Chern-Simons Lie-3-Algebra Inside Derivations of String Lie-2-Algebra

November 7, 2006

Chern-Simons Lie-3-Algebra Inside Derivations of String Lie-2-Algebra

Posted by Urs Schreiber

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The $n$ -Category Café started with a discussion of the Lie 3-group underlying 11-dimensional supergravity #. In a followup #, I discussed a semistrict Lie 3-algebra $\mathrm{cs}(g)$ with the property that 3-connections taking values in it $d\mathrm{tra} : \mathrm{Lie}(P_1(X)) \to \mathrm{cs}(g)$ are Chern-Simons 3-forms with values in $g$ , giving the local gauge structure of heterotic string backgrounds.

At that time I guessed that $\mathrm{cs}(g)$ is in fact equivalent to the Lie-3-algebra of inner derivations of the $\mathrm{string}_g = (\hat \Omega_k g \to P g)$ Lie-2-algebra, using the fact that there is only a 1-parameter family of possible Lie 3-algebra structures on the underlying 3-vector space.

It would be quite nice if this were indeed true.

While I still have no full proof that $\mathrm{cs}(g)$ is equivalent (tri-equivalent, if you like) to $\mathrm{inn}(\mathrm{string}_g)$ , I have now checked at least one half of this statement:

there is a morphism $\mathrm{cs}(g) \to \mathrm{inn}(\mathrm{string}_g)$ and one going the other way $\mathrm{inn}(\mathrm{string}_g) \to \mathrm{cs}(g)$ such that the composition $\mathrm{cs}(g) \to \mathrm{inn}(\mathrm{string}_g) \to \mathrm{cs}(g)$ is the identity on $\mathrm{cs}(g)$ :

So at least $\mathrm{cs}(g)$ sits inside $\mathrm{inn}(\mathrm{string}_g)$ : $\mathrm{cs}(g) \subset \mathrm{inn}(\mathrm{string}_g) \,.$

The details can be found here:

$\;\;$ Chern-Simons and $\mathrm{string}_G$ Lie-3-algebras

This (rather unpleasant) computation is a generalization of that in the last section of From Loop Groups to 2-Groups, which shows the equivalence $\mathrm{string}_g \simeq g_k$ .

I take this as further indication # that the structure 3-group of $G$ -Chern-Simons theory is (a subgroup of) $\mathrm{INN}(\mathrm{String}_G)$ .

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The above notes make use of some previous computations.

The inner automorphism 3-group $\mathrm{INN}(H\to G)$ of any strict Lie 2-group $(H \to G)$ is computed in the first part of these notes on non fake flat surface transport (discussed here). The FDA description of the corresponding Lie 3-algebra is derived in the second part, in terms of the 2- and 3-form curvature of a 2-connection with values in $\mathrm{INN}(H \to G)$ .

(This is a general mechanism: the equations defining the graded differential of the FDA that describes some Lie $n$ -algebra are precisely the (Bianchi- and other) identities satisfied by a connection with values in that Lie $n$ -algebra.)

The details of the FDA description of $\mathrm{cs}(g)$ and $\mathrm{inn}(h\to g)$ , together with my notational conventions, can be found in the fda laboratory, example 10 and 12, respectively.

Introduction

For any Lie algebra $g$ , there is a semistrict Lie-3-algebra $\mathrm{cs}(g)$ such that 3-connections $d\mathrm{tra} : \mathrm{Lie}(P_1\of{X}) \to \mathrm{cs}(g)$ (i.e. algebroid morphisms from the pair algebroid of $X$ to the 3-algebroid $\mathrm{cs}(g)$ ) are given by a $g$ -valued 1-form $A$ , its curvature 2-form and its Chern-Simons 3-form $\mathrm{CS}(A) = \langle A \wedge dA\rangle + \frac{1}{3} \langle A \wedge [A \wedge A]\rangle$ on $X$ .

