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Symmetries of Quantum Spaces – Podlés

June 9, 2019 in Quantum Groups, Research | Leave a comment

Just some notes on section 1 of this paper. Flags and notes are added but mistakes are mine alone.

Definition

Let C(G) be the algebra of continuous functions on a compact matrix quantum group. Such an object is given by a matrix u=\{u_{ij}\}_{i,j=1}^N which generates C(G) as a C*-algebra. Furthermore, there exists a C*-algebra homomorphism \Delta:C(G)\rightarrow C(G)\otimes C(G) such that

\displaystyle \Delta(u_{ij})=\sum_{k=1}^N u_{ik}\otimes u_{kj},

and both u and u^T are invertible in M_N(C(G)).

Any subgroup G\subset \text{GL}(N,\mathbb{C}) is such an object, with the u_{ij}\in C(G) given by u_{ij}(g)=g_{ij}\in\mathbb{C}. Furthermore

\mathrm{C}_{\text{comm}}\langle u_{ij}\rangle \cong C(G).

We say that \rho=(\rho_{ij})_{i,j=1}^{d_\rho}\in M_{d_{\rho}}(C(G)) is a representation if it is invertible and

\displaystyle \Delta(\rho_{ij})=\sum_{k=1}^{d_\rho}\rho_{ik}\otimes\rho{kj}.

The transpose \rho^T=(\rho_{ji})_{i,j=1}^N\in M_{d_{\rho}}(C(G)) is also invertible and so we have:

Proposition

The C*algebra generated by the \rho_{ij} is also the algebra of continuous functions on a compact matrix quantum group.

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Periodicity & Aperiodicity for Random Walks on Quantum Groups

June 7, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research | Leave a comment

Background

I am trying to prove an Ergodic Theorem for Random Walks on Finite Quantum Groups. A random walk on a quantum group driven by \nu\in M_p(G) is ergodic if the convolution powers (\nu^{\star k})_{k\geq 0} converge to the Haar state \int_G.

The classical theorem for finite groups:

Ergodic Theorem for Random Walks on Finite Groups

A random walk on a finite group G driven by a probability \nu\in M_p(G) is ergodic if and only if \nu is not concentrated on a proper subgroup nor the coset of a proper normal subgroup.

Not concentrated on a proper subgroup gives irreducibility. A random walk is irreducible if for all g\in G, there exists k\in\mathbb{N} such that \nu^{\star k}(\{g\})>0.

Not concentrated on the coset of a proper normal subgroup gives aperiodicity. Something which should be equivalent to aperiodicity is if

p:=\gcd\{k>0:\nu^{\star k}(e)>0\}

is equal to one (perhaps via invariance \mathbb{P}[\xi_{i+1}=t|\xi_{i}=s]=\mathbb{P}[\xi_{i+1}=th|\xi_{i}=sh]).

If \nu is concentrated on the coset a proper normal subgroup N\rhd G, specifically on Ng\neq Ne, then we have periodicity (Ng\rightarrow Ng^2\rightarrow \cdots\rightarrow Ng^{o(g)}=Ne\rightarrow Ng\rightarrow \cdots), and p=o(g), the order of g.

In Markov chain theory, ergodicity is equivalent to irreduciblity and aperiodicity.

The theorem in the quantum case should look like:

Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group G driven by a state \nu\in M_p(G) is ergodic if and only if “X”.

Irreducibility

At the moment I have some part of X;the irreducibility bit. As is well known since Pal (1996), it is possible to have a probability not concentrated on a quantum subgroup be reducible. This led Franz & Skalski to generalise quantum subgroups to group-like-projections, which I will say correspond to quasi-subgroups following Kasprzak & Sołtan.

I have shown that if \nu is concentrated on a proper quasi-subgroup S, in the sense that \nu(P_S)=1 for a group-like-projection P_S, that so are the \nu^{\star k}. The analogue of irreducible is that for all q projections in F(G), there exists k\in\mathbb{N} such that \nu^{\star k}(q)>0. If \nu is concentrated on a quasi-subgroup S, then for all k, \nu^{\star k}(Q_S)=0, where Q_S=\mathbf{1}_G -P_S.

