The String Coffee Table

Besides Bouwknegt’s talk on D-branes and generalized geometry, which Marnie and David are writing about, and Mueger’s talk on modular tensor categories, which I already wrote about, there were two more talks down here in Sydney. Both involved operads! One was Jean-Louis Loday’s talk on “Generalized bialgebras and triples of operads”, and the other was Jonathan Scott’s talk on “Bimodules of operads: encoding deep structure of morphisms”.

So, I want to talk about these.

To my limited intelligence, the most interesting thing about Loday’s talk was some stuff I should have already known. Suppose you have a coalgebra. Then we say an element $x$ is “primitive” if its coproduct is $x \otimes 1 + 1 \otimes x$. In the case where our coalgebra is the universal enveloping algebra of a Lie algebra, the primitive elements are just the elements of the original Lie algebra. I knew this. But, Quillen made a cool definition that goes like this:

Given a coalgebra $C$, we can give it a filtration where $C_{n+1}$ consists of those elements $x$ for which the difference of the coproduct of x and $x \otimes 1 + 1 \otimes x$ lives in $C_n \otimes C_n$. We start things off by saying $C_0 = 0$, so $C_1$ consists of the primitive elements and so on. For the universal enveloping algebra of a Lie algebra this trick gives the usual filtration. Quillen says a coalgebra is “connected” if the union of all the $C_n$ is all of $C$.

We then have the following wonderful theorem.

Suppose $H$ is a cocommutative bialgebra over a field of characteristic zero (say the complex numbers, if you’re a physicist). Then the following three are equivalent:

1. H is connected.

2. H is the universal enveloping algebra of the Lie algebra of primitive elements of H. (The primitive elements of a bialgebra form a Lie algebra with $[x,y] = xy - yx$).

3. H is cofree among connected cocommutative coalgebras.

The fact that 1 implies 2 is a cool characterization of connected cocommutative bialgebras: they’re just universal enveloping algebras! This generalizes a result of Milnor and Hopf, who showed that connected cocommutative bialgebras that are also COMMUTATIVE must be symmetric algebras.

The fact that 2 implies 3 is a slicker-than-hell statement of the Poincare-Birkhoff-Witt theorem. You start with the universal enveloping algebra of some Lie algebra L and you conclude that as a coalgebra it’s the same as the symmetric algebra of L, with its usual shuffle coproduct. (This is what a “cofree connected cocommutative coalgebra” is: a symmetric algebra, with its usual coproduct!)

As a corollary, we get the fact that the tensor algebra TV of a vector space V is the same as the symmetric algebra of the free Lie algebra on V. (I already knew this….)

Also, we get the fact that the space of primitive elements of TV is the free Lie algebra of V.

Loday’s talk consisted of a massive generalization of these facts, replacing commutative, associative and Lie algebras by algebras of other threesomes of operads!

Jonathan Scott’s talk was mainly about a trick for automatically defining a notion of “weak morphism” between algebras of operads like the $A_\infty$ and $L_\infty$ operads. This is part of some work he’s doing with Kathryn Hess.

The key technical trick involved looking an $O$-bimodule $R$ for an operad $O$, assuming that it’s a comonoid in the category of $O$-bimodules, and then defining a weak morphism from an $O$-module $L$ to an $O$-module $M$ to be an $O$-module morphism from $R \otimes L$ to $M$. Using the comonoid structure on $R$ you can compose these things.

I like this kind of stuff, but it takes a while to develop a taste for it!

John Baez