I enjoyed [probably because I actually understood] the following post on group theory from Terry Tao. I thought I'd share a link:

http://terrytao.wordpress.com/2009/10/19/grothendiecks-definition-of-a-group/

## Friday, July 30, 2010

### $\mathbb{Q}$-lattices and commensurability

My supervisor, Professor Kim, helped clear up some of my confusion on the homomorphism in $\mathbb{Q}$ lattices. Lets go over it again: A $\mathbb{Q}$-lattice is a normal lattice $\Lambda \subset \mathbb{R}^n$ together with a homomorphism $\phi : \mathbb{Q}^n / \mathbb{Z}^n \to \mathbb{Q}\Lambda / \Lambda$. Lets break down that notation a bit.

$\mathbb{Q}^n / \mathbb{Z}^n$ looked a bit frightening to me at first, but its really not so bad. Consider the far more neighbourly quotient $\mathbb{R} / \mathbb{Z}$, which of course is just a circle. $\mathbb{Q} / \mathbb{Z}$ clearly is a subset of the circle, but only including the points on the circle that go back to the origin when multiplied by an integer. $(\mathbb{Q} / \mathbb{Z})^2$ is a subset of the torus $(\mathbb{R} / \mathbb{Z})^2$, and so on, and that gives me a reasonable way to think about the more general looking $\mathbb{Q}^n / \mathbb{Z}^n$. Not that bad after all, right?

Now for $\mathbb{Q}\Lambda / \Lambda$. $\mathbb{Q}\Lambda$ turns out to be exactly what it sounds like. For a lattice $\Lambda$, we can consider it as the $\mathbb{Z}$-span of basis vectors $\{ v_1,...,v_n \}$. For $\mathbb{Q}\Lambda$, just take the $\mathbb{Q}$-span. But what about the quotient? Its really not that different from $\mathbb{Q}^n / \mathbb{Z}^n$. Consider that $\Lambda$ is like a "tilted" version of $\mathbb{Z}^n$. From there its not hard to imagine $\mathbb{Q}\Lambda / \Lambda$ as being a "tilted" version of the "hyper-rational-torus" $\mathbb{Q}^n / \mathbb{Z}^n$ (I call it "hyper" because it may be more than 2 dimensions, and "rational" because it only includes the points that get back to the origin when multiplied by an integer.) In fact, its not so hard to see that there's an isomorphism between $\mathbb{Q}^n / \mathbb{Z}^n$ and $\mathbb{Q}\Lambda / \Lambda$ that comes just from figuring out how much $\Lambda$ is titled from $\mathbb{Z}^n$, id est, expressing the basis for $\Lambda$ in terms of the standard basis of $\mathbb{Z}^n$.

So why then does the the definition of $\mathbb{Q}$-lattices include a homomorphism and not a full, invertible isomorphism? In fact, why do we even need a definition for $\mathbb{Q}$-lattices at all? It just seems to be a lattice with an obvious homomorphism attached to it. (this question is really quite moot. commensurability and the commensurability classes yield plenty results, among them is the topic of my 4th year project!)

Finally, we also have a notion of what it means for 2 $\mathbb{Q}$-lattices to be commensurable: $\mathbb{Q} \Lambda_1 = \mathbb{Q} \Lambda_2$ and $\phi_1 - \phi_2\, =\, 0\, mod\, \Lambda_1 + \Lambda_2$. What does this mean? As best I can tell [I'm really not too sure], the homomorphism labels some of our tilted-torus points as coming from our nice hyper-rational-torus. The statement that $\phi_1 - \phi_2\, =\, 0\, mod\, \Lambda_1 + \Lambda_2$ says that they have the same points labelled in the tilted-torus. It's pretty easy to see that if the homomorphisms are isomorphism, then 2 $\mathbb{Q}-lattices are only commensurable if they are in fact, equal. Its also not too hard to see that commensurability is an equivalence relation (Connes has a proof of this fact in Non-commutative geometry, Quantum Fields and motives, it was from that prove that I gathered the "same label" idea.) We denote all the commensurability classes of $\mathbb{Q}$-lattices in $\mathbb{R}^n$ as $\mathcal{L}_n$.

