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Random Walks on Finite Quantum Groups

January 8, 2012 in Random Walks on Finite Quantum Groups, Research | Leave a comment

Taken from Random Walks on Finite Quantum Groups by Franz & Gohm.

The most important special case of the construction from here is obtained when we choose B=A and \beta=\Delta. Then we have a random walk on a finite group A. Let us first show that this indeed a generalisation of a left-invariant random walk (I must be careful to remember this — I have always worked with right-invariant walks that multiply on the left: these multiply on the right.) Using the coassociativity of \Delta we see that the transition operator T_\phi=(I_A\otimes\phi)\circ\Delta satisfies the formula (\checkmark)

\Delta\circ T_\phi=(I_A\otimes T_\phi)\circ\Delta.

Suppose now that B=A consists of functions on a finite group G and \beta=\Delta is the comultiplication which encodes the group multiplication, i.e.

\displaystyle\Delta\left(\mathbf{1}_{\{g\}}\right)=\sum_{h\in G}\mathbf{1}_{\{gh^{-1}\}}\otimes\mathbf{1}_{\{h\}}=\sum_{h\in G}\mathbf{1}_{\{h^{-1}\}}\otimes\mathbf{1}_{\{hg\}}.

We also have

\displaystyle T_\phi\left(\mathbf{1}_{\{g\}}\right)=\sum_{h\in G}p(h,g)\mathbf{1}_{\{h\}},

where [p(h,g)] is the stochastic matrix.

This calculation makes perfect sense when p(h,g)=\phi\left(\mathbf{1}_{h^{-1}g}\right) and since we’ve said nothing about what \phi should be this makes perfect sense — and with linearity we get all the properties of the stochastic operator we could possibly want. 

Now calculate

\displaystyle (\Delta\circ T_\phi)\mathbf{1}_{\{g\}}=\Delta\left(\sum_{h\in G}p(h,g)\mathbf{1}_{\{h\}}\right)=\sum_{t\in G}\mathbf{1}_{\{t^{-1}\}}\otimes\sum_{h\in G}p(h,g)\mathbf{1}_{\{th\}},

\displaystyle[(I_{F(G)}\otimes T_\phi)\circ\Delta]\mathbf{1}_{\{g\}}=(I_{F(G)}\otimes T_\phi)\sum_{t\in G}\mathbf{1}_{\{t^{-1}\}}\otimes\mathbf{1}_{\{tg\}}

\displaystyle =\sum_{t\in G}\mathbf{1}_{\{t^{-1}\}}\otimes \sum_{s\in G} p(s,tg)\mathbf{1}_{\{s\}}.

Now reindex the second sum here s\rightarrow h and at the same time map s\rightarrow th and we may then conclude that p(h,g)=p(th,tg) for all g,\,h\,t\in G. This is the left invariance of the random walk.

For random walks on a finite quantum groups there are some natural special choices for the initial distribution \psi. On the one hand, one may choose \psi=\varepsilon (the counit) which in the commutative case (i.e. for a group) corresponds to starting at the identity. Then the time evolution of the distributions is given by \varepsilon\star\phi^{\star n}=\star^{\star n}. In other words, we get a convolution semigroup of states.

Inasmuch as I can tell, we have

\psi\star \phi=(\psi\otimes\phi)\circ\Delta.

On the other hand, stationarity of the random walk can be obtained if \psi is chosen such that \psi\star\phi=\star. In particular, we may choose the unique Haar state \eta of the finite quantum group A.

Proposition 4.1

The random walks on a finite quantum group are stationary for all choices of \phi if and only if \psi=\eta.

Proof : This follows from Proposition 3.2 together with the fact that the Haar state is characterised by its right invariance.

Conditional Expectation in Quantum Probability

January 8, 2012 in Probability Theory, Research | Leave a comment

Taken from Condition Expectation in Quantum Probabilty by Denes Petz.

In quantum probability there are a number of fundamental questions that ask how faithfully can one quantise classical probability. Suppose that (\Omega,\mathcal{S},P) is a (classical) probability space and \mathcal{G}\subset\mathcal{A} a sub-\sigma-algebra. The conditional expectation of some integrable function f (with respect to some L-space) relative to \mathcal{G} is the orthogonal projection onto the closed subspace L(\mathcal{G}):

\mathbb{E}^{\mathcal{G}}:L(\mathcal{A})\rightarrow L(\mathcal{G}), f\mapsto \mathbb{E}(f|\mathcal{G}).

Suppose now that (A,\rho) is a quantum probability space and that B is some C*-subalgebra of A. Can we always define a conditional expectation with respect to B? The answer turns out to be not always, although this paper gives sufficient conditions for the existence of such a projection. Briefly, things work the other way around. Distinguished states give rise to quantum conditional expectations — and these conditional expectations define a subalgebra. We can’t necessarily start with a subalgebra and find the state which gives rise to it — Theorem 2 gives necessary conditions in which this approach does work.

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Infinite Products of Probability Spaces

January 8, 2012 in Probability Theory, Research | 1 comment

Taken from Real Analysis and Probability by R.M. Dudley.