Another Lie 3-algebra canonically associated to $g$ is obtained as follows: The semistrict Baez-Crans Lie 2-algebra $g_k$ is equivalent to the strict Lie 2-algebra $\mathrm{string}_g := (\hat \Omega g \to P g) \,.$ For any strict Lie 2-algebra $(r \stackrel{\delta}{\to} s)$ , the Lie-3-algebra $\mathrm{inn}(r \stackrel{\delta}{\to} s)$ of its inner derivations is characterized by the fact that 3-connections $d\mathrm{tra} : \mathrm{Lie}(P_1\of{X}) \to \mathrm{inn}(r\to s)$ are given by an $s$ -valued 1-form $A$ , an $r$ -valued 2-form $B$ such that with $\beta := F_A + \delta(B)$ and $H = d_A B$ we have $d_A \beta = \delta(H)$ and $d_A H + \beta\wedge B = 0 \,.$ Had we in addition required that $\beta = 0$ , then this would characterize $(r \to s)$ itself. For $r = \hat \Omega_{k} g$ and $s = P g$ we write $\mathrm{string}_g := (\hat \Omega_k g \to P g) \,.$ It is known that $\mathrm{string}_g \simeq g_k \,.$ Here we are after a generalization of this equivalence when passing from $\mathrm{string}_g$ to $\mathrm{inn}(\mathrm{string}_g)$ . We fall short of actually proving an equivalence. Instead we construct a morphism $\mathrm{cs}(g) \to \mathrm{inn}(\mathrm{string}_g)$ and a morphism $\mathrm{inn}(\mathrm{string}_g) \to \mathrm{cs}(g)$ such that $\mathrm{cs}(g) \to \mathrm{inn}(\mathrm{string}_g) \to \mathrm{cs}(g)$ is the identity on $\mathrm{cs}(g)$ . This is done for $k = -1$ .

We will work throughout in terms of the Koszul dual description of semistrict Lie- $n$ -algebras. Every Lie- $n$ -algebra is encoded in a free differential graded algebra, and morphisms of Lie- $n$ -algebras are given by maps between FDAs that are at the same time chain maps and algebra homomorphisms.

Posted at November 7, 2006 8:08 PM UTC

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draft commentary on fda lab

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For Lie algebras and their generalizations, the graded symmetric algebra version of cochains with field coefficients is actually the Hom dual of the graded symmetric coalgebra version. The advantage of the algebra version, in addition to familiarity, is that properties of a derivation need only be checked on generators.

The graded symmetric coalgebra $S V$ is naturally a subcoalgebra of the tensor space $T V$ i.e. is spanned by graded symmetric tensors $x_1 \vee x_2 \vee \cdots \vee x_p$ for $x_i in V$ where I use $\vee$ ratrher than $\wedge$ to emphasize the coalgebra aspect. e.g. $x \vee y = x \otimes y \pm y \otimes x$ notice: no factor of $1/2$ needed.

Instead of checking a coderivation on cogenerators we check on elements with image in $V$ .

So let $g$ be a Lie algebra. Define a codifferential $d: S g \to S g$ with $g$ regarded as of degree 1 by

(1)

d(x \vee y) = [x,y]

and extend as a coderivation that means

(2)

d( x_1 \vee x_2 \vee \cdots \vee x_p) = \sum \pm [x_i, x_j] \vee x_1 \vee \cdots \vee x_p

where the sum is over $i \lt j$ and $x_i$ and $x_j$ are omitted in the $\vee$ if they are bracketed.

to check $d^2 = 0$ , we need only check it on $x_1 \vee x_2 \vee x_3$

where it is readily seen to correspond to the Jacobi identity.

Now if $V$ is a dg Lie algebra, i.e. $V = V_0 \oplus V_1 \oplus \cdots$ with a differential derivation $t: V_i \to V_{i-1}$ then regrade $V$ before taking $S$ i.e. $S s V$ , where $s$ denotes the shift or if you prefer $S V[\pm 1]$ - I can never remember this notation.

Define $d$ by

(3)

d(x \vee y) = [x,y]

and

(4)

d x = \pm t x

Now $d^2=0$ follows from jacobi and $t$ being a derivation and $t^2 = 0$ .

Modulo an issue about one of the axioms for a Lie crossed module the latter is a dg Lie algebra.

Turning to example 7 of Urs’ fda lab, the additonal term is

(5)

d(b_1 \vee b_2) = \langle b_1,b_2\rangle

The additional verification for $d^2 = 0$ follows from the invariance of $\langle \cdot,\cdot\rangle$

Now Example 8 has a surprise there is a term $d(a_1 \vee a_2 \vee a_3) = b$ in other words, we have an $L_\infty$ algebra with $l_n = 0$ for $n \gt 3$ !

