Tag: Connes

Bost-Connes for ringtheorists

Published January 23, 2008 by lieven

Over the last days I’ve been staring at the Bost-Connes algebra to find a ringtheoretic way into it. Ive had some chats about it with the resident graded-guru but all we came up with so far is that it seems to be an extension of Fred’s definition of a ‘crystalline’ graded algebra. Knowing that several excellent ringtheorists keep an eye on my stumblings here, let me launch an appeal for help :

What is the most elegant ringtheoretic framework in which the Bost-Connes Hecke algebra is a motivating example?

Let us review what we know so far and extend upon it with a couple of observations that may (or may not) be helpful to you. The algebra $H$ is the algebra of $Q$ -valued functions (under the convolution product) on the double coset-space $Γ_{0} ∖ Γ / Γ_{0}$ where

$Γ = [\begin{matrix} 1 & b \\ 0 & a \end{matrix}] : a, b \in Q, a > 0$ and $Γ_{0} = [\begin{matrix} 1 & n \\ 0 & 1 \end{matrix}] : n \in N_{+}$

We have seen that a $Q$ -basis is given by the characteristic functions $X_{γ}$ (that is, such that $X_{γ} (γ^{'}) = δ_{γ, γ^{'}}$ ) with $γ$ a rational point represented by the couple $(a, b)$ (the entries in the matrix definition of a representant of $γ$ in $Γ$ ) lying in the fractal comb

defined by the rule that $b < \frac{1}{n}$ if $a = \frac{m}{n}$ with $m, n \in N, (m, n) = 1$ . Last time we have seen that the algebra $H$ is generated as a $Q$ -algebra by the following elements (changing notation)

${\begin{cases} X_{m} = X_{α_{m}} & with α_{m} = [\begin{array}{c} 1 & 0 \\ 0 & m \end{array}] \forall m \in N_{+} \\ X_{n}^{*} = X_{β_{n}} & with β_{n} = [\begin{array}{c} 1 & 0 \\ 0 & \frac{1}{n} \end{array}] \forall n \in N_{+} \\ Y_{γ} = X_{γ} & with γ = [\begin{array}{c} 1 & γ \\ 0 & 1 \end{array}] \forall λ \in Q / Z \end{cases}$

Using the tricks of last time (that is, figuring out what functions convolution products represent, knowing all double-cosets) it is not too difficult to prove the defining relations among these generators to be the following (( if someone wants the details, tell me and I’ll include a ‘technical post’ or consult the Bost-Connes original paper but note that this scanned version needs 26.8Mb ))

(1) : $X_{n}^{*} X_{n} = 1, \forall n \in N_{+}$

(2) : $X_{n} X_{m} = X_{n m}, \forall m, n \in N_{+}$

(3) : $X_{n} X_{m}^{*} = X_{m}^{*} X_{n}, whenever (m, n) = 1$

(4) : $Y_{γ} Y_{μ} = Y_{γ + μ}, \forall γ, m u \in Q / Z$

(5) : $Y_{γ} X_{n} = X_{n} Y_{n γ}, \forall n \in N_{+}, γ \in Q / Z$

(6) : $X_{n} Y_{λ} X_{n}^{*} = \frac{1}{n} \sum_{n δ = γ} Y_{δ}, \forall n \in N_{+}, γ \in Q / Z$

Simple as these equations may seem, they bring us into rather uncharted ringtheoretic territories. Here a few fairly obvious ringtheoretic ingredients of the Bost-Connes Hecke algebra $H$

the group-algebra of $Q / Z$

The equations (4) can be rephrased by saying that the subalgebra generated by the $Y_{γ}$ is the rational groupalgebra $Q [Q / Z]$ of the (additive) group $Q / Z$ . Note however that $Q / Z$ is a torsion group (that is, for all $γ = \frac{m}{n}$ we have that $n . γ = (γ + γ + \dots + γ) = 0$ ). Hence, the groupalgebra has LOTS of zero-divisors. In fact, this group-algebra doesn’t have any good ringtheoretic properties except for the fact that it can be realized as a limit of finite groupalgebras (semi-simple algebras)

