 2-algebra g ,[,]g_{\langle \cdot,[\cdot,\cdot]\rangle} (you find the δS=J,dJ\delta S = \langle J , d J \rangle on p. 30 here for instance).

So for this case it is pretty clear what’s going on. But there must be a more general statement lurking here which hasn’t been made explicit yet.

(Of course we talked about this before. Last time here I think.)

Posted by: Urs Schreiber on September 12, 2007 11:30 AM | Permalink | Reply to this

Re: Current algebra cohomology

We cannot integrate S except in 1D. Indeed, de Rham 1-forms can only be integrated in 1D without extra structure.

AFAIU, δ’s lack of nilpotency is not a defect, because the corresponding Lie algebra does satisfy the Jacobi identities. However, Jacobi would fail unless we impose a closedness condition, which is the last line of eq (5) of math-ph/0501023. The closedness condition translates into δ2 != 0. I was quite surprised when I realized this.

That a differential is only nilpotent modulo something exact of another differential is standard, although I don’t remember the correct terminology (relative cohomology?). This happens in BRST, where the KT differential d (which imposes the constraints) and the longitudinal differential δ (which identifies points on gauge orbits) combine into the BRST differential s = d + δ + more. Evidently the existence of such a nilpotent s is enough.

Posted by: Thomas Larsson on September 12, 2007 12:16 PM | Permalink | Reply to this

Re: Current algebra cohomology

We cannot integrate SS except in 1D. Indeed, de Rham 1-forms can only be integrated in 1D without extra structure.

Sorry, I was thinking of letting JJ be an nn form in 2n+12n +1 dimensions, such that JdJ\langle J \wedge d J\rangle is degree 2n+12n +1.

As in the discussion here and here.

Posted by: Urs Schreiber on September 12, 2007 12:24 PM | Permalink | Reply to this
Read the post Obstructions for n-Bundle Lifts
Weblog: The n-Category Café
Excerpt: On obstructions to lifting the structure n-group of n-bundles.
Tracked: September 12, 2007 11:28 PM


What I wrote yesterday was not quite right. To get a nilpotent differential, we must introduce yet another thing R, which is a de Rham 0-form and a Lie 2-form valued in the trivial g rep. The correct DGA should read

δ J = J ∧ J
δ S = < J, dJ > - dR
δ R = < J ∧ J, dJ >

We check that this δ satisfies δ2 = 0, modulo any sign errors in my formulas.

If we dualize this and work in a Fourier basis, we get the nonzero brackets

[Jam, Jbn] = fabc Jcm+n + kab mμ Sμm+n
[Rm] = mμ Sμm
[Jam, Jbn, Jcr] = fabcRm+n+r

This is a 2-algebra, right? Since it comes from dualizing a DGA with a properly nilpotent differential, it should be. I have always imposed the condition Rm = 0, which makes the Jacobi identities hold on the nose.

Another question is whether we gain anything by viewing this structure as a 2-algebra. In all representations that I am aware of, Rm = 0 holds automatically. One must expand all fields around a privileged curve (“the observer’s trajectory”), and the closedness condition states that the integral of a closed 1-form along this curve vanishes. Note that the above 2-algebra in particular holds in 1D, where the condition leads to S1m = Kronecker delta and we recover the usual affine algebra. Is the 2-algebra reformulation useful here?

Posted by: Thomas Larsson on September 13, 2007 5:14 AM | Permalink | Reply to this

Correction to correction

If we don’t require Rm = 0, the cocycle term must be

kab ( mμ - nμ) Sμm+n

to ensure that the JJ bracket is antisymmetric.

Posted by: Thomas Larsson on September 13, 2007 5:20 AM | Permalink | Reply to this

Re: Correction

This is a 2-algebra, right?

Here is the general rule:

the degree nn of the Lie nn-algebra corresponding to a given quasi-free differential graded algebra (qDGCA) is determined like this:

the graded commutative algebra underlying the qDGCA (i.e. that what remains after forgetting the existence of the differential) must be a free graded commutative algebra. Hence a Grassman algebra of some \mathbb{Z}-graded vector space V *=V 1 *V 2 *V n * V^* = V_1^* \oplus V_2^* \oplus \cdots \oplus V_n^* which is nontrivial at most in degree 1dn1 \leq d \leq n.

(Strictly speaking this is defined only for VV finite dimensional. Whereas for the case you consider it is infinite-dimensional. But as long as we consistently behave like physicists, this doesn’t cause problems and can be ignored for the present purpose.)

