Constructing the Bost-Connes system. Attempt 1.

At the end of my last blog on lattices I described how the space of 1-dimensional $\mathbb{Q}$-lattices can be identified with $\hat{\mathbb{Z}} \times \mathbb{R}_{+}^{*}$. It's not too hard to see that if we mod out scaling by real factors, then we just end up with $\hat{\mathbb{Z}}$. Essentially, in one dimension, if we treat $\mathbb{Q}$-lattices that are multiples of each other as the same, all we have left is the homomorphism.

Now I want to describe a few more descriptions of 1 dimensional $\mathbb{Q}$-lattices (I'll henceforth use the symbol "1dQL" to mean 1 dimensional $\mathbb{Q}$-lattices), culminating in my describing a natural time evolution, and thus constructing the Quantum Statistical Mechanical system of Bost and Connes. First off, I want to describe the structure of the commensurability relation of 1dQL. But to do that, I first need to talk about groupoids.

Without delving too deep into category theory (which I am just now getting used to) let me suffice to say that a groupoid $\mathcal{G}$ is a collection of objects which will we call $\mathcal{G}^0$, and a collection of morphisms/mappings/functions (or just "arrows") $\mathcal{G}^1$ on $\mathcal{G}^0$ . We require every morphism in $\mathcal{G}^1$ to be invertible. Without category theory, you can think of a groupoid as being a lot like a normal group, except the group operation is not a binary operation. Id est, while every element has an inverse, not every pair of elements can be composed.

Now when I talk about the structure of the commensurability relation of 1dQL, its important to note that I don't mean equivalence classes. I'm actually talking about pairs $(\Lambda_1, \phi_1)$ and $(\Lambda_2, \phi_2)$ of 1dQLs that are commensurable. In Lemma 3.21 of Connes' monograph, he posits that this is isomorphic to the étale groupoid

$\mathcal{G}_1 = \{ (r, \rho, \lambda) : r \in \mathbb{Q}_{+}^{*}, \rho \in \hat{\mathbb{Z}}, \lambda \in \mathbb{R}_{+}^{*}$ such that $r \rho \in \hat{\mathbb{Z}}\}$

with the composition $(r_1,\rho_1,\lambda_1) \circ (r_2,\rho_2,\lambda_2) = (r_1 r_2,\rho_2,\lambda_2)$ if $r_2 \rho_2 = \rho_1$ and $r_2 \lambda_2 = \lambda_1$.

Étale groupoids are just groupoids with a specific topological structure (I'm still working out the details on that.) You can see this isomorphism from the map $l: (\rho,\lambda) \mapsto (\Lambda,\phi) $. In particular:

$(r,\rho,\lambda) \in \mathcal{G}_1 \mapsto (l(r \rho, r \lambda), l(\rho, \lambda))$

Finally, via Proposition 3.22, we can go even further. The following groupoid is isomorphic to the quotient $\mathcal{G}_1/\mathbb{R}_{+}^{*}$, hence we can also describe the commensurability classes of 1dQL up to scaling.

$\mathcal{U}_1 = \{(r,\rho) : r \in \mathbb{Q}_{+}^{*}, \rho \in \hat{\mathbb{Z}} $ such that $r \rho \in \hat{\mathbb{Z}} \}$

The isomorphism is via the map $\gamma : \mathcal{U}_1 \to \mathcal{G}_1/\mathbb{R}_{+}^{*}$ by $(r,\rho) \mapsto ((r^{-1}\mathbb{Z},\rho),(\mathbb{Z},\rho)) \in\mathcal{G}_1/\mathbb{R}_{+}^{*} $. The proof that this is an isomorphism isn't too hard (at least it doesn't look to hard) but it is yet still one of the details I've not finished.

Now to a locally compact groupoid (and I'm trusting Connes here when he says its locally compact) $\mathcal{G}$ we can construct the convolution algebra $\mathcal{A}_{c}(\mathcal{G})$. This convolution algebra is a space of functionals (into $\mathbb{C}$) on the groupoid. In the case of $\mathcal{U}_1$, the convolution product is

$f_1 * f_2 (L_1, L_2) = \sum_{L_1 \sim K \sim L_2} f_1(L_1, K) f_2(L, L_2)$

The involution is $f*(L_1, L_2) = \overline{f(L_1, L_2)}$. The symbols $L_1, L_2, K$ are 1dQLs and $\sim$ denotes commensurability.

(Note to readers: an algebra is essentially a vector space with a multiplication. The convolution product is our multiplication. Convolution is just a fancy word for an operation on two functions that produces a third. Involution is what makes C* algebras into C* algebras; its an abstraction of complex conjugation. Indeed, $\mathbb{C}$ is not only an algebra, it's a C* algebra.)
By completing the appropriate norm (which I do not fully understand, so I won't mention it here.) we can upgrade the convolution algebra $\mathcal{A}_{c}(\mathcal{U}_1)$ to the C* groupoid algebra $C^{*}(\mathcal{U}_1)$.

Connes has another description of the C* algebra in terms of generators and relations. However, I do not yet understand the proposition or the proof. It involves semigroup cross products, Morita equivalences, adeles, and notation I've not deciphered. I'll return to it soon.

The last thing left to do before we have a full Quantum Statistical Mechanical system is to define the time evolution (Lemma 3.24 in Connes' monograph). We can actually do this using data from an element in $\mathcal{G}_1/\mathbb{R}_{+}^{*}$. Recall that an element there is actually a pair of 1dQLs $L_1 = (\Lambda_1,\phi_1)$ and $L_2 = (\Lambda_2,\phi_2)$. Since they are commensurable, the lattices $\Lambda_1$ and $\Lambda_2$ differ by a scaling factor $r$. This $r$ happens to be the ratio of covolumes of two lattices.

Recall that the covolume of a lattice is the "area" of its fundamental parallelogram (or equivalent hyper-dimensional shape). And that the we can find this area by taking the determinate of the basis vectors of the lattice. In one dimension, we only have a single basis vector (some member of $\mathbb{R}$). Now when we take the ratio of the covolumes of lattices of commensurable 1 dimensional $\mathbb{Q}$-lattices, we simply get the scaling factor between the two, id est, $r$.

Now members of $C^{*}(\mathcal{G}_1/\mathbb{R}_{+}^{*})$ are simply functions on pairs of commensurable 1dQLs, e.g., $f(\Lambda_1, \Lambda_2)$. We define the time evolution as

$\sigma_{t}(f)(\Lambda_1, \Lambda_2) = r^{i t} f(\Lambda_1, \Lambda_2)$ where $r$ is the scaling factor.

Or in terms of the equivalent $C^{*}(\mathcal{U}_1)$ (functions $f(r,\rho)$ for $(r,\rho) \in \mathcal{U}_1$) we have

$\sigma_{t}(f)(r,\rho) = r^{i t} f(r,\rho)$.

And there you have it, folks, a bona fide Quantum Statistical Mechanical system. Bit embarrassing that it took me til September!


Details I've missed and Things I don't understand

Étale groupoids
How the groupoid inversion and operation works on $\mathcal{G}_1$ and $\mathcal{U}_1$. Plus how it's locally compact.
The proof of the isomorphism.
Proof of Proposition 3.22
All of Proposition 3.23 in Connes' monograph
Construction of C* group algebras and C* groupoid algebras.

Whats next

All of the above, plus the Hecke algebra description and its equivalence to the above. Then on to symetries of the system and the class field theory of $\mathbb{Q}$.