For a sequence of n repeated, independent trials of an experiment, some probability distributions and variables converge as n tends to infinity. In proving such limit theorems, it is useful to be able to construct a probability space on which a sequence of independent random variables is defined in a natural way; specifically, as coordinates for a countable Cartesian product.

The Cartesian product of finitely many \sigma-finite measure spaces gives a \sigma-finite measure space. For example, Cartesian products of Lesbesgue measure on the line give Lesbesgue measure on finite-dimensional Euclidean spaces. But suppose we take a measure space \{0,1\} with two points each having measure 1, \mu(\{0\})=1=\mu(\{1\}), and form a countable Cartesian product of copies of this space, so that the measure of any countable product of sets equals the product of their measures. Then we would get an uncountable space in which all singletons have measure 1, giving the measure usually called counting measure. An uncountable set with counting measure is not a \sigma-finite space, although in this example it was a countable product of finite measure spaces. By contrast, the the countable product of probability measures will again be a probability space. Here are some definitions.

For each n=1,2,\dots let (\Omega_n,S_n,P_n) be a probability space. Let \Omega be the Cartesian product \displaystyle \prod_{n\geq 1}\Omega_n, that is, the set of all sequences \{\omega_n\}_{n\geq 1} with \omega_n\in\Omega_n for all n. Let \pi_n be the natural projection of \Omega onto \Omega_n for each n: \pi_n\left(\{\omega_m\}_{m\geq 1}\right)=\omega_n for all n. Let S be the smallest \sigma-algebra of subsets of \Omega such that for all m, \pi_m is measurable from (\Omega,S) to (\Omega_m,S_m). In other words, S is the smallest \sigma-algebra containing all sets \pi_n^{-1}(A) for all n and all A\in S_n.

Let \mathcal{R} be the collection of all sets \displaystyle \prod_{n\geq 1}A_n\subset\Omega where A_n\in \mathcal{S}_n for all n and A_m=\Omega_m except for at most finitely many values of n. Elements of \mathcal{R} will be called rectangles. Now recall the notion of semiring. \mathcal{R} has this property.

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Direct Limits and Tensor Products: Direct Limits of C*-algebras

January 7, 2012 in C*-Algebras & Operator Theory, Research | Leave a comment

Taken from C*-Algebras and Operator Theory by Gerald J. Murphy.

Although the principal aim of this section is to construct direct limits of C*-algebras, we begin with direct limits of groups.

If \{G_n\}_{n=1}^\infty is a sequence of groups, and if for each n we have a homomorphism \varphi_n:G_n\rightarrow G_{n+1}, then we call \{G_n\}_{n\geq1} a direct sequence of groups. Given such a sequence and positive integers n\leq m, we set \varphi_{nn}=I_{G_n} and we define \varphi_{nm}:G_n\rightarrow G_m inductively on m by setting

\varphi_{n,m+1}=\varphi_m\varphi_{nm}.

If n\leq m\leq k, we have \varphi_{nk}=\varphi_{mk}\varphi_{nm}.

If G' is a group and we have homomorphisms \theta^n:G^n\rightarrow G' such that the diagram

commutes for each n, that is \theta^n=\theta^{n+1}\varphi_n, then \theta^n=\theta^m\varphi_{nm} for all m\geq n.

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PhD Progression Report

December 2, 2011 in Research, Research Updates | Leave a comment

The following runs a thread through what I’ve looked at over the past year: Progression Report.

Hopf Algebras: Algebras

October 25, 2011 in Quantum Groups, Research | 1 comment

Taken from Hopf Algebras by Abe. I am not doing this over a commutative ring as he is doing but just over the field of complex numbers, \mathbb{C} — therefore some of my constructions will be simplified. Some of them might even be wrong.

In this section the notion of algebras will be introduced; and we will present some examples of such algebras. An algebra over \mathbb{C} is defined by giving a structure map.

Algebras

Suppose A=\{e_\lambda,0\}_{\lambda\in\Lambda} is a ring and we are given a ring morphism \eta_A:\mathbb{C}\rightarrow A. Then A can be viewed as a complex vector space. If we have

(ax)y=x(ay)=a(xy) for all a\in\mathbb{C} and x,\,y\in A

then A is said to be an algebra. Now the map defined by f(x,y)=xy turns out to be bilinear. Thus we obtain a morphism

\nabla_A:A\otimes A\rightarrow A.

From this fact, we see that an algebra A can be defined in the following manner. For a complex vector space A and morphisms \eta_A:\mathbb{C}\rightarrow A, \nabla_A:A\otimes A\rightarrow A, we have the following:

\nabla_A\circ(\nabla_A\otimes I_A)=\nabla_A\circ(I_A\otimes\nabla_A)

(the associative law)

\nabla_A\circ(\eta_A\otimes I_A)=\nabla_A\circ (I_A\otimes \eta_A)=I_A

(the unitary property)

Here, \nabla_A is said to be the multiplicative map of A, \eta_A the unit map of A, and together we call \nabla_A, \eta_A the structure maps of the algebra A.