$Q [Q / Z] = \underset{\to}{l i m} Q [Z / n Z]$

and hence is a quasi-free (or formally smooth) algebra, BUT far from being finitely generated…

the grading group $Q_{\times}^{+}$

The multiplicative group of all positive rational numbers $Q_{\times}^{+}$ is a torsion-free Abelian ordered group and it follows from the above defining relations that $H$ is graded by this group if we give

$d e g (Y_{γ}) = 1, d e g (X_{m}) = m, d e g (X_{n}^{*}) = \frac{1}{n}$

Now, graded algebras have been studied extensively in case the grading group is torsion-free abelian ordered AND finitely generated, HOWEVER $Q_{\times}^{+}$ is infinitely generated and not much is known about such graded algebras. Still, the ordering should allow us to use some tricks such as taking leading coefficients etc.

the endomorphisms of $Q [Q / Z]$

We would like to view the equations (5) and (6) (the latter after multiplying both sides on the left with $X_{n}^{*}$ and using (1)) as saying that $X_{n}$ and $X_{n}^{*}$ are normalizing elements. Unfortunately, the algebra morphisms they induce on the group algebra $Q [Q / Z]$ are NOT isomorphisms, BUT endomorphisms. One source of algebra morphisms on the group-algebra comes from group-morphisms from $Q / Z$ to itself. Now, it is known that

$H o m_{g r p} (Q / Z, Q / Z) ≃ \hat{Z}$ , the profinite completion of $Z$ . A class of group-morphisms of interest to us are the maps given by multiplication by n on $Q / Z$ . Observe that these maps are epimorphisms with a cyclic order n kernel. On the group-algebra level they give us the epimorphisms

$Q [Q / Z] ⟶^{ϕ_{n}} Q [Q / Z]$ such that $ϕ_{n} (Y_{λ}) = Y_{n λ}$ whence equation (5) can be rewritten as $Y_{λ} X_{n} = X_{n} ϕ_{n} (Y_{λ})$ , which looks good until you think that $ϕ_{n}$ is not an automorphism…

There are even other (non-unital) algebra endomorphisms such as the map $Q [Q / Z] \to^{ψ_{n}} R_{n}$ defined by $ψ_{n} (Y_{λ}) = \frac{1}{n} (Y_{\frac{λ}{n}} + Y_{\frac{λ + 1}{n}} + \dots + Y_{\frac{λ + n - 1}{n}})$ and then, we can rewrite equation (6) as $Y_{λ} X_{n}^{*} = X_{n}^{*} ψ_{n} (Y_{λ})$ , but again, note that $ψ_{n}$ is NOT an automorphism.

almost strongly graded, but not quite…

Recall from last time that the characteristic function $X_{a}$ for any double-coset-class $a \in Γ_{0} ∖ Γ / Γ_{0}$ represented by the matrix $a = [\begin{matrix} 1 & λ \\ 0 & \frac{m}{n} \end{matrix}]$ could be written in the Hecke algebra as $X_{a} = n X_{m} Y_{n λ} X_{n}^{*} = n Y_{λ} X_{m} X_{n}^{*}$ . That is, we can write the Bost-Connes Hecke algebra as

$H = \oplus_{\frac{m}{n} \in Q_{\times}^{+}} Q [Q / Z] X_{m} X_{n}^{*}$

Hence, if only the morphisms $ϕ_{n}$ and $ψ_{m}$ would be automorphisms, this would say that $H$ is a strongly $Q_{\times}^{+}$ -algebra with part of degree one the groupalgebra of $Q / Z$ .