I have also shown on the other hand that if the random walk is reducible that it must be concentrated on a proper quasi-subgroup. Franz & Skalski show that group-like-projections also have a correspondence with idempotent states. The Césaro Means

\displaystyle \nu_n:=\frac{1}{n}\sum_{k=1}^n\nu^{\star k},

converge to an idempotent state \nu_\infty. If \nu^{\star k}(q)=0 for all k then the \nu_{\infty}(q)=0 also, so that \nu_\infty\neq \int_G (as the Haar state is faithful). I was able to prove that \nu is supported on the quasi-subgroup given by the idempotent \nu_\infty.

I believe this result — irreducible if and only if not concentrated on a quasi-subgroup — holds more generally than just in finite quantum groups.

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Some Unresolved and Unexplored Aspects of Random Walks on Quantum Groups

May 14, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research | 1 comment

Slides of a talk given to the Functional Analysis seminar in Besancon.

Some of these problems have since been solved.

“e in support” implies convergence

Consider a \nu\in M_p(G) on a finite quantum group such that where

M_p(G)\subset \mathbb{C}\varepsilon \oplus (\ker \varepsilon)^*,

\nu=\nu(e)\varepsilon+\psi with \nu(e)>0. This has a positive density of trace one (with respect to the Haar state \int_G\in M_p(G)), say

\displaystyle a_\nu=\nu(e)\eta+b_\psi\in \mathbb{C}\eta\oplus \ker \varepsilon,

where \eta is the Haar element. 

An element in a direct sum is positive if and only if both elements are positive. The Haar element is positive and so b_\psi\geq 0. Assume that b_\psi\neq 0 (if b_\psi=0, then \psi=0\Rightarrow \nu=\varepsilon\Rightarrow \nu^{\star k}=\varepsilon for all k and we have trivial convergence)

Therefore let

\displaystyle a_{\tilde{\psi}}:=\frac{b_\psi}{\int_G b_\psi}

be the density of \tilde{\psi}\in M_p(G).

Now we can explicitly write

\displaystyle \nu=\nu(e)\varepsilon+(1-\nu(e))\tilde{\psi}.

This has stochastic operator

P_\nu=\nu(e)I_{F(G)}+(1-\nu(e))P_{\tilde{\psi}}.

Let \lambda be an eigenvalue of P_\nu of eigenvector a. This yields

\nu(e)a+(1-\nu(e))P_{\tilde{\psi}}(a)=\lambda a

and thus

\displaystyle P_{\tilde{\psi}}a=\frac{\lambda-\nu(e)}{1-\nu(e)}a.

Therefore, as a is also an eigenvector for P_{\tilde{\psi}}, and P_{\tilde{\psi}} is a stochastic operator (if a is an eigenvector of eigenvalue |\lambda|>1, then \|P_\nu a\|_1=|\lambda|\|a\|_1\leq \|a\|_1, contradiction), we have

\displaystyle \left|\frac{\lambda-\nu(e)}{1-\nu(e)}\right|\leq 1

\Rightarrow |\lambda-\nu(e)|\leq 1-\nu(e).

This means that the eigenvalues of P_\nu lie in the ball B_{1-\nu(e)}(\nu(e)) and thus the only eigenvalue of magnitude one is \lambda=1, which has (left)-eigenvector the stationary distribution of P_\nu, say \nu_\infty.

If \nu is symmetric/reversible in the sense that \nu=\nu\circ S, then P_\nu is self-adjoint and has a basis of (left)-eigenvectors \{\nu_\infty=:u_1,u_2,\dots,u_{|G|}\}\subset \mathbb{C}G and we have, if we write \nu=\sum_{t=1}^{|G|}a_tu_t,

\displaystyle \nu^{\star k}=\sum_{t=1}^{|G|}a_t\lambda_t^ku_t,

which converges to a_1\nu_\infty (so that a_1=1).

If \nu is not reversible, it is a standard argument to show that when put in Jordan normal form, that the powers P_{\nu}^k converge and thus so do the \nu^{\star k} \bullet

Total Variation Decrasing

Uses Simeng Wang’s \|a\star_Ab\|_1\leq \|a\|_1\|b\|_1. Result holds for compact Kac if the state has a density.