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Hopf Algebras: Bialgebras and Hopf Algebras

October 14, 2011 in Quantum Groups, Research | Leave a comment

Taken from Hopf Algebras by Abe. This is not even nearly finished however I pressed publish instead of save draft… oh well. 

In this section, we give the definition of Hopf algebras and present some examples. We begin by defining coalgebras, which are in a dual relationship with algebras., then bialgebras and Hopf algebras as algebraic systems in which the structures of algebras and coalgebras are interrelated by certain laws.

1.1 Coalgebras

We define a coalgebra dually to an algebra. Given an algebra A and algebra homomorphisms  \Delta:A\rightarrow A\otimes A and \varepsilon:A\rightarrow\mathbb{C}, we call (A,\Delta,\varepsilon) or just A a coalgebra when we have:

(\Delta\otimes I_A)\circ \Delta=(I_A\otimes\Delta)\circ\Delta,

(the coassociative law).

(\varepsilon\otimes I_A)\circ\Delta=(I_A\otimes\varepsilon)\circ\Delta,

(the counitary property)

The maps \Delta and \varepsilon are called the comultiplication map and the counit map of A, and together they are said to be the structure maps of the coalgebra A.

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Representations of C*-Algebras: Irreducible Representations and Pure States

October 5, 2011 in C*-Algebras & Operator Theory, Research | 1 comment

Taken from C*-Algebras and Operator Theory by Gerald Murphy.

This section is concerned with positive linear functionals and representations. Pure states are introduced and shown to be the extreme points of a certain convex set, and their existence is deduced from the Krein-Milman theorem. From this the existence of  irreducible representations is proved by establishing a correspondence between them and the pure states.

If (H,\varphi) is a representation of a C*-algebra A, we say x\in H is a cyclic vector for (H,\varphi) if x is cyclic for the C*-algebra \varphi(A) (This means that cyclic vector is a vector x\in H such that the closure of the linear span of \{\varphi(a)x\,:\,a\in A\} equals H). If (H,\varphi) admits a cyclic vector, then we say that it is a cyclic representation.

We now return to the GNS construction associated to a state to show that the representations involved are cyclic.

Theorem 5.1.1

Let A be a C*-algebra and \rho\in S(A). Then there is a unique vector x_\rho\in H_\rho\in H such that

\rho(a)=\langle a+N_\rho,x_\rho\rangle, for a\in A.

Moreover, x_\rho is a unit cyclic vector for (H_\rho,\varphi_\rho) and

\varphi_\rho(a)x_\rho=a+N_\rho, for a\in A.

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An alternative quantisation of a Markov chain? or Why do we need Coalgebras?

September 21, 2011 in Random Walks on Finite Quantum Groups, Research | Leave a comment

Quantisation

I had been of the understanding that a quantisation looks as follows. There is some process or property P(X) of a space X which we want to examine. Depending on the type of space X, from a suitable algebra of complex functions on X, F(X), we can recover and examine many of the properties of P(X) by instead looking at F(X): we essentially have the identification X\leftrightarrow F(X). When the process/ property P(X) is about a space then it is said to be classical or commutative because for any x\in X and f,\,g\in F(X) we have that fg=gf because fg(x)=f(x)g(x)=gf(x) as the f(x),\,g(x) lie in the commutative algebra \mathbb{C}.

Now from X we know about F(X) and vice versa. Now a suitably chosen F(X) is just a commutative C*-algebra so what about a non-commutative C*-algebra F — can we examine it’s “underlying space” in the same way?

So essentially, this means that I thought you quantised objects, such as Markov chains, by replacing a commutative C*-algebra F(X) with a not-necessarily commutative one. (This roughly follows my interpretation as per this)

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Random Walks on Finite Quantum Groups: Random Walks on Comodule Algebras

September 15, 2011 in Random Walks on Finite Quantum Groups, Research | 1 comment

Taken from Franz & Gohm.

Let return to the map b:M\times G\rightarrow M considered in the beginning of the previous section. If G is a group, then b:M\times G\rightarrow M is called a left action of G on M, if it satisfies the following axioms expressing associativity and unit (\checkmark0),

b(b(x,g),h)=b(x,hg), and b(x,e)=x

for all x\in M, g,h\in G; and where e\in G is the identity. As before we have the unital *-homomorphisms \alpha_g:F(M)\rightarrow F(M). Actually, in order to get a representation of G on F(M), i.e. \alpha_g\alpha_h=\alpha_{gh} for all g,\,h\in G we modify  the definition and use \alpha_g(f)(x):=f(b(x,g^{-1})). (Otherwise we get an anti-representation. But this is a minor point at this stage). In the associated coaction \beta:F(M)\rightarrow F(M)\otimes F(G) the axioms above are turned into the coassociativity and counit properties. These make perfect sense not only for groups but also for quantum groups and we state them at once in this more general setting. We are rewarded with a particular interesting class of quantum Markov chains associated to quantum groups which we call random walks and are the subject of this lecture.

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