However, they are not. But there is an extension of the notion of strongly graded algebras which Fred has dubbed crystalline graded algebras in which it is sufficient that the algebra maps are all epimorphisms. (maybe I’ll post about these algebras, another time). However, this is not the case for the $ψ_{m}$ …

So, what is the most elegant ringtheoretic framework in which the algebra $H$ fits??? Surely, you can do better than generalized crystalline graded algebra…

the Bost-Connes Hecke algebra

Published January 19, 2008 by lieven

As before, $Γ$ is the subgroup of the rational linear group $G L_{2} (Q)$ consisting of the matrices

$[\begin{matrix} 1 & b \\ 0 & a \end{matrix}]$ with $a \in Q_{+}$ and $Γ_{0}$ the subgroup of all matrices $[\begin{matrix} 1 & n \\ 0 & 1 \end{matrix}]$ with $n \in N$ . Last time, we have seen that the double coset space $Γ_{0} ∖ Γ / Γ_{0}$ can be identified with the set of all rational points in the fractal comb consisting of all couples $(a, b)$ with $a = \frac{m}{n} \in Q_{+}$ and $b \in [0, \frac{1}{n}) \cap Q$

The blue spikes are at the positive natural numbers $a = 1, 2, 3, \dots$ . Over $a = 1$ they correspond to the matrices $[\begin{matrix} 1 & γ \\ 0 & 1 \end{matrix}]$ with $γ \in [0, 1) \cap Q$ and as matrix-multiplication of such matrices corresponds to addition of the $γ$ we see that these cosets can be identified with the additive group $Q / Z$ (which will reappear at a later stage as the multiplicative group of all roots of unity).

The Bost-Connes Hecke algebra $H = H (Γ, Γ_{0})$ is the convolution algebra of all comlex valued functions with finite support on the double coset space $Γ_{0} ∖ Γ / Γ_{0}$ . That is, as a vector space the algebra has as basis the functions $e_{X}$ with $X \in Γ_{0} ∖ Γ / Γ_{0}$ (that is, $X$ is a point of the fractal comb) and such that $e_{X} (X) = 1$ and $e_{X} (Y) = 0$ for all other double cosets $Y \neq X$ . The algebra product on $H$ is the convolution-product meaning that if $f, f^{'}$ are complex functions with finite support on the Bost-Connes space, then they can also be interpreted as $Γ_{0}$ -bi-invariant functions on the group $Γ$ (for this just means that the function is constant on double cosets) and then $f * f^{'}$ is the function defined for all $γ \in Γ$ by

$f * f^{'} (γ) = \sum_{μ \in Γ / Γ_{0}} f (μ) f^{'} (μ^{- 1} γ)$

Last time we have seen that the coset-space $Γ / Γ_{0}$ can be represented by all rational points $(a, b)$ with $b < 1$ . At first sight, the sum above seems to be infinite, but, f and f’ are non-zero only at finitely many double cosets and we have see last time that $Γ_{0}$ acts on one-sided cosets with finite orbits. Therefore, $f * f$ is a well-defined $Γ_{0}$ -bi-invariant function with finite support on the fractal comb $Γ_{0} ∖ Γ / Γ_{0}$ . Further, observe that the unit element of $H$ is the function corresponding to the identity matrix in $Γ$ .

Looking at fractal-comb picture it is obvious that the Bost-Connes Hecke algebra $H$ is a huge object. Today, we will prove the surprising result that it can be generated by the functions corresponding to the tiny portion of the comb, shown below.

That is, we will show that $H$ is generated by the functions $e (γ)$ corresponding to the double-coset $X_{γ} = [\begin{matrix} 1 & γ \\ 0 & 1 \end{matrix}]$ (the rational points of the blue line-segment over 1, or equivalently, the elements of the group $Q / Z$ ), together with the functions $ϕ_{n}$ corresponding to the double-coset $X_{n} = [\begin{matrix} 1 & 0 \\ 0 & n \end{matrix}]$ for all $n \in N_{+}$ (the blue dots to the right in the picture) and the functions $ϕ_{n}^{*}$ corresponding to the double cosets $X_{1 / n} = [\begin{matrix} 1 & 0 \\ 0 & \frac{1}{n} \end{matrix}]$ (the red dots to the left).

Take a point in the fractal comb $X = [\begin{matrix} 1 & γ \\ 0 & \frac{m}{n} \end{matrix}]$ with $(m, n) = 1$ and $γ \in [0, \frac{1}{n}) \cap Q \subset [0, 1) \cap Q$ . Note that as $γ < \frac{1}{n}$ we have that $n γ < 1$ and hence $e (n γ)$ is one of the (supposedly) generating functions described above.