Posted by: Jim Stasheff on November 14, 2006 12:19 PM | Permalink | Reply to this

Re: draft commentary on fda lab

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The graded symmetric coalgebra $S V$ …

I am aware # of your paper with Tom Lada, where this is discussed nicely.

The reason I keep doing computations like those in the FDA lab # in terms of the dual FDAs and working in terms of chosen bases is that, personally, this seemed to be the most efficient way to actually do computations. But quite possibly there is a better way.

Now Example 8 has a surprise […] we have an $L_\infty$ algebra with $l_n = 0$ for $n \gt 3$ !

Yes, $l_3$ is nonvanishing here. This means at the level of Lie $n$ -algebras that we have a nontrivial “Jacobiator”.

My example 7 is a slight extension of the “Baez-Crans” Lie-2-algebra $g_k$ , which has $g$ as the Lie algebra of objects, has $\mathrm{Lie}(U(1))$ as the Lie algebra of morphisms, and a trivial target map. The only nontrivial thing about $g_k$ is a nontrivial Jacobiator (which, however, due to the triviality of the target map, does not affect the Lie algebra structure on $g$ )

(1)

l_3(\cdot,\cdot,\cdot) = k \langle \cdot, [\cdot,\cdot] \rangle \,.

This Lie algebra $g_k$ is in fact the “string” Lie 2-algebra. It is equivalent to the strict (vanishing Jacobiator) Lie 2-algebra $(\hat \Omega g \to P g)$ .

What I do in the above entry is to show how this equivalence generalizes as we move from the Lie 2-algebra $g_k$ to the Lie 3-algebra $cs(g)$ from example 7.

Posted by: urs on November 14, 2006 12:32 PM | Permalink | Reply to this

Read the post 2-Monoid of Observables on String-G
Weblog: The n-Category Café
Excerpt: Rep(L G) from 2-sections.
Tracked: November 24, 2006 5:43 PM

Higher Morphisms of Lie n-Algebras

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In the last couple of days I tried to see if the inclusion

(1)

\mathrm{cs}(g) \stackrel{\subset}{\to} \mathrm{inn}(\mathrm{string}(g))

extends in fact to an equivalence, as I was expecting it would.

But now it seems that I have proven that there is no way to turn that particular inclusion into an equivalence using the surjection

(2)

\mathrm{inn}(\mathrm{string}(g)) \to \mathrm{cs}(g)

that I considered.

(Notice that am not saying that I have proven that there is no equivalence at all - I still expect there to be one - just that the morphisms going back and forth that I did come up with don’t yield an equivalence.)

But there is one issue here, which I might want to check the literature against.

In my derivations, I am making use of the fact that semistrict Lie $n$ -algebras are “the same” as $n$ -term $L_\infty$ algebras (following Baez/Crans), which in turn are the same as cofree counital coassociative graded cocommutaice coalgebras with nilpotent coderivation

(3)

D

of degree -1, which in turn are “the same” as their duals, namely graded commutative differential algebras that are free as graded commutative algebras. The relation between the codifferential and the differential is the usual one:

(4)

(d\omega)(e_1,e_2,\cdots e_n) = \omega(D(e_1 \vee e_2 \vee \cdots \vee e_n)) \,.

While at the level of these objects this is clear, I haven’t seen much discussion at all of what the higher morphisms between these structures should be.

Baez/Crans write down 1- and 2-morphisms of 2-term $L_\infty$ -algebras in HDA VI. One can check (now included as example 8 in my fda Laboratory) that these correspond to chain maps and “linearized” chain homotopies of the corresponding (quasi-)free differential algebras.

If just presented with FDAs, one might guess that the right notion of morphisms are simply chain maps, chain homotopies, homotopies of homotopies, and so on. But - already for duals of 2-term $L_\infty$ algebras - it is a little more subtle than that. There is that “linearization” going on. (I tried to describe that in more detail in the last section of my thesis.)

Anyway, all this makes me want to compare what I think the right $p$ -morphisms of $n$ -term $L_\infty$ algebras should be with what other people have done.

But so far I haven’t found much literature.

Danny Stevenson kindly pointed me to:

Marco Grandis, On the homotopy structure of strongly homotopy associative algebras, Journal of Pure and Applied Algebra 134 (1999) 15-8 I

Maybe the answer I am looking for is in there, but I need to read more in order to find out.

But in any case, if anyone here can say anything about higher morphisms of $n$ -term $L_\infty$ -algebras (in any of their equivalent incarnations), I’d be very grateful.

