Eric Forgy Noncommutative Kähler Differentials

Hi,

I will probably take this to my personal nLab page to work things out, but wanted to point something out here in the hopes that someone might find it interesting enough to help turn it into something worth putting on the main nLab.

In this discussion we have considered (commutative) Kahler differentials on the real line $\Omega_K(R)$ . We can also consider (noncommutative) Kahler differentials on the integers $\Omega_K(Z)$ .

To do this, we want to turn $Z$ into a directed graph so that we can think of each integer $i$ as sitting over a node $\mathbf{e}^i$ with a directed edge denoted by $\mathbf{e}^i\otimes\mathbf{e}^{i+1}$ between each consecutive integer.

Let $A$ denote a commutative unital $k$ -algebra generated by $\mathbf{e}^i$ with product defined on basis elements by

\mathbf{e}^i\mathbf{e}^j = \delta_{i,j} \mathbf{e}^i,

where $\delta_{i,j}$ is the Kronecker delta. The unit element is given by

1 = \sum_i \mathbf{e}^i.

Let $M$ denote the bimodule generated by $\mathbf{e}^i\otimes\mathbf{e}^{i+1}$ with left multiplication defined on basis elements via

\mathbf{e}^i\left(\mathbf{e}^j\otimes\mathbf{e}^{j+1}\right) = \left(\mathbf{e}^i\mathbf{e}^j\right)\otimes\mathbf{e}^{j+1}

and right multiplication defined on basis elements by

\left(\mathbf{e}^j\otimes\mathbf{e}^{j+1}\right)\mathbf{e}^i = \mathbf{e}^j\otimes\left(\mathbf{e}^{j+1}\mathbf{e}^i\right).

Note that for $a\in A$ and $m\in M$ we have

a m\ne m a.

Define a $k$ -linear map

d:A\to M

given by

d a = [\mathbf{G},a],

where

\mathbf{G} = 1\otimes 1 = \sum_i \mathbf{e}^i\otimes\mathbf{e}^{i+1}

and we’ve set

\mathbf{e}^i\otimes\mathbf{e}^j = \delta_{i+1,j}\mathbf{e}^i\otimes\mathbf{e}^{i+1}.

The map $d:A\to M$ is a derivation since

d(a b) = [\mathbf{G},a b] = [\mathbf{G},a] b + a [\mathbf{G},b] = (d a)b + a(d b).

This differential may be recognized as the universal differential when expressed in the more familiar way

d a = [\mathbf{G},a] = (1\otimes 1)a - a(1\otimes 1) = 1\otimes a - a\otimes 1.

Let $t\in A$ be a “coordinate” given by

t = \sum_i i\Delta\mathbf{e}^i.

It follows that

d t = \sum_i \Delta\mathbf{e}^i\otimes\mathbf{e}^{i+1},

any $\alpha\in M$ can be written as

\alpha = \alpha_t d t

for some $\alpha_t\in A$ , and $t$ and $d t$ satisfy the commutative relation

[d t,t] = \Delta d t.

This commutative relation separates these finitary (noncommutative) Kahler differentials from the continuum (commutative) Kahler differentials we’ve been discussing and the “finiteness” is measured by the lack of commutativity. In fact, commuting the differential $d t$ from the left to the right involves a translation in $t$ , i.e.

(d t) t = (t+\Delta) d t.

Observation

I just tried to compute $d(t^n)$ in this finitary framework and found something interesting (but probably totally well known). By simply using Leibniz plus the commutative relation we get

\begin{aligned} d(t^n) &= (d t) t^{n-1} + t d(t^{n-1}) \\ &= (t+\Delta) (d t) t^{n-2} + t d(t^{n-1}) \\ &= \dots \\ &= \left[\sum_{r=0}^{n-1} t^r (t+\Delta)^{n-r-1}\right] d t. \end{aligned}

That term in brackets looks like a convolution, which reminds me of something I recently read on the nLab, but can’t find it now :|

I’m pretty sure my algebra is correct because

\lim_{\Delta\to 0} \sum_{r=0}^{n-1} t^r (t+\Delta)^{n-r-1} = n t^{n-1}

as expected.

I found it! It was a recent comment by Urs:

Concerning that table:

since I wrote it, I understood a few more things. There might be a better story to be told here:

it’s all about taking “algebras of functions on an $\infty$ -groupoid”, using pointwise or convolution product.

take a Lie $\infty$ -groupoid $A$ , let $\mathfrak{a}\subset A$ be its sub-object of infinitesimal morphisms. Take degreewise functions on this, equipped with the pointwise product. The resulting cosimplicial algebra has as its complex of chains a commutative dg-algebra: the Chevalley-Eilenberg algebra of the $L_\infty$ -algebroid $\mathfrak{a}$ .

But take instead functions on $A$ equipped not with the pointwise, but with the convolution product, i.e. the $\infty$ -version of the category algebra. This should be the quantization of the previously mentioned CE-algebra (hence account for the entries labeled “Clifford” in the above table).

Could this be related to my convolution appearing above?

That would be kind of cool. It would support my ages old gut feeling that going from continuum differential geometry to finitary (abstract) differential geometry was akin to quantization, where going in the opposite direction, the “continuum limit” is analogous to the “classical limit”.

Created on December 31, 2009 at 15:22:43 by Eric Forgy