Roytenberg on Weak Lie 2-Algebras

December 21, 2007

Posted by John Baez

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It’s been circulating informally since October, but now it’s available on the arXiv — a proposal for the definition of a fully general categorified Lie algebra!

Dmitry Roytenberg, On weak Lie 2-algebras.

Abstract: A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2-algebras can also be defined, yielding a 2-category. Passing to the normalized chain complex gives an equivalence of 2-categories between Lie 2-algebras and 2-term “homotopy everything” Lie algebras; for strictly skew-symmetric Lie 2-algebras, these reduce to $L_\infty$ -algebras, by a result of Baez and Crans. Lie 2-algebras appear naturally as infinitesimal symmetries of solutions of the Maurer–Cartan equation in some differential graded Lie algebras and $L_\infty$ -algebras. In particular, (quasi-) Poisson manifolds, (quasi-) Lie bialgebroids and Courant algebroids provide large classes of examples.

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When categorifying an algebraic notion, we have various choices, since we can “weaken” any equational law, replacing it by an isomorphism, or demand that it still hold “strictly” as an equation. For Lie algebras the two main equational laws are the antisymmetry of the bracket:

$[x,y] = -[y,x]$

and the Jacobi identity

$[x,[y,z]] = [[x,y],z] + [y, [x,z]]$

When Alissa Crans and I categorified the concept of Lie algebra in HDA6, we weakened the Jacobi identity, replacing it by an isomorphism we called the Jacobiator — but we kept the antisymmetry holding strictly. We called the resulting gadgets ‘semistrict Lie 2-algebras’. We hoped that these might be ‘sufficiently general’, i.e. that any fully weak Lie 2-algebra, when these were finally defined, would turn out to be equivalent to a semistrict one.

This appears not to be true! In this paper, Dmitry Roytenberg weakens not only the Jacobi identity but also the antisymmetry, replacing the equation

$[x,y] = -[y,x]$

by an isomorphism he calls the alternator. He classifies the resulting ‘weak Lie 2-algebras’, and it seems not all of them are equivalent to our old ‘semistrict’ ones… although they can all be made semistrict by a certain potentially destructive process.

There’s a lot more to say… but for now, if you’re a fan of higher gauge theory, 2-groups, and the like, I urge you to read the paper yourself!

I’m very happy this paper is out, since my grad student Alex Hoffnung and Chris Rogers are already running into weak Lie 2-algebras that aren’t semistrict in their work on categorified classical mechanics.

Posted at December 21, 2007 6:55 PM UTC

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12 Comments & 1 Trackback

Re: Roytenberg on Weak Lie 2-Algebras

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In case anyone is interested, I had discussed aspects of this in Categorified Clifford Algebra and weak Lie $n$ -Algebras.

Lie $\infty$ -algebras with strict skew symmetry and weak Jacobi identity are the same as graded commutative codifferential coalgebras.

In the above entry I suggested that the right generalization to fully weak Lie $\infty$ -algebras should correspond to allowing arbitrary codifferential coalgebras, not necessarily graded commutative.

I still don’t have much more (but a little bit more) than the gut feeling “hey, that must be right” to support this, but my recent conversation with Todd Trimble about the general nice properties of the category of codifferential coalgebras (which need not be not restricted to commutative coalgebras) just strenghtened my belief.

I am willing to take high bets that this suggestion is right. Anyone?

Posted by: Urs Schreiber on December 22, 2007 10:34 AM | Permalink | Reply to this

Re: Roytenberg on Weak Lie 2-Algebras

I’ll take that bet. Differential graded coalgebras (assoc not gc) are known as
A_\infty algebras, at least if either non-pos or non-neg graded. No symm even up to homotopy.

Also be careful about how weak is weak Jacobi. As you say, dgc coalgebras correspond to L_\infty (assuming no `constant’ term = backgrounf) but homtopy Jacobi does not imply any coherence of the homotopy.

Posted by: jim stasheff on December 22, 2007 2:08 PM | Permalink | Reply to this

Re: Roytenberg on Weak Lie 2-Algebras

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Differential graded coalgebras (assoc not gc) are known as $A_\infty$ algebras

Right, I should have been more precise. As I say at that entry, we want something right in between $A_\infty$ and $L_\infty$ : “Clifford infinity”, where graded commutativity holds up higher generators.

Notice, for instance, that a “graded Clifford algebra” in degre 0 is a symplectic vector space.

Posted by: Urs Schreiber on December 23, 2007 12:35 PM | Permalink | Reply to this

Re: Roytenberg on Weak Lie 2-Algebras

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Graded-commutative differential algebras can, to some extent, all be interpreted as the algebra of differential forms on some smooth space. (I talk more about that here).

Similarly, I’d expect that the differential-graded but non-necessarily graded-commutative algebras corresponding to Lie $\infty$ -algebras with weakly skew-symmetric bracket $\infty$ -functor are modeled on the “differential Clifford algebra” $\Gamma( \mathrm{Cl}(T X \oplus T^* X))$ on which, for $X$ Riemannian, the canonical deRham-Kähler Dirac operator $D = d \pm \star d \star$ acts.

Somehow such that the statement “every weakly graded-commutative dg-algebra is the algebra of sections of a Clifford bundle on some smooth space” makes sense and becomes true.

Posted by: Urs Schreiber on December 27, 2007 2:32 PM | Permalink | Reply to this

Re: Roytenberg on Weak Lie 2-Algebras

Anyone know an operad or mild generalization for Clifford algebras?