So whats next? Apparently, the Bost-Connes quantum statistical mechanical system a C*-algebra somehow generated from the quotient $\mathcal{L}_n / \mathbb{R}^{*}_{+}$ (I don't recognize the notation $\mathbb{R}^{*}_{+}$, $\mathbb{R}^{*}$ I thought was the real numbers excluding 0, and $\mathbb{R}_{+}$ the positive reals. so is $\mathbb{R}^{*}_{+}$ all the strictly positive reals?) So I suppose my next step is to 1) understand how to form that C*-algebra [which implicitly involves understanding what a C*-algebra is, I think I'll probably spend some time reading the first chapter to Averson's Invitation to C*-algebras] and 2) understand how that C*-algebra is a quantum statistical mechanical system. After that point, I can figure out how this is the same as the Hecke algebra mentioned in Connes' and Bost's work and what it has to do with $Gal(\mathbb{Q}^{cycl} / \mathbb{Q})$.

$\mathbb{Q}^n / \mathbb{Z}^n$ looked a bit frightening to me at first, but its really not so bad. Consider the far more neighbourly quotient $\mathbb{R} / \mathbb{Z}$, which of course is just a circle. $\mathbb{Q} / \mathbb{Z}$ clearly is a subset of the circle, but only including the points on the circle that go back to the origin when multiplied by an integer. $(\mathbb{Q} / \mathbb{Z})^2$ is a subset of the torus $(\mathbb{R} / \mathbb{Z})^2$, and so on, and that gives me a reasonable way to think about the more general looking $\mathbb{Q}^n / \mathbb{Z}^n$. Not that bad after all, right?

Now for $\mathbb{Q}\Lambda / \Lambda$. $\mathbb{Q}\Lambda$ turns out to be exactly what it sounds like. For a lattice $\Lambda$, we can consider it as the $\mathbb{Z}$-span of basis vectors $\{ v_1,...,v_n \}$. For $\mathbb{Q}\Lambda$, just take the $\mathbb{Q}$-span. But what about the quotient? Its really not that different from $\mathbb{Q}^n / \mathbb{Z}^n$. Consider that $\Lambda$ is like a "tilted" version of $\mathbb{Z}^n$. From there its not hard to imagine $\mathbb{Q}\Lambda / \Lambda$ as being a "tilted" version of the "hyper-rational-torus" $\mathbb{Q}^n / \mathbb{Z}^n$ (I call it "hyper" because it may be more than 2 dimensions, and "rational" because it only includes the points that get back to the origin when multiplied by an integer.) In fact, its not so hard to see that there's an isomorphism between $\mathbb{Q}^n / \mathbb{Z}^n$ and $\mathbb{Q}\Lambda / \Lambda$ that comes just from figuring out how much $\Lambda$ is titled from $\mathbb{Z}^n$, id est, expressing the basis for $\Lambda$ in terms of the standard basis of $\mathbb{Z}^n$.

So why then does the the definition of $\mathbb{Q}$-lattices include a homomorphism and not a full, invertible isomorphism? In fact, why do we even need a definition for $\mathbb{Q}$-lattices at all? It just seems to be a lattice with an obvious homomorphism attached to it. (this question is really quite moot. commensurability and the commensurability classes yield plenty results, among them is the topic of my 4th year project!)

