Posts Tagged ‘formally smooth algebra’



abc on adelic Bost-Connes

Saturday, February 2nd, 2008

Noncommutative geometry and the Riemann zeta function

  1. the Bost-Connes coset space
  2. the Bost-Connes Hecke algebra
  3. Bost-Connes for ringtheorists
  4. BC stands for Bi-Crystalline graded
  5. adeles and ideles
  6. Chinese remainders and adele classes
  7. abc on adelic Bost-Connes
  8. “God given time”
  9. KMS, Gibbs & zeta function

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

  1. 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 )

  2. 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} = \prod_p \mathbb{Z}_p and where \mathcal{G} is the group (!) \prod_p \mathbb{Z}_p^*. ( details )

  3. 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 a‘ 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 = \prod_p p^{d(a)} solutions. In particular, for any a \in \mathcal{G} = \prod_p \mathbb{Z}_p^*, multiplication by a 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}} X_a \mathbb{Q}[\mathbb{Q}/\mathbb{Z}] with commutation relation Y_{\lambda}X_a = 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} X_a^* = X_a^* (\frac{1}{n_a})(\sum_{a.\mu = \lambda} Y_{\mu}). Let us call this bi-crystalline graded algebra \mathcal{H}_{big}, then we have the following facts

  1. For every a \in \mathcal{G}, the element X_a is a unit in \mathcal{H}_{big} and X_a^{-1}=X_a^*. Conjugation by X_a induces on the subalgebra \mathbb{Q}[\mathbb{Q}/\mathbb{Z}] the map Y_{\lambda} \rightarrow Y_{a \lambda}.

  2. 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 X_a for any a \in \mathcal{G} sends \mathcal{H} to itself. However, as the X_a \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 X_m
  • 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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