Posted by: jim stasheff on December 27, 2007 5:25 PM | Permalink | Reply to this

Lie superalgebras and beyond

Without having read the paper, it sounds like the notion of weak 2-Lie algebras is general enough to encompass some Lie superalgebras. Denote even elements by a, b, … and odd elements by α, β, …, and let the only non-zero alternators and Jacobiators be

Alt(α, β) = 2[α, β]

Jac(a, α, β) = 2[[α, a], β]

Then I suppose that there are some coherence laws that these must satisfy. The graded Heisenberg algebra,

[A, B] = (A, B) e,

where (.,.) is the graded-skew symplectic metric, should satisfy these laws, since all Jacobiators vanish and the alternators are proportional to the unit e, which commutes with everything.

More generally, one can presumably define weak n-Lie algebras by relaxing coherence laws up to order n. An example is given by any non-commutative, non-associative nilpotent product of length n. By product I mean that the distributive law and multiplication by scalars hold as usual, which means that we can introduce a basis uⁱ, such that

[uⁱ, u^j] = f^ij_ku^k.

A priori, no assumption is made on the structure constants, and e.g.

Alt(uⁱ, u^j) = (f^ij_k + f^ji_k)u^k.

Finally, nilpotency of length n means that we have a grading, 1 <= deg uⁱ <= n, such that f^ij_k = 0 unless deg uⁱ + u^j = deg u^k. The point is that a coherence law of order n+1 has at least degree n+1 and is thus zero.

Taking this philosophy to the extreme, one ends up with weak infinity-Lie algebras, which seems to be the same as non-commutative non-associative algebras.

Posted by: Thomas Larsson on January 5, 2008 8:10 AM | Permalink | Reply to this

Re: Lie superalgebras and beyond

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one ends up with weak infinity-Lie algebras, which seems to be the same as non-commutative non-associative algebras.

If you add “differential” to the list of qualifiers and qualify the non-commutativity, that sound like the kind of conjecture we were discussing above.

In fact, I know that Roytenberg has tried to find out a class of non-associative dg-algebras relevant here, but I don’t know if I am allowed to talk about this in public.

Posted by: Urs Schreiber on January 5, 2008 10:10 AM | Permalink | Reply to this

Re: Lie superalgebras and beyond

`by relaxing coherence laws up to order n’

A crucial issue si to what extent, i.e. whihc coherence laws. L_\infty relaxes Jacobi up to homotopy to all orders but keeps strict graded anticomunativity

WARNING! there are L_n-algebras (L_\infty up to a point) Lie n-agebras (of more than one flavor), n-Lie algebras. Nomenclature does matter!!

Posted by: jim stasheff on January 5, 2008 2:21 PM | Permalink | Reply to this

Lie Y-algebras

It is certainly bad style to reuse old nomenclature for new concepts, as Roytenberg does. Perhaps one should instead talk about Lie (m,n)-algebras if one has alternators up to order m and Jacobiators up to order n. In particular, a (2,1)-algebra is a hemistrict 2-algebra, a (1,2)-algebra is a semistrict 2-algebra, and a (1,1)-algebra is an ordinary 1-algebra.

Moreover, I doubt that the notion of (m,n)-algebras is unique, since there might be several consistent ways to relax mixed alternator-Jacobiator coherence laws. It seems natural to consider Lie Y-algebras, where Y is a Young diagram. If Y has a box in position (m,n), we replace coherence laws involving m alternators and n Jacobiators by the corresponding coherators. In particular, a (m,1)-algebra is given by a Young diagram with one row and m columns, and a (1,n)-algebra has one column and n rows.

Of course, it is just guesswork that something like this can be done in a self-consistent way. In principle, it shouldn’t be too hard to figure out the right definitions, although to explicitly write them down will rapidly become tedious with growing m and n. However, I doubt that such Lie Y-algebras would be very interesting, given how general the construction seems to be.

Posted by: Thomas Larsson on January 6, 2008 9:31 AM | Permalink | Reply to this

Re: Lie Y-algebras

Thanks for a first step toward some decent nomenclature.

Posted by: jim stasheff on January 6, 2008 7:45 PM | Permalink | Reply to this

Re: Lie Y-algebras

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Thomas wrote:

It is certainly bad style to reuse old nomenclature for new concepts, as Roytenberg does.

I’m not sure what you think he did wrong! Alissa Crans and I described ‘semistrict’ Lie 2-algebras and made it clear that these were a stopgap — a step en route to the full-fledged Lie 2-algebras that Roytenberg has now found. So, I fully support his use of the term ‘Lie 2-algebra’.

One can imagine various schemes of notation for partially strict algebras, but they need to have a least 3 parameters, not just two: the $n$ -categorical cutoff, a parameter measuring the weakening of the Jacobi identity, and a parameter measuring the weakening of the antisymmetry.

Such notations will become solidified if and when people ever need them… no need to rush it. We’re still in the ‘Wild West’ stage of the subject, when things are a bit disorganized, not yet boring.

Posted by: John Baez on January 7, 2008 7:23 AM | Permalink | Reply to this

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 12:42 PM

who was Maurer?

I haven’t found even his (Maurer’s) first name! The Wiki attributes Maurer-Cartan equation only to Cartan 1904.

Posted by: jim stasheff on March 15, 2009 4:05 PM | Permalink | Reply to this

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December 21, 2007