Because $X = [\begin{matrix} 1 & γ \\ 0 & \frac{m}{n} \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & m \end{matrix}] [\begin{matrix} 1 & n γ \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & \frac{1}{n} \end{matrix}] = X_{m} X_{n γ} X_{1 / n}$ we are aiming for a relation in the Hecke algebra $ϕ_{m} * e (n γ) * ϕ_{n}^{*} = e_{X}$ . This is ‘almost’ true, except from a coefficient.

Let us prove first the equality of functions $e_{X} * ϕ_{n} = n ϕ_{m} * e (n γ)$ . To do this we have to show that they have the same value for all points $Y \in Γ_{0} ∖ Γ / Γ_{0}$ in the fractal comb. Let us first study the function on the right hand side.

$ϕ_{m} * e (n γ) = \sum_{g \in Γ / Γ_{0}} ϕ_{m} (g) e (n γ) (g^{- 1} Y)$ . Because $X_{m} Γ_{0}$ is already a double coset (over $m$ we have a comb-spike of length one, so all rational points on it determine at the same time a one-sided and a double coset. Therefore, $ϕ_{m} (g)$ is zero unless $g = X_{m}$ and then the value is one.

Next, let us consider the function on the left-hand side. $e_{X} * ϕ_{n} (Y) = \sum_{g \in Γ / Γ_{0}} e_{X} (g) ϕ_{m} (g^{- 1} Y)$ . We have to be a bit careful here as the double cosets over $a = \frac{m}{n}$ are different from the left cosets. Recall from last time that the left-cosets over a are given by all rational points of the form $(a, b)$ with $b < 1$ whereas the double-cosets over a are represented by the rational points of the form $(a, b)$ with $b < \frac{1}{n}$ and hence the $Γ_{0}$ -orbits over a all consist of precisely n elements g.
That is, $e_{X} (g)$ is zero for all $g \in Γ / Γ_{0}$ except when g is one of the following matrices

$g \in [\begin{matrix} 1 & γ \\ 0 & \frac{m}{n} \end{matrix}], [\begin{matrix} 1 & γ + \frac{1}{n} \\ 0 & \frac{m}{n} \end{matrix}], [\begin{matrix} 1 & γ + \frac{2}{n} \\ 0 & \frac{m}{n} \end{matrix}], \dots, [\begin{matrix} 1 & γ + \frac{n - 1}{n} \\ 0 & \frac{m}{n} \end{matrix}]$

Further, $ϕ_{n} (g^{- 1} Y)$ is zero unless $g^{- 1} Y \in Γ_{0} [\begin{matrix} 1 & 0 \\ 0 & n \end{matrix}] Γ_{0}$ , or equivalently, that $Y \in Γ_{0} g Γ_{0} [\begin{matrix} 1 & 0 \\ 0 & n \end{matrix}] Γ_{0} = Γ_{0} g [\begin{matrix} 1 & 0 \\ 0 & n \end{matrix}] Γ_{0}$ and for each of the choices for g we have that

$[\begin{matrix} 1 & γ + \frac{k}{n} \\ 0 & \frac{m}{n} \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & n \end{matrix}] = [\begin{matrix} 1 & n γ + k \\ 0 & m \end{matrix}] \sim [\begin{matrix} 1 & n γ \\ 0 & m \end{matrix}]$

Therefore, the function $e_{X} * ϕ_{n}$ is zero at every point of the fractal comb unless at $[\begin{matrix} 1 & n γ \\ 0 & m \end{matrix}]$ where it is equal to $n$ . This proves the claimed identity of functions and as one verifies easily that $ϕ_{n}^{*} * ϕ_{n} = 1$ , it follows that all base vectors $e_{X}$ of $H$ can be expressed in the claimed generators

