abc on adelic Bost-Connes

By lieven

Noncommutative geometry and the Riemann zeta function

The adelic interpretation of the Bost-Connes Hecke algebra $\mathcal{H}$ is based on three facts we’ve learned so far :

The diagonal embedding of the rational numbers $\delta~:~\mathbb{Q} \rightarrow \prod_p \mathbb{Q}p$ has its image in the adele ring $\mathcal{A}$ . ( details )
There is an exact sequence of semigroups $1 \rightarrow \mathcal{G} \rightarrow \mathcal{I} \cap \mathcal{R} \rightarrow \mathbb{N}^+_{\times} \rightarrow 1$ where $\mathcal{I}$ is the idele group, that is the units of $\mathcal{A}$ , where $\mathcal{R} = \prodp \mathbb{Z}_p$ and where $\mathcal{G}$ is the group (!) $\prodp \mathbb{Z}_p^*$ . ( details )
There is an isomorphism of additive groups $\mathbb{Q}/\mathbb{Z} \simeq \mathcal{A}/\mathcal{R}$ . ( details )

Because $\mathcal{R}$ is a ring we have that $a\mathcal{R} \subset \mathcal{R}$ for any $a=(a_p)p \in \mathcal{I} \cap \mathcal{R}$ . Therefore, we have an induced ‘multiplication by ‘ morphism on the additive group $\mathcal{A}/\mathcal{R} \rightarrow^{a.} \mathcal{A}/\mathcal{R}$ which is an epimorphism for all $a \in \mathcal{I} \cap \mathcal{R}$ .

In fact, it is easy to see that the equation a.x = y for $y \in \mathcal{A}/\mathcal{R}$ has precisely $n_a = \prodp p^{d(a)}$ solutions. In particular, for any $a \in \mathcal{G} = \prod_p \mathbb{Z}p^*$ , multiplication by is an isomorphism on $\mathcal{A}/\mathcal{R} = \mathbb{Q}/\mathbb{Z}$ .

But then, we can form the crystalline semigroup graded skew-group algebra $\mathbb{Q}(\mathbb{Q}/\mathbb{Z}) \bowtie (\mathcal{I} \cap \mathcal{R})$ . It is the graded vectorspace $\oplus_{a \in \mathcal{I} \cap \mathcal{R}} Xa \mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ with commutation relation $Y_{\lambda}Xa = X_a Y{a \lambda}$ for the base-vectors $Y_{\lambda}$ with $\lambda \in \mathbb{Q}/\mathbb{Z}$ . Recall from last time we need to use approximation (or the Chinese remainder theorem) to determine the class of $a \lambda$ in $\mathbb{Q}/\mathbb{Z}$ .

We can also extend it to a bi-crystalline graded algebra because multiplication by $a \in \mathcal{I} \cap \mathcal{R}$ has a left-inverse which determines the commutation relations $Y_{\lambda} Xa^* = X_a^* (\frac{1}{na})(\sum_{a.\mu = \lambda} Y{\mu})$ . Let us call this bi-crystalline graded algebra $\mathcal{H}_{big}$ , then we have the following facts

For every $a \in \mathcal{G}$ , the element is a unit in $\mathcal{H}{big}$ and $X_a^{-1}=Xa^*$ . Conjugation by induces on the subalgebra $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ the map $Y{\lambda} \rightarrow Y_{a \lambda}$ .
Using the diagonal embedding $\delta$ restricted to $\mathbb{N}^+_{\times}$ we get an embedding of algebras $\mathcal{H} \subset \mathcal{H}{big}$ and conjugation by for any $a \in \mathcal{G}$ sends $\mathcal{H}$ to itself. However, as the $Xa \notin \mathcal{H}$ , the induced automorphisms are now outer!

Summarizing : the Bost-Connes Hecke algebra $\mathcal{H}$ encodes a lot of number-theoretic information :

the additive structure is encoded in the sub-algebra which is the group-algebra $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$
the multiplicative structure in encoded in the epimorphisms given by multiplication with a positive natural number (the commutation relation with the
the automorphism group of $\mathbb{Q}/\mathbb{Z}$ extends to outer automorphisms of $\mathcal{H}$

That is, the Bost-Connes algebra can be seen as a giant mashup of number-theory of $\mathbb{Q}$ . So, if one can prove something specific about this algebra, it is bound to have interesting number-theoretic consequences.

But how will we study $\mathcal{H}$ ? Well, the bi-crystalline structure of it tells us that $\mathcal{H}$ is a ‘good’-graded algebra with part of degree one the group-algebra $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$ . This group-algebra is a formally smooth algebra and we study such algebras by studying their finite dimensional representations.

Hence, we should study ‘good’-graded formally smooth algebras (such as $\mathcal{H}$ ) by looking at their graded representations. This will then lead us to Connes’ “fabulous states”…

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Bost-Connes, Connes, formally smooth algebra, groups, Hecke algebra, representations

2 comments
Posted in geometry
Written on Sat, 02 February 2008 at 2:52 pm
Tags: Bost-Connes, Connes, formally smooth algebra, groups, Hecke algebra, representations
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February 2nd, 2008 at 2:55 pm
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February 2nd, 2008 at 2:59 pm
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