Finally, we also have a notion of what it means for 2 $\mathbb{Q}$-lattices to be commensurable: $\mathbb{Q} \Lambda_1 = \mathbb{Q} \Lambda_2$ and $\phi_1 - \phi_2\, =\, 0\, mod\, \Lambda_1 + \Lambda_2$. What does this mean? As best I can tell [I'm really not too sure], the homomorphism labels some of our tilted-torus points as coming from our nice hyper-rational-torus. The statement that $\phi_1 - \phi_2\, =\, 0\, mod\, \Lambda_1 + \Lambda_2$ says that they have the same points labelled in the tilted-torus. It's pretty easy to see that if the homomorphisms are isomorphism, then 2 $\mathbb{Q}-lattices are only commensurable if they are in fact, equal. Its also not too hard to see that commensurability is an equivalence relation (Connes has a proof of this fact in Non-commutative geometry, Quantum Fields and motives, it was from that prove that I gathered the "same label" idea.) We denote all the commensurability classes of $\mathbb{Q}$-lattices in $\mathbb{R}^n$ as $\mathcal{L}_n$.

So whats next? Apparently, the Bost-Connes quantum statistical mechanical system a C*-algebra somehow generated from the quotient $\mathcal{L}_n / \mathbb{R}^{*}_{+}$ (I don't recognize the notation $\mathbb{R}^{*}_{+}$, $\mathbb{R}^{*}$ I thought was the real numbers excluding 0, and $\mathbb{R}_{+}$ the positive reals. so is $\mathbb{R}^{*}_{+}$ all the strictly positive reals?) So I suppose my next step is to 1) understand how to form that C*-algebra [which implicitly involves understanding what a C*-algebra is, I think I'll probably spend some time reading the first chapter to Averson's Invitation to C*-algebras] and 2) understand how that C*-algebra is a quantum statistical mechanical system. After that point, I can figure out how this is the same as the Hecke algebra mentioned in Connes' and Bost's work and what it has to do with $Gal(\mathbb{Q}^{cycl} / \mathbb{Q})$.

## 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?

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?

## Monday, July 26, 2010

### non-commutative geometry and the bost connes system...getting started

So I'm working through Matilde Marcolli's monograph on Arithmetic Noncommutative Geometry, particularly Chapter 3 "Quantum Statistical Mechanics and Galois Theory" (That's also the title of my 4th year project with Professor Kim at UCL.) Due to ongoing personal and financial problems, I don't have nearly as much time to be doing maths as I should like, but I was able to do some reading today, and I thought I should write something done to denote what I think I might understand [not much!] and what I definitely do not understand [most of it!]

I was able to take a first glance at the short introduction chapter as well as the meat-and-potatoes in Chapter 3. Here's what I know about non-commutative geometry so far:

Non-commutative geometry tries to extend the tools and methods of ordinary geometry [I'm not sure what "ordinary geometry" is, to be honest] to work on nasty quotient spaces where the "Ring of functions" [I believe that's related to Morse theory?] is too small in some sense. Particularly, if X is a space and ~ an equivalence relation on it, then the "Ring of functions" for the quotient X/~ is defined as a subset of the ring of functions of X where each function is invariant on the induced partitions. From what I know about quotient spaces, it seems very unlikely that the quotient ring of functions is anything but a collection of boring constance functions. But apparently, and I don't understand how at all, we can use a non-commutative algebra of coordinates (much like using non-commuting variables in relativistic quantum mechanics) to get an interesting ring of functions, and this is somehow related to an extension of the Gel'fand-Naimark theorem (Marcolli mentions this as "X is a locally compact Hausdorff space <=> C_0(X) is an abelian C*-algebra, but looking the names up in Arveson's "An invitation to C*-algebras" says that the theorem says "Every abstract C* algebra with identity is isometrically *-isomorphic to a C*-algebra of operators", I think I have an idea what that theorem means - its a bit like saying any linear operator on a finite vector space can be expressed as a matrix...I think). I don't know how the two are related, if at all.