$e_{X} = n ϕ_{m} * e (n γ) * ϕ_{n}^{*}$

Bost and Connes use slightly different generators, namely with $μ_{n} = \frac{1}{\sqrt{n}} ϕ_{n}$ and $μ_{n}^{*} = \sqrt{n} ϕ_{n}^{*}$ in order to have all relations among the generators being defined over $Q$ (as we will see another time). This will be important later on to have an action of the cyclotomic Galois group $G a l (Q^{c y c l} / Q)$ on certain representations of $H$ .

the Bost-Connes coset space

Published January 17, 2008 by lieven

By now, everyone remotely interested in Connes’ approach to the Riemann hypothesis, knows the _one line mantra_

one can use noncommutative geometry to extend Weil’s proof of the Riemann-hypothesis in the function field case to that of number fields

But, can one go beyond this sound-bite in a series of blog posts? A few days ago, I was rather optimistic, but now, after reading-up on the Connes-Consani-Marcolli project, I feel overwhelmed by the sheer volume of their work (and by my own ignorance of key tools in the approach). The most recent account takes up half of the 700+ pages of the book Noncommutative Geometry, Quantum Fields and Motives by Alain Connes and Matilde Marcolli…

So let us set a more modest goal and try to understand one of the first papers Alain Connes wrote about the RH : Noncommutative geometry and the Riemann zeta function. It is only 24 pages long and relatively readable. But even then, the reader needs to know about class field theory, the classification of AF-algebras, Hecke algebras, etc. etc. Most of these theories take a book to explain. For example, the first result he mentions is the main result of local class field theory which appears only towards the end of the 200+ pages of Jean-Pierre Serre’s Local Fields, itself a somewhat harder read than the average blogpost…

Anyway, we will see how far we can get. Here’s the plan : I’ll take the heart-bit of their approach : the Bost-Connes system, and will try to understand it from an algebraist’s viewpoint. Today we will introduce the groups involved and describe their cosets.

For any commutative ring $R$ let us consider the group of triangular $2 \times 2$ matrices of the form

$P_{R} = [\begin{matrix} 1 & b \\ 0 & a \end{matrix}] | b \in R, a \in R^{*}$

(that is, $a$ in an invertible element in the ring $R$ ). This is really an affine group scheme defined over the integers, that is, the coordinate ring

$Z [P] = Z [x, x^{- 1}, y]$ becomes a Hopf algebra with comultiplication encoding the group-multiplication. Because

$[\begin{matrix} 1 & b_{1} \\ 0 & a_{1} \end{matrix}] [\begin{matrix} 1 & b_{2} \\ 0 & a_{2} \end{matrix}] = [\begin{matrix} 1 & 1 \times b_{2} + b_{1} \times a_{2} \\ 0 & a_{1} \times a_{2} \end{matrix}]$

we have $Δ (x) = x \otimes x$ and $Δ (y) = 1 \otimes y + y \otimes x$ , or $x$ is a group-like element whereas $y$ is a skew-primitive. If $R \subset R$ is a subring of the real numbers, we denote by $P_{R}^{+}$ the subgroup of $P_{R}$ consisting of all matrices with $a > 0$ . For example,

$Γ_{0} = P_{Z}^{+} = [\begin{matrix} 1 & n \\ 0 & 1 \end{matrix}] | n \in Z$

which is a subgroup of $Γ = P_{Q}^{+}$ and our first job is to describe the cosets.

The left cosets $Γ / Γ_{0}$ are the subsets $γ Γ_{0}$ with $γ \in Γ$ . But,

$[\begin{matrix} 1 & b \\ 0 & a \end{matrix}] [\begin{matrix} 1 & n \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & b + n \\ 0 & a \end{matrix}]$

so if we represent the matrix $γ = [\begin{matrix} 1 & b \\ 0 & a \end{matrix}]$ by the point $(a, b)$ in the right halfplane, then for a given positive rational number $a$ the different cosets are represented by all $b \in [0, 1) \cap Q = Q / Z$ . Hence, the left cosets are all the rational points in the region between the red and green horizontal lines. For fixed $a$ the cosets correspond to the rational points in the green interval (such as over $\frac{2}{3}$ in the picture on the left.

