I’m going up to Ottawa for a few days, in part to talk about spans and groupoids (basically, some cross section of the material in these posts here) at a conference put on by the Ottawa U math department, primarily for grad students and postdocs in the general vicinity. This is nice – gives me a chance to visit my parents and friends there (the fraction of my life I lived in Ottawa is now creeping down toward a mere third, but it probably has as strong a claim to “home” as anywhere). May is also one of the most tolerable months to be there. One of the grad students in our department is also going. Enxin Wu recently decided to start working with Dan Christensen too, so probably in future we’ll have various things to talk about. Last week, he gave a seminar talk on algebra deformation that was a long version of the one he’ll be giving in Ottawa.

Enxin is one of those guys who seems to really understand – it’s tempting to say grok- algebra, which I always find impressive. I’m a predominantly visual thinker, and the kind of symbolic computations common in algebra always seem a little mysterious to me at first until I can find a picture, or at least practice them a lot. Lie groups, for instance, make some sense to me – you can picture rotation groups, or at least keep a geometric picture of a manifold in mind. Lie algebras, being infinitesimal versions of Lie groups, are also not so hard to visualize. General associative algebras? Harder.

The talk was about associative algebras, to give some background on deformation, but the things whose deformations Enxin has been thinking about are $A_{\infty}$-algebras (see this brief intro, for instance), an “invention” of Stasheff. The talk was about deformation of these algebras – the kind of deformation that pertains to deformation quantization. This has been studied by Kontsevich. Deformation quantization has to do with replacing things valued in some algebra $A$ by new things, valued in the bigger algebra $A[[t]]$ of formal power series in $t$ with coefficients in $A$, so that the original structure you started with is just the constant part that appears when you set $t=0$. (The term “quantization” applies when you consider algebras of functions on a manifold, with a Poisson bracket – in other words, algebras of observables of a physical system).

Some of the main results have to do with the Hochschild cohomology for some complex associated to the algebra you start with, and the fact that this cohomology classifies obstructions to the deformation. I expected to get lost in a maze of notation – and there certainly is a lot – but as it turns out, I had some mental pictures to attach to these things, because related things came up a few years ago in the quantum gravity seminar at UCR (week 8 on that page especially), which provides a few pictures that helped a lot. Diagrammatic notation makes algebra a lot more comprehensible to me.

So let’s get more specific.

The point is to replace a multiplication operator $m : A \otimes A \rightarrow A$ with a power series whose coefficients are “multiplication” operators. That is, a deformation of an associative algebra $(A,m)$ (where $m : A \otimes A \rightarrow A$ is the multiplication for $A$) is $(A[[t]],m_t)$, where the new multiplication $m_t$ is defined (by linearity) by its action on elements of $A$, which works like this:

$m_t(a,b) = \sum_{i=0}^{\infty} {\alpha_i}(a,b){t^i}$

for some operators $\alpha_i : A \otimes A \rightarrow A$. Then there are a bunch of conditions on the $\alpha$ that are needed to make $m_t$ associative. There’s one condition for each power of $t$, since the coefficients in the associator should be zero:

$\sum_{i+j=n\\i,j>0} \alpha_i( (\alpha_j \otimes 1) - (1 \otimes \alpha_j)) = 0$

The $n=0$ condition just says that $\alpha_0$ is associative – so it’s the $m$ from the original algebra, which you get back when $t=0$.

Then given an algebra $A$, you can create the deformation category $\mathcal{D}$ of $A$ whose objects are its deformations. The morphisms are continuous algebra homomorphisms that get along with the multiplication operations. It turns out that since formal power series with nonzero $n=0$ term are invertible (a consequence of the Lagrange theorem) this $\mathcal{D}$ is actually a groupoid. Then the question is to classify the isomorphism classes of deformations – that is, $\Pi_0(\mathcal{D})$. One can easily imagine that there might be no nontrivial deformations of some algebra – that is, every one is isomorphic to the deformation where all the $\alpha_i$ are trivial except $\alpha_0 = m$. So when does this happen? More generally, how can one classify the deformations up to isomorphism?