Periodic e^2 is concentrated on a coset of a proper normal subgroup of \mathfrak{G}_0

e_2+e_4 is a minimal projection (coset) in the quotient space of the normal subgroup (to be double checked) given by \langle e_1,e_3\rangle

Supported on Subgroup implies Reducible

I have a proof that reducible is equivalent to supported on a pre-subgroup.

Publication

February 25, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | Leave a comment

Diaconis–Shahshahani Upper Bound Lemma for Finite Quantum Groups, Journal of Fourier Analysis and Applications, doi: 10.1007/s00041-019-09670-4 (earlier preprint available here)

Abstract

A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses Fourier analysis on the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The Fourier analysis of quantum groups is remarkably similar to that of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for finite quantum groups. The Upper Bound Lemma is used to study the convergence of ergodic random walks on the dual group \widehat{S_n} as well as on the truly quantum groups of Sekine.

A Condition for Convergence of Random Walks on Quantum Groups

November 13, 2018 in Probability Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research | Leave a comment

In a recent preprint, Haonan Zhang shows that if \nu\in M_p(Y_n) (where Y_n is a Sekine Finite Quantum Group), then the convolution powers, \nu^{\star k}, converges if

\nu(e_{(0,0)})>0.

The algebra of functions F(Y_n) is a multimatrix algebra:

F(Y_n)=\left(\bigoplus_{i,j\in\mathbb{Z}_n}\mathbb{C}e_{(i,j)}\right)\oplus M_n(\mathbb{C}).

As it happens, where a=\sum_{i,j\in\mathbb{Z}_n}x_{(i,j)}e_{(i,j)}\oplus A, the counit on F(Y_n) is given by \varepsilon(a)=x_{(0,0)}, that is \varepsilon=e^{(0,0)}, dual to e_{(0,0)}.

To help with intuition, making the incorrect assumption that Y_n is a classical group (so that F(Y_n) is commutative — it’s not), because \varepsilon=e^{(0,0)}, the statement \nu(e_{(0,0)})>0, implies that for a real coefficient x^{(0,0)}>0,

\nu=x^{(0,0)}\varepsilon+\cdots= x^{(0,0)}\delta^e+\cdots,

as for classical groups \varepsilon=\delta^e.

That is the condition \nu(e_{(0,0)})>0 is a quantum analogue of e\in\text{supp}(\nu).

Consider a random walk on a classical (the algebra of functions on G is commutative) finite group G driven by a \nu\in M_p(G).

The following is a very non-algebra-of-functions-y proof that e\in \text{supp}(\nu) implies that the convolution powers of \nu converge.

Proof: Let H be the smallest subgroup of G on which \nu is supported:

\displaystyle H=\bigcap_{\underset{\nu(S_i)=1}{S_i\subset G}}S_i.

We claim that the random walk on H driven by \nu is ergordic (see Theorem 1.3.2).

The driving probability \nu\in M_p(G) is not supported on any proper subgroup of H, by the definition of H.

If \nu is supported on a coset of proper normal subgroup N, say Nx, then because e\in \text{supp}(\nu), this coset must be Ne\cong N, but this also contradicts the definition of H.

Therefore, \nu^{\star k} converges to the uniform distribution on H \bullet

Apart from the big reason — that this proof talks about points galore — this kind of proof is not available in the quantum case because there exist \nu\in M_p(G) that converge, but not to the Haar state on any quantum subgroup. A quick look at the paper of Zhang shows that some such states have the quantum analogue of e\in\text{supp}(\nu).

So we have some questions:

  • Is there a proof of the classical result (above) in the language of the algebra of functions on G, that necessarily bypasses talk of points and of subgroups?
  • And can this proof be adapted to the quantum case?
  • Is the claim perhaps true for all finite quantum groups but not all compact quantum groups?

Probabilities concentrated on Quantum Subgroups

November 6, 2018 in Probability Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research | Leave a comment

Quantum Subgroups

Let C(G) be a the algebra of functions on a finite or perhaps compact quantum group (with comultiplication \Delta) and \nu\in M_p(G) a state on C(G). We say that a quantum group H with algebra of function C(H) (with comultiplication \Delta_H) is a quantum subgroup of G if there exists a surjective unital *-homomorphism \pi:C(G)\rightarrow C(H) such that:

\displaystyle \Delta_H\circ \pi=(\pi\otimes \pi)\circ \Delta.