Similarly, the right cosets $Γ_{0} ∖ Γ$ are the subsets $Γ_{0} γ$ and as

$[\begin{matrix} 1 & n \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & b \\ 0 & a \end{matrix}] = [\begin{matrix} 1 & b + n a \\ 0 & a \end{matrix}]$

we see similarly that the different cosets are precisely the rational points in the region between the lower red horizontal and the blue diagonal line. So, for fixed $a$ they correspond to rational points in the blue interval (such as over $\frac{3}{2}$ ) $[0, a) \cap Q$ . But now, let us look at the double coset space $Γ_{0} ∖ Γ / Γ_{0}$ . That is, we want to study the orbits of the action of $Γ_{0}$ , acting on the right, on the left-cosets $Γ / Γ_{0}$ , or equivalently, of the action of $Γ_{0}$ acting on the left on the right-cosets $Γ_{0} ∖ Γ$ . The crucial observation to make is that these actions have finite orbits, or equivalently, that $Γ_{0}$ is an almost normal subgroup of $Γ$ meaning that $Γ_{0} \cap γ Γ_{0} γ^{- 1}$ has finite index in $Γ_{0}$ for all $γ \in Γ$ . This follows from

$[\begin{matrix} 1 & n \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & b \\ 0 & a \end{matrix}] [\begin{matrix} 1 & m \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & b + m + a n \\ 0 & a \end{matrix}]$

and if $n$ varies then $a n$ takes only finitely many values modulo $Z$ and their number depends only on the denominator of $a$ . In the picture above, the blue dots lying on the line over $\frac{2}{3}$ represent the double coset

$Γ_{0} [\begin{matrix} 1 & \frac{2}{3} \\ 0 & \frac{2}{3} \end{matrix}]$ and we see that these dots split the left-cosets with fixed value $a = \frac{2}{3}$ (that is, the green line-segment) into three chunks (3 being the denominator of a) and split the right-cosets (the line-segment under the blue diagonal) into two subsegments (2 being the numerator of a). Similarly, the blue dots on the line over $\frac{3}{2}$ divide the left-cosets in two parts and the right cosets into three parts.

This shows that the $Γ_{0}$ -orbits of the right action on the left cosets $Γ / Γ_{0}$ for each matrix $γ \in Γ$ with $a = \frac{2}{3}$ consist of exactly three points, and we denote this by writing $L (γ) = 3$ . Similarly, all $Γ_{0}$ -orbits of the left action on the right cosets $Γ_{0} ∖ Γ$ with this value of a consist of two points, and we write this as $R (γ) = 2$ .

For example, on the above picture, the black dots on the line over $\frac{2}{3}$ give the matrices in the double coset of the matrix

$γ = [\begin{matrix} 1 & \frac{1}{7} \\ 0 & \frac{2}{3} \end{matrix}]$

and the gray dots on the line over $\frac{3}{2}$ determine the elements of the double coset of

$γ^{- 1} = [\begin{matrix} 1 & - \frac{3}{14} \\ 0 & \frac{3}{2} \end{matrix}]$

and one notices (in general) that $L (γ) = R (γ^{- 1})$ . But then, the double cosets with $a = \frac{2}{3}$ are represented by the rational b’s in the interval $[0, \frac{1}{3})$ and those with $a = \frac{3}{2}$ by the rational b’s in the interval $\frac{1}{2}$ . In general, the double cosets of matrices with fixed $a = \frac{r}{s}$ with $(r, s) = 1$ are the rational points in the line-segment over $a$ with $b \in [0, \frac{1}{s})$ .

That is, the Bost-Connes double coset space $Γ_{0} ∖ Γ / Γ_{0}$ are the rational points in a horrible fractal comb. Below we have drawn only the part of the dyadic values, that is when $a = \frac{r}{2^{t}}$ in the unit inverval

and of course we have to super-impose on it similar pictures for rationals with other powers as their denominators. Fortunately, NCG excels in describing such fractal beasts…

UPDATE : here is a slightly beter picture of the coset space, drawing the part over all rational numbers contained in the 15-th Farey sequence. The blue segments of length one are at 1,2,3,…