## 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.

Question

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

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

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

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

More details on the Lie $n$-algebraic aspects are in

Lie $n$-algebra cohomology

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

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

Lie $n$-algebras in terms of differential algebra

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

Lie $n$-algebras are easier to understand:

a (semistrict) Lie $n$-algebra is an $n$-category $g_{(n)}$ internal to $\mathrm{Vect}$ equipped with an antisymmetric bracket functor $[\cdot, \cdot ] : g_{(n)} \times g_{(n)} \to g_{(n)}$ 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 $V$ to be the vector space of $n$-morphisms of $g_{(n)}$. This is naturally $\mathbb{Z}$-graded: $V = \oplus_{k = 0}^{n-1} V_{k} \,.$ The subspaces $V_0 \oplus \cdots \oplus V_{k-1}$ are the space of $k$-morphisms of $g_{(n)}$. Alternatively, we may think of $V_k$ alone as the space of $k$-morphisms that start at the vanishing $(k-1)$-morphism.

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

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

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

- source and target maps for all $k$-morphisms

- identity-assigning maps for all $k$-morphisms

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

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

So

(Semistrict) Lie $n$-algebras are the same as free graded-co-commutative coalgebras on generators in degree $1 \leq k \leq n$ 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 $L_{\infty}$-algebras. So we may equivalently say

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

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 $(S^c(s V), D)$ gives rise to a quasi-free differential algebra $(\wedge^\bullet (s V^*), d)$ simply by setting $d \omega := - \omega(D(\cdot))$ for all dual elements $\omega \in sV^*$.

Hence finally we get

(Semistrict) Lie $n$-algebras are the same as free graded-commutative algebras on generators in degree $1 \leq k \leq n$ 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 $n$-Bundles in terms of groupoids

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

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

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

- the nonabelian $G_{(n)}$-cocycle data $g$

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

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

This is the situation in the world of Lie $n$-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 $n$-algebras and their differential algebra description, we shall arrive at an understanding of characteristic classes of these $n$-bundles.

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

Universal $n$-bundles in terms of Lie $n$-algebras

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

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

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

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

- $G_{(n)}$ is an $n$-group. Its Lie $n$-algebra is $g_{(n)}$

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

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

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

(Compare slide 135-140.)

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

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

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

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

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

Hence we write

$\Omega^\bullet( X ) \stackrel{\{K_i\}}{\leftarrow} b g_{(n)}^* \,.$

Completing the cone

$\array{ && \mathrm{inn}(g_{(n)})^* \\ && \uparrow \\ \Omega^\bullet( X ) &\stackrel{\{K_i\}}{\leftarrow}& b g_{(n)}^* }$

to the left amounts to choosing a bundle

$p : P \to X$

over $X$ together with a $g_{(n)}$-valued connection $A$ on its total space

$\array{ \Omega^\bullet(P) &\stackrel{(A,F_A)}{\leftarrow}& \mathrm{inn}(g_{(n)})^* \\ \uparrow^{p^*} && \uparrow \\ \Omega^\bullet( X ) &\stackrel{\{K_i\}}{\leftarrow}& b g_{(n)}^* }$

which is such that feeding its $n$-curvatures $F_A$ into all invariant $g_{(n)}$ polynomials $k_i$ produces the previously fixed characteristic classes

$K_i = k_i(F_A) \in \Omega^\bullet(X) \,.$

Demanding furthermore that the pushout of

$\array{ && g_{(n)}^* \\ && \uparrow \\ \Omega^\bullet(P) &\stackrel{(A,F_A)}{\leftarrow}& \mathrm{inn}(g_{(n)})^* \\ \uparrow^{p^*} && \uparrow \\ \Omega^\bullet( X ) &\stackrel{\{K_i = k_i(F_A)\}}{\leftarrow}& b g_{(n)}^* }$

exists,

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

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

(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 $A$. But I am not sure yet how to see this.)

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

Characteristic classes of String and $g_{\mu}$-$n$-bundles

It is then an not hard to see that

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

For $g$ simple and $k = \langle \cdot, \cdot\rangle$ 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 $n$-algebra cohomology.

Posted at September 7, 2007 2:08 PM UTC

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### 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 $A$ on the total space $P$ of some $G$-bundle $P \to X$ 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

$Y \to X$

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

$P = Y$.

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 $n$-groupoid and the Lie $n$-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

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

to be a pushout is the first condition on the Ehresmann connection $A$: it says that $A$ pulled back to a fiber has to become the canonical $g$-valued 1-form on $G$.

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

### Re: Current algebra cohomology

Thanks, Thomas!

It’s good that you mention this again.

Maybe if we’d slightly modified this setup by letting $S$ be a mere scalar and setting $\delta S = \int \langle J, d J \rangle$ 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 $n$-algebras, in a sense.

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

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.

### 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.

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

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
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Excerpt: On obstructions to lifting the structure n-group of n-bundles.
Tracked: September 12, 2007 11:28 PM

### Correction

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?

### 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.

### Re: Correction

This is a 2-algebra, right?

Here is the general rule:

the degree $n$ of the Lie $n$-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 $V^* = V_1^* \oplus V_2^* \oplus \cdots \oplus V_n^*$ which is nontrivial at most in degree $1 \leq d \leq n$.

(Strictly speaking this is defined only for $V$ 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 $\{\delta(x,\cdot)\sigma^a\}$ a “basis” for $V_1$ (with $x$ running over parameter space and $a$ 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 $\Omega^\bullet(X)\otimes \wedge^\bullet(s g^* \oplus s u(1)^*)$ equipped with the obvious total degree. Here $g$ is the given Lie algebra, $s$ the shift operator, which simply says that we regard an element in $g^*$ as in degree 1, and “$s u(1)^*$” the thing spanned by what you denote $e$.

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 $n$-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 $n$ is nothing but a Lie $n$-algebra, we realize that there are things like Lie $n$-algebra cocycles and Lie $n$-algebra invariant polynomials and the like associated with it (hence the title of this entry).

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

Then, of course, realizing a hiddn $n$-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
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