The Classical Case

In the classical case, where the algebras of functions on G and H are commutative,

\displaystyle \pi(\delta_g)=\left\{\begin{array}{cc}\delta_g & \text{ if }g\in H \\ 0 & \text{ otherwise}\end{array}\right..

There is a natural embedding, in the classical case, if H is open (always true for G finite) (thanks UwF) of \imath: C(H) \xrightarrow\, C(G),

\displaystyle \sum_{h\in H}a_h \delta_h \mapsto \sum_{g\in G} a_g \delta_g,

with a_g=a_h for h\in G, and a_g=0 otherwise.

Furthermore, \pi is has the property that

\pi\circ\imath\circ \pi=\pi,

which resembles \pi^2=\pi.

In the case where \nu is a probability on a classical group G, supported on a subgroup H, it is very easy to see that convolutions \nu^{\star k} remain supported on H. Indeed, \nu^{\star k} is the distribution of the random variable

\xi_k=\zeta_k\cdots \zeta_2\cdot \zeta_1,

where the i.i.d. \zeta_i\sim \nu. Clearly \xi_k\in H and so \nu^{\star k} is supported on H.

We can also prove this using the language of the commutative algebra of functions on G, C(G). The state \nu\in M_p(G) being supported on H implies that

\nu\circ\imath\circ \pi=\nu\imath\pi=\nu.

Consider now two probabilities on G but supported on H, say \mu,\,\nu. As they are supported on H we have

\mu=\mu\imath\pi and \nu=\nu\imath\pi.

Consider

(\mu\star \nu)\imath\pi=(\mu\otimes \nu)\circ \Delta\circ \imath\pi

=((\mu\imath\pi)\otimes(\nu\imath\pi))\circ \Delta\circ\imath\pi =(\mu\imath\otimes \nu\imath)(\pi\circ \pi)\Delta\circ\imath\pi

=(\mu\imath\otimes\nu\imath)(\Delta_H\circ \pi\circ \imath\circ \pi)=(\mu\imath\otimes\nu\imath)(\Delta_H\circ \pi)

=(\mu\imath\otimes \nu\imath)\circ (\pi\circ \pi)\circ\Delta=(\mu\imath\pi\otimes \nu\imath\pi)\circ\Delta

=(\mu\otimes\nu)\circ\Delta=\mu\star \nu,

that is \mu\star \nu is also supported on H and inductively \nu^{\star k}.

Some Questions

Back to quantum groups with non-commutative algebras of functions.

  • Can we embed C(H) in C(G) with a map \imath and do we have \pi\circ \imath\circ \pi=\pi, giving the projection-like quality to \pi?
  • Is \nu\circ\imath\circ \pi=\nu a suitable definition for \nu being supported on the subgroup H.

If this is the case, the above proof carries through to the quantum case.

  • If there is no such embedding, what is the appropriate definition of a \nu\in M_p(G) being supported on a quantum subgroup H?
  • If \pi does not have the property of \pi\circ \imath\circ \pi=\pi, in this or another definition, is it still true that \nu being supported on H implies that \nu^{\star k} is too?

Edit

UwF has recommended that I look at this paper to improve my understanding of the concepts involved.

General Mathematical Audience Talk: The Diaconis-Shahshahani Upper Bound Lemma for Finite Quantum Groups

September 1, 2018 in Probability Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | Leave a comment

Slides of a talk given at the Irish Mathematical Society 2018 Meeting at University College Dublin, August 2018.

Abstract Four generalisations are used to illustrate the topic. The generalisation from finite “classical” groups to finite quantum groups is motivated using the language of functors (“classical” in this context meaning that the algebra of functions on the group is commutative). The generalisation from random walks on finite “classical” groups to random walks on finite quantum groups is given, as is the generalisation of total variation distance to the quantum case. Finally, a central tool in the study of random walks on finite “classical” groups is the Upper Bound Lemma of Diaconis & Shahshahani, and a generalisation of this machinery is used to find convergence rates of random walks on finite quantum groups.