Posted by: Urs on December 17, 2006 3:08 PM | Permalink | Reply to this

Re: Higher Morphisms of Lie n-Algebras

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Urs wrote:

But in any case, if anyone here can say anything about higher morphisms of $n$ -term $L_\infty$ -algebras (in any of their equivalent incarnations), I’d be very grateful.

Unlike our definition of Lie 2-algebra, which we could check against the definition of ‘2-term $L_\infty$ -algebra’, Alissa and I couldn’t find anything to check our definition of Lie 2-algebra morphisms and 2-morphisms against. So, we decided to just figure out definitions of morphisms and 2-morphisms for 2-term $L_\infty$ -algebras.

Similarly, you and Jim Stasheff may need to figure out the right concept of $n$ -morphism for $L_\infty$ -algebras.

Luckily, there’s some general abstract nonsense that can guide you.

For any operad in $\mathrm{Vect}$ , say $O$ , there’s an operad in $\mathrm{ChainComplexes}$ called $O_\infty$ which ‘ $\infty$ -categorifies’ the concept conveyed by $O$ . For example, if $O$ is the operad for Lie algebras, $O_\infty$ is the operad for $L_\infty$ -algebras. You probably know this; I talked about it and listed some references in week239.

In fact, this construction goes back to this classic book:

J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures, Lecture Notes in Mathematics 347, Springer, 1973.

(They did the construction in $\mathrm{Top}$ instead of $\mathrm{ChainComplexes}$ ; they used PROPs instead of operads, and they called it the ‘cherry tree’ construction, or ‘ $W$ ’. But, one can adapt their work to get what I’m saying.)

The cool thing is, Boardman and Vogt did more than $\infty$ -categorify any algebraic structure described by an operad. They also figured out decent concepts of morphisms, 2-morphisms, 3-morphisms etc. for these structures. And, they showed they got a ‘quasicategory’ of these structures - that is, a simplicial set that acts like an $\infty$ -category where all the $n$ -morphisms for $n > 1$ are weakly invertible.

(Of course they did this long before Joyal introduced the term ‘quasicategory’: they used the term ‘restricted Kan complex’.)

So, I believe the machinery already exists to instantly cook up a quasicategory of $L_\infty$ -algebras!

However, understanding this abstract machinery well enough to see what it really does in concrete examples is not something you can do instantly… unless you’re really good at this stuff.

Posted by: John Baez on December 17, 2006 6:37 PM | Permalink | Reply to this

Re: Higher Morphisms of Lie n-Algebras

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Thanks for the reference to Boardman and Vogt!

However, understanding this abstract machinery well enough to see what it really does in concrete examples is not something you can do instantly… unless you’re really good at this stuff.

Yeah, I guess so.

Similarly, I believe I am able to write down the definition of a Lie 3-algebra and of 1-, 2- and 3-morphisms of such beasts and see what that implies for my quasiFDAs. It should be more or less straightforward. But it will require quite a lot of time.

Posted by: urs on December 18, 2006 12:21 PM | Permalink | Reply to this

Re: Higher Morphisms of Lie n-Algebras

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There is a category of $L_\infty$ algebras cf. the corresponding category of strict dgc algebras (differerential graded commutative) given by the Chevalley-Eilenberg chain complex. Remind me what a quasicat is. The appropriate up to homotopy analog?

IF a Lie n-algebra is just an $L_\infty$ algebra with $l_k = 0$ for $g \gt n$ , then do you want strict $n$ -morphisms to be strict $L_\infty$ algebra morphisms OR the strong homtopy $L_\infty$ algebra morphisms

There exist several specific examples of small $L_\infty$ algebras, e.g. 2-term and even 3-term of finite dim. Any interest in a compendium?

jim

Posted by: jim stasheff on December 19, 2006 3:15 PM | Permalink | Reply to this

Read the post Higher Morphisms of Lie n-Algebras and L-infinity Algebras
Weblog: The n-Category Café
Excerpt: On higher morphisms of Lie n-algebras, and higher homotopies between L_oo algebras.
Tracked: February 15, 2007 4:10 PM

Read the post Zoo of Lie n-Algebras
Weblog: The n-Category Café
Excerpt: A menagerie of examples of Lie n-algebras and of connections taking values in these, including the String 2-connection and the Chern-Simons 3-connection.
Tracked: May 10, 2007 6:08 PM

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November 7, 2006