Yesterday, I found out a friend of mine back home in Colorado died and have since not had the focus to do much work. I haven't done that review of algebra topics like I said I would. But it's important to keep good habits, so I thought I'd review and write a bit about \mathbb{Q}-lattices a bit more.
Connes and Marcolli's monographs define a \mathbb{Q}-lattice as a lattice \Lambda \in \mathbb{R}^n together with a homomorphism \phi : \mathbb{Q}^n / \mathbb{Z}^n \to \mathbb{Q} \Lambda / \Lambda. We call two \mathbb{Q}-lattices \Lambda_1 and \Lambda_2 commensurable (\Lambda_1 \sim \Lambda_2) if:
1) \mathbb{Q} \Lambda_1 =\mathbb{Q} \Lambda_2 and
2) \phi_1 = \phi_2 mod \Lambda_1 + \Lambda_2.
Condition 1) means that there exists some \mathbb{Q}-span in \mathbb{R}^n such that \Lambda_1 and \Lambda_2 both sit inside it. For instance, both \frac{1}{2} \mathbb{Z} and \frac{1}{5} \mathbb{Z} sit inside \mathbb{Q}, so they meet condition 1. But if we take < (1,0), (0,\sqrt{2}) > and <(\frac{1}{2},0), (0,0) > we cannot find any vectors in \mathbb{R}^2 that, when spanned over \mathbb{Q}, would contain both of those lattices.
Condition 2) is actually how I originally understood it. The homomorphism must "label the same points". More formally, Lets say X =\mathbb{Q} \Lambda_1 =\mathbb{Q} \Lambda_2. For each \Lambda_i we have a projection map \pi_i : X/\Lambda_i \to X/(\Lambda_1 + \Lambda_2) by r+\Lambda_i \mapsto r + (\Lambda_1 + \Lambda_2) (of course, r may not be the best representative element in \Lambda_1 + \Lambda_2). Condition two says that \pi_1 \circ \phi_1 = \pi_2 \circ \phi_2.
1-dim Q-lattices
We can describe the Bost-Connes system with \mathbb{Q}-lattices in \mathbb{R}, and 1-dim \mathbb{Q}-lattices can be described by \hat{\mathbb{Z}} \times \mathbb{R}_{+}^{*}, where \hat{\mathbb{Z}} = \varprojlim_{n} \mathbb{Z} / n\mathbb{Z}. First note any lattice \Lambda \in \mathbb{R} can be described by \lambda \mathbb{Z}, where \lambda \in \mathbb{R}_{+}^{*}. So the only data we need is the homomorphism \phi : \mathbb{Q} / \mathbb{Z} \to \mathbb{Q} / \lambda \mathbb{Z}. But apparently \hat{\mathbb{Z}} \cong Hom (\mathbb{Q} / \mathbb{Z},\mathbb{Q} / \mathbb{Z}), that is, \hat{\mathbb{Z}} is isomorphic to the set of all endomorphisms of \mathbb{Q} / \mathbb{Z} (I'm not yet sure how! Something to do with Pontryagin duality?), so all that remains is a choice of \rho \in \hat{\mathbb{Z}} and we have (\Lambda,\phi) \mapsto (\rho,\lambda). One should note that the inverse of this map is well defined, so we can identify the set of all 1-dim \mathbb{Q}-lattices with \hat{\mathbb{Z}} \times \mathbb{R}_{+}^{*}.
I'll continue reading about 1-dim \mathbb{Q}-lattices for the next several days, though I do still plan to take some time to catch up on some algebra (algebras over a field, group rings, group algebras, C* algebras, their representations, group C* algebras, Pontryagin duels, etale groupoids.)
Thanks to Professor Kim and the folks at mathoverlow for helping me understand this.
Also, for the next few days I will all but abandon Marcolli's monograph; that text goes straight into a description of the Bost-Connes system via Shimura varieties, which is a bit above my level. I do intent to understand the Shimura variety and the Hecke algebra description of Bost-Connes eventually, however.
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