The Cut-Off Phenomenon in Random Walks on Compact Quantum Groups

February 13, 2018 in Quantum Groups, Research, Research Updates | 1 comment

Amaury Freslon has put a pre-print on the arXiv, Cut-off phenomenon for random walks on free orthogonal quantum groups, that answers so many of these questions, some of which appeared as natural further problems in my PhD thesis.

It really is a fantastic paper and I am delighted to see my PhD work cited: it appears that while I may have taken some of the low hanging fruit, Amaury has really extended these ideas and has developed some fantastic examples: all beyond my current tools.

This pre-print gives me great impetus to draft a pre-print of my PhD work, hopefully for publication. I am committed to improving my results and presentation, and Amaury’s paper certainly provides some inspiration is this direction.

As things stand I do not have to tools to develop results as good as Amaury’s. Therefore I am trying to develop my understanding of compact quantum groups and their representation theory. Afterwards I can hopefully study some of the remaining further problems mentioned in the thesis.

As suggested by Uwe Franz, representation theoretic methods (such as presented by Diaconis (1988) for the classical case), might be useful for analysing random walks on quantum homogeneous spaces.

Projection Distance Equal to Total Variation Distance?

November 9, 2017 in C*-Algebras & Operator Theory, Probability Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research | 2 comments

Distances between Probability Measures

Let G be a finite quantum group and M_p(G) be the set of states on the \mathrm{C}^\ast-algebra F(G).

The algebra F(G) has an invariant state \int_G\in\mathbb{C}G=F(G)^\ast, the dual space of F(G).

Define a (bijective) map \mathcal{F}:F(G)\rightarrow \mathbb{C}G, by

\displaystyle \mathcal{F}(a)b=\int_G ba,

for a,b\in F(G).

Then, where \|\cdot\|_1^{F(G)}=\int_G|\cdot| and \|\cdot\|_\infty^{F(G)}=\|\cdot\|_{\text{op}}, define the total variation distance between states \nu,\mu\in M_p(G) by

\displaystyle \|\nu-\mu\|=\frac12 \|\mathcal{F}^{-1}(\nu-\mu)\|_1^{F(G)}.

(Quantum Total Variation Distance (QTVD))

Standard non-commutative \mathcal{L}^p machinary shows that:

\displaystyle \|\nu-\mu\|=\sup_{\phi\in F(G):\|\phi\|_\infty^{F(G)}\leq 1}\frac12|\nu(\phi)-\mu(\phi)|.

(supremum presentation)

In the classical case, using the test function \phi=2\mathbf{1}_S-\mathbf{1}_G, where S=\{\nu\geq \mu\}, we have the probabilists’ preferred definition of total variation distance:

\displaystyle \|\nu-\mu\|_{\text{TV}}=\sup_{S\subset G}|\nu(\mathbf{1}_S)-\mu(\mathbf{1}_S)|=\sup_{S\subset G}|\nu(S)-\mu(S)|.

In the classical case the set of indicator functions on the subsets of the group exhaust the set of projections in F(G), and therefore the classical total variation distance is equal to:

\displaystyle \|\nu-\mu\|_P=\sup_{p\text{ a projection}}|\nu(p)-\mu(p)|.

(Projection Distance)

In all cases the quantum total variation distance and the supremum presentation are equal. In the classical case they are equal also to the projection distance. Therefore, in the classical case, we are free to define the total variation distance by the projection distance.

Quantum Projection Distance \neq Quantum Variation Distance?

Perhaps, however, on truly quantum finite groups the projection distance could differ from the QTVD. In particular, a pair of states on a M_n(\mathbb{C}) factor of F(G) might be different in QTVD vs in projection distance (this cannot occur in the classical case as all the factors are one dimensional).

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Talk: The Diaconis-Shahshahani Upper Bound Lemma for Finite Quantum Groups

May 21, 2017 in Probability Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | 1 comment

Slides of a talk given at the Topological Quantum Groups and Harmonic Analysis workshop at Seoul National University, May 2017.

Abstract A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis & Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. These ideas are generalised to the case of finite quantum groups.