In the case you have in mind, I think we have {δ(x,)σ a}\{\delta(x,\cdot)\sigma^a\} a “basis” for V 1V_1 (with xx running over parameter space and aa running over a chosen basis of the Lie algebra) and then something in degree 2.

It may require a little more care saying precisely what it is in degree 2. I had something slightly different in mind than what you seem to be doing now.

I guess the graded-commutative algebra which you are dealing with is the algebra generated from the degree 1 and 2 elements of Ω (X) (sg *su(1) *) \Omega^\bullet(X)\otimes \wedge^\bullet(s g^* \oplus s u(1)^*) equipped with the obvious total degree. Here gg is the given Lie algebra, ss the shift operator, which simply says that we regard an element in g *g^* as in degree 1, and “su(1) *s u(1)^*” the thing spanned by what you denote ee.

So, yes, I think what you are describing is a (infinite dimensional) Lie 2-algebra.

Is the 2-algebra reformulation useful here?

Making an implicit nn-categorical structure manifest is useful whenever one wants to understand how the given gadget one already knows and love fits together with other concepts.

For instance, in the topic of the thread here: once we realize that a qDGCA generated in degree nn is nothing but a Lie nn-algebra, we realize that there are things like Lie nn-algebra cocycles and Lie nn-algebra invariant polynomials and the like associated with it (hence the title of this entry).

And it tells you that there are nn-bundles on whose fibers there are differential forms which mimic the original qDGCA and that these nn-bundles have certain properties which can be read off from the Lie nn-algebra cocycles and invariant polynomials of the given qDGCA.

Then, of course, realizing a hiddn nn-categorical structure tells us more about the right notion of morphisms and of higher morphisms which to use.

For instance, in the variation of the qDGCA which you describe that I considered above, it turns out that the Lie 2-algebra picture, which gives one the right notion of equivalence of these beasts, shows that the infinite dimensional Lie 2-algebra was in fact equivalent to a much smaller finite dimensional one. Things like that can be very useful.

Posted by: Urs Schreiber on September 13, 2007 11:51 AM | Permalink | Reply to this
Read the post Cohomology of the String Lie 2-Algebra
Weblog: The n-Category Café
Excerpt: On the Lie 2-algebra cohomlogy of the String Lie 2-algebra and its relation to twisted K-theory.
Tracked: October 8, 2007 3:18 AM
Read the post Categorified Clifford Algebra and weak Lie n-Algebras
Weblog: The n-Category Café
Excerpt: On weak Lie n-algebras, differential graded Clifford algebra and Roytenberg's work on weak Lie 2-algebras.
Tracked: October 9, 2007 4:49 PM
Read the post BV-Formalism, Part IV
Weblog: The n-Category Café
Excerpt: Lie algebroids of action groupoids and their relation to BRST formalism.
Tracked: October 11, 2007 9:47 PM
Read the post On Lie N-tegration and Rational Homotopy Theory
Weblog: The n-Category Café
Excerpt: On the general ideal of integrating Lie n-algebras in the context of rational homotopy theory, and about Sullivan's old article on this issue in particular.
Tracked: October 20, 2007 4:14 PM
Read the post Modules for Lie infinity-Algebras
Weblog: The n-Category Café
Excerpt: On modules for Lie infinity algebras in general and the definition given by Lars Kjeseth in particular.
Tracked: November 13, 2007 9:23 PM
Read the post Lie oo-Connections and their Application to String- and Chern-Simons n-Transport
Weblog: The n-Category Café
Excerpt: A discussion of connections for general L-infinity algebras and their application to String- and Chern-Simons n-transport.
Tracked: December 25, 2007 7:37 PM
Read the post Differential Forms and Smooth Spaces
Weblog: The n-Category Café
Excerpt: On turning differential graded-commutative algebras into smooth spaces, and interpreting these as classifying spaces.
Tracked: January 30, 2008 10:16 AM
Read the post What I learned from Urs
Weblog: The n-Category Café
Excerpt: Bruce Bartlett talks about some aspects of the program of systematically understanding the quantization of Sigma-models in terms of sending parallel transport n-functors to the cobordism representations which encode the quantum field theory of the n-pa...
Tracked: February 26, 2008 10:41 AM
Read the post Thoughts (mostly on super infinity-things)
Weblog: The n-Category Café
Excerpt: Thoughts while travelling and talking.
Tracked: April 22, 2008 4:06 AM
Read the post Differential Graded Clifford Algebra
Weblog: The n-Category Café
Excerpt: On Clifford-defomrations of Chevalley-Eilenberg and Weil algebras.
Tracked: August 17, 2008 12:43 PM

Post a New Comment