I'm interested in non-commutative geometry because it relates spontaneous symmetry breaking in a quantum mechanical system to $Gal(\mathbb{Q}^{cycl} / \mathbb{Q})$. So lets talk about what I know of this relationship:

Connes, et al, constructed quantum statistical mechanical systems that recover the class field theory of $\mathbb{Q}$ and imaginary fields, and possibly even quadratic fields (I don't know what class field theory is, but its apparently related to $Gal(\mathbb{Q}^{cycl} / \mathbb{Q})$ as best I can tell.) A quantum statistical mechanical system is a C* dynamical system, which is a C* algebra A together with a family of automorphism $\sigma_t$ of A that can be described with a single parameter: t for time. The construction of system has something to do with $\mathbb{Q}$-lattices, which is a lattice $\Lambda$ of $\mathbb{R}^n$ together with a homomorphism $\phi : \mathbb{Q}^n / \mathbb{Z}^n \to \mathbb{Q} \Lambda / \Lambda$ (I have no idea what that means.) Two $\mathbb{Q}$-lattices are commensurable if their torsion labels interact in some way I don't understand. Commesurability is an equivalence relation, and the C* algebra on the set of commensurability classes of $\mathbb{Q}$-lattices in $\mathbb{R}$ is the Bost-Connes system.

I think my goal for next couple of weeks is to understand that construction.

I was able to take a first glance at the short introduction chapter as well as the meat-and-potatoes in Chapter 3. Here's what I know about non-commutative geometry so far:

Non-commutative geometry tries to extend the tools and methods of ordinary geometry [I'm not sure what "ordinary geometry" is, to be honest] to work on nasty quotient spaces where the "Ring of functions" [I believe that's related to Morse theory?] is too small in some sense. Particularly, if X is a space and ~ an equivalence relation on it, then the "Ring of functions" for the quotient X/~ is defined as a subset of the ring of functions of X where each function is invariant on the induced partitions. From what I know about quotient spaces, it seems very unlikely that the quotient ring of functions is anything but a collection of boring constance functions. But apparently, and I don't understand how at all, we can use a non-commutative algebra of coordinates (much like using non-commuting variables in relativistic quantum mechanics) to get an interesting ring of functions, and this is somehow related to an extension of the Gel'fand-Naimark theorem (Marcolli mentions this as "X is a locally compact Hausdorff space <=> C_0(X) is an abelian C*-algebra, but looking the names up in Arveson's "An invitation to C*-algebras" says that the theorem says "Every abstract C* algebra with identity is isometrically *-isomorphic to a C*-algebra of operators", I think I have an idea what that theorem means - its a bit like saying any linear operator on a finite vector space can be expressed as a matrix...I think). I don't know how the two are related, if at all.

I'm interested in non-commutative geometry because it relates spontaneous symmetry breaking in a quantum mechanical system to $Gal(\mathbb{Q}^{cycl} / \mathbb{Q})$. So lets talk about what I know of this relationship:

Connes, et al, constructed quantum statistical mechanical systems that recover the class field theory of $\mathbb{Q}$ and imaginary fields, and possibly even quadratic fields (I don't know what class field theory is, but its apparently related to $Gal(\mathbb{Q}^{cycl} / \mathbb{Q})$ as best I can tell.) A quantum statistical mechanical system is a C* dynamical system, which is a C* algebra A together with a family of automorphism $\sigma_t$ of A that can be described with a single parameter: t for time. The construction of system has something to do with $\mathbb{Q}$-lattices, which is a lattice $\Lambda$ of $\mathbb{R}^n$ together with a homomorphism $\phi : \mathbb{Q}^n / \mathbb{Z}^n \to \mathbb{Q} \Lambda / \Lambda$ (I have no idea what that means.) Two $\mathbb{Q}$-lattices are commensurable if their torsion labels interact in some way I don't understand. Commesurability is an equivalence relation, and the C* algebra on the set of commensurability classes of $\mathbb{Q}$-lattices in $\mathbb{R}$ is the Bost-Connes system.

I think my goal for next couple of weeks is to understand that construction.

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