Wednesday, July 28, 2010

$\mathbb{Q}$-Lattices, KMS condition, Hecke algebras, and other things I don't understand

Connes, Mercolli, et al, give the following definition for a $\mathbb{Q}$-Lattice:

A normal lattice $\Lambda$ in $\mathbb{R}^n$ together with a group homomorphism:
$\phi : \mathbb{Q}^n / \mathbb{Z}^n \to \mathbb{Q} \Lambda / \Lambda$

Mercolli describes this homomorphism as giving "a system of labelling [the lattices] torsion points". Two $\mathbb{Q}$-Lattices (the underlying lattices are $\Lambda_1$ and $\Lambda_2$) are called commensurable if

$\mathbb{Q} \Lambda_1 = \mathbb{Q} \Lambda_2$ and $\phi_1 = \phi_2$ mod $\Lambda_1 + \Lambda_2$

Commensurability is said to be an equivalence relation amoung $\mathbb{Q}$-Lattices, and we can denote the set of all equivalence classes of $\mathbb{Q}$-Lattices in $\mathbb{R}^n$ by $\mathcal{L}_n$. The topology for this space is "encoded" [whatever that means] in a C* Algebra denoted $C^{*} (\mathcal{L}_n)$ for n=1 we have the Bost-Connes system [I think].

But in the 1995 paper, the Bost-Connes system was built up from Heck algebras. these two are apparently isomorphic, as best as I understand things.

The problem with all the above is that I don't understand it. $\phi$ is a homomorphism between two groups, neither of which I understand (I barely know what the notation means.) Ditto for the definition of Commensurability.

Now, these $C* (\mathcal{L}_n)$ are only the beginning. This system is suppose to describe a Quantum Statistical Mechanical system (C* dynamical system...or a C* algebra with a one-parameterized group of automorphism) and all that has something to do with another thing I don't understand...the KMS condition. Which is used to define equilibrium states of the QSM system. Conne's "Non-commutative geometry, quantum fields, and motives" has a pretty long section on QSM that might just get me up to speed on the KMS condition and such, if not, I have R Haag's book on "Local Quantum Physics", which should take another 2-3 months to digest (:S) before I'll get anywhere. I am still fuzzy on the idea of a C* algebra, to be honest.

But first thing first. I need to know what a $\mathbb{Q}$-Lattice is. So I need to understand $\phi$ and its groups. Anyone know of any good reading material on the subject?

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