The answer has to do with Hochschild cohomology, which is related to a complex you can make from $A$. Taking $C^n(A) = hom(A^{\otimes n},A)$, the space of $n$-ary multilinear operations on $A$, you build this complex:

$0 \stackrel{d_0}{\longrightarrow} C^0(A) \stackrel{d_1}{\longrightarrow} C^1(A) \stackrel{d_2}{\longrightarrow} \dots$

where the differential maps are $d_n : C^n(A) \rightarrow C^{n+1}(A)$ defined by an alternating sum:

$d(f)(a_1, \dots, a_n) = a_1 f(a_2, \dots, a_{n+1}) + \sum_{i=1}^{n} (-1)^i f(a_1, \dots, a_i a_{i+1}, \dots, a_{n+1}) + (-1)^{n+1} f(a_1, \dots,a_n) a_{n+1}$

(Intuitively: there are too many arguments, so you start with the extra one on the left, push it into the middle as a “lump under the rug” where two arguments are combined, and push the lump all the way to the right. To ensure that $d^2 = 0$, you do this with alternating signs. This kind of algebraic manipulation is the kind of thing I can do, and clearly works, but I don’t exactly grok.)

Then you take the Hochschild cohomology groups in the standard cohomology way: $HH^i = \frac{ker(d_{i+1})}{Im(d_i)}$. A cohomology class in one of these groups is a class of multilinear maps from $n$ copies of $A$ to $A$ (up to a factor which is $d_n$ of something). As usual with cohomology, they describe obstructions to something – to exactness. Exactness, in this setting, would mean that $A$ has no interesting deformations at the $n^{th}$ level.

What does “level” mean here? Well, for example, at level 2 we’re talking about maps $A \otimes A \rightarrow A$, such as the multiplication map. In fact, we have $d_3(m) = 0$ for an associative algebra – you can check that $d(m)$ is twice the associator $a_1(a_2a_3) - (a_1a_2)a_3$, which is zero. So $m$ is a cochain. Is it a coboundary? Sure – it’s $d_2(1)$. So $m$ is in the trivial class in $HH^2(A)$. The point then is that it turns out that if this is the only class – if $HH^2(A) = 0$ – then there are no interesting deformations of the multiplication of $A$ in the sense described above. The groupoid $\mathcal{D}$ has just one object. (One thing that occurs to me is that this makes it a group – which group is something Enxin didn’t discuss. My algebra instincts aren’t quite up to answering that off the top of my head.) For example, if $A = \mathbb{C}$ (as an algebra over $\mathbb{R}$), there are no nontrivial deformations: $HH^2(\mathbb{C}) = 0$.

What do the other levels mean? Really, this is where you’d want to look at the generalization from associative algebras to $A_{\infty}$-algebras. Whereas for an associative algebra $A$, the associator $a(x,y,z) = x(yz) – (xy)z$ is zero, in general an $A_{\infty}$-algebra will have an associator map $a : A^{\otimes 3} \rightarrow A$ (that is, $a \in C^3$ in the complex above), which might not be zero, but which is $d_3(m)$.

This is the beginning of a story relating $A_{\infty}$-algebras to weak $\infty$-categories: a bicategory, for example, has an associator for composition of morphisms. In a bicategory, you expect the associator to satisfy a certain identity – the Pentagon identity – but in general you’d just ask for a “pentagonator” (something in $C^4$), and so on (this is where those seminar notes above help me think in pictures, by the way). An $A_{\infty}$-algebra is a vector space equipped with maps at all these levels – described by Stasheff’s associahedra – satisfying some relations. The general story of deformation relates the Hochschild cohomology groups at different levels to deformations of $A_{\infty}$-algebras. Enxin didn’t go into this in his talk, but he did say a little something about the next level:

An infinitesimal deformation of $A$ is a deformation not in $A[[t]]$, but in the quotient $A[[t]]/(t^2=0)$. This only needs two maps, $\alpha_0 , \alpha_1$. The third Hochschild cohomology measures obstructions to extending an infinitesimal deformation to a full deformation in $A[[t]]$ – if $HH^3(A) = 0$, then any infinitesimal deformation can be extended to a full deformation.

All in all, I thought the talk was interesting – it tied in much more closely to things I already knew about TQFTs and higher categories than I’d expected. I’ll be really impressed if he can condense it into a 25-minute version…