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.

Read the rest of this entry »

Advertisements

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.

Read the rest of this entry »

Lattice of Idempotent States on a Locally Compact Quantum Group — Kasprzak & Sołtan

May 30, 2019 in C*-Algebras & Operator Theory, Quantum Groups | Leave a comment

Just some notes on the pre-print. I am looking at this paper to better understand this pre-print. In particular I am hoping to learn more about the support of a probability on a quantum group. Flags and notes are added but mistakes are mine alone.

Abstract

From this paper I will look at:

  • lattice operations on \mathcal{I}(G), for G a LCQG (analogues of intersection and generation)

1. Introduction

Idempotent states on quantum groups correspond with “subgroup-like” objects. In this work, on LCQG, the correspondence is with quasi-subgroups (the work of Franz & Skalski the correspondence was with pre-subgroups and group-like projections).

Let us show the kind of thing I am trying to understand better.

Let F(G) be the algebra of function on a finite quantum group. Let \nu,\,\mu\in M_p(G) be concentrated on a pre-subgroup S. We can associate to S a group like projection p_S.

Let, and this is another thing I am trying to understand better, this support, the support of \nu be ‘the smallest’ (?) projection p\in F(G) such that \nu(p)=1. Denote this projection by p_\nu. Define p_\mu similarly. That \mu,\,\nu are concentrated on S is to say that p_\nu\leq p_S and p_\mu\leq p_S.

Define a map T_\nu:F(G)\rightarrow F(G) by 

a\mapsto p_\nu a (or should this be ap_\nu or p_\nu a p_\nu?)

We can decompose, in the finite case, F(G)\cong \text{Im}(T_\nu)\oplus \ker(T_\nu). 

Claim: If \nu is concentrated on S, \nu(ap_S)=\nu(a)… I don’t have a proof but it should fall out of something like p_\nu\leq p_S\Rightarrow \ker p_\nu\subseteq \ker p_S together with the decomposition of F(G) above. It may also require that \int_G is a trace, I don’t know. Something very similar in the preprint.

From here we can do the following. That p_S is a group-like projection means that:

\Delta (p_s)(\mathbf{1}_G\otimes p_S)=p_S\otimes p_S

\Rightarrow \sum p_{S(1)}\otimes (p_{S(2)}p_S)=p_S\otimes p_S

Hit both sides with \nu\times \mu to get:

\sum \nu(p_{S(1)})\mu(p_{S(2)}p_S)=\nu(p_S)\mu(p_S).

By the fact that \nu,\,\mu are supported on S, the right-hand side equals one, and by the as-yet-unproven claim, we have

\sum \nu(p_{S(1)})\mu(p_{S(2)})=1.

However this is the same as

(\nu\otimes\mu)\Delta(p_S)=1\Rightarrow (\nu\star \mu)(p_S)=1,

in other words p_{\nu\star \mu}\leq p_S, that is \nu\star \mu remains supported on S. As a corollary, a random walk driven by a probability concentrated on a pre-subgroup S\subset G remains concentrated on S.

Read the rest of this entry »

Some Unresolved and Unexplored Aspects of Random Walks on Quantum Groups

May 14, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research | Leave a 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 believe I have a full proof that reducible is equivalent to supported on a pre-subgroup.

MATH7021: Spring 2019, Week 12

May 1, 2019 in MATH7021 | Leave a comment

Mathematics Exam Advice

  • The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
  • Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect. 
  • For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

  1. have I answered the question that was asked?
  2. does my answer make sense?
  3. check your answer (e.g. differentiate/antidifferentiate an antiderivative/derivative, substitute your solution into equations, check your answer against a rough estimate, or what a picture is telling you, etc)

Week 12

On Monday, and Wednesday PM, we finished the module by looking at triple integrals.

The Wednesday 09:00 lecture was a tutorial along with most of Wednesday PM and the Thursday class.

Read the rest of this entry »

MATH6040: Spring 2019, Week 12

May 1, 2019 in MATH6040 | Leave a comment

Mathematics Exam Advice

  • The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
  • Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect. 
  • For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

  1. have I answered the question that was asked?
  2. does my answer make sense?
  3. check your answer (e.g. differentiate/antidifferentiate an antiderivative/derivative, substitute your solution into equations, check your answer against a rough estimate, or what a picture is telling you, etc)

Read the rest of this entry »

MATH7021: Spring 2019, Week 11

April 11, 2019 in MATH7021 | Leave a comment

Assignment 2

Has been corrected and results emailed to you.

Some remarks on common mistakes here.

Week 11

We had a systems of differential equations tutorial Monday and before looking at double integrals.

Week 12

We will look at triple integrals and then have one or two tutorials on. Possibly Wednesday 09:00 for double integrals and Thursday for triple integrals.

Week 13

We will review the Summer 2018 paper.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Exam Papers

These are not always found in your programme selection — most of the time you will have to look here.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

MATH6040: Spring 2019, Week 11

April 10, 2019 in MATH6040 | 6 comments

Test 2

Test 2, worth 15% and based on Chapter 3, will now take place Week 12, 30 April. There is a sample test [to give an indication of length and layout only] in the notes (marking scheme) and the test will be based on Chapter 3 only.

More Q. 1s (on the test) can be found on p.112; more Q. 2s on p. 117; more Q. 3s on p.125 and p.172, Q.1; more Q. 4s on p.136, and more Q. 5s on p. 143.

Chapter 3 Summary p. 144.

Please feel free to ask me questions via email or even better on this webpage.

Homework

Once you are prepared for Test 2 you can start looking at Chapter 4:

  • Revision of Integration, p.161.
  • p.167, Q. 1-5
  • p.182

Week 11

We had some tutorial time from 18:00-19:00 for further differentiation (i.e. for Test 2, after Easter).

We completed our review of antidifferentiation before starting Chapter 4 proper.

We looked at Integration by Parts and centroids.

For those who could not make it here is some video and slides from what we did after the video died.

Week 12

We will have some tutorial time from 18:00-19:00 for further differentiation.

We will have Test 2 from 19:00-20:05.

Then we will look at completing the square, centres of gravity, and work.

Week 13

We will look the Winter 2018 paper at the back of your manual.

CIT Mathematics Exam Papers

These are not always found in your programme selection — most of the time you will have to look here.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

MATH7021: Spring 2019, Week 10

April 4, 2019 in MATH7021 | Leave a comment

Week 10

We had two additional tutorials… actually four tutorials in total and two lectures; the lectures focused on Systems of Differential Equations.

Assignment 2

Assignment 2 now has a pushed back deadline of 12:00, 12 April: the Friday of Week 11. Assignment 2 is in the manual, P. 164. Usual warnings about copying apply.

Week 11

We will have a systems of differential equations tutorial Monday and then look at double integrals.

Week 12

We will look at triple integrals and then have one or two tutorials. Possibly Monday for double integrals and Thursday for triple integrals.

Week 13

We will review the Summer 2018 paper.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Exam Papers

These are not always found in your programme selection — most of the time you will have to look here.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

MATH6040: Spring 2019, Week 10

April 3, 2019 in MATH6040 | 2 comments

Test 2

Test 2, worth 15% and based on Chapter 3, will now take place Week 12, 30 April. There is a sample test in the notes and the test will be based on Chapter 3 only.

Homework Exercises

If you do any of the suggested exercises you can give them to me for correction. Please feel free to ask me questions about the exercises via email or even better on this webpage.

I recommend strongly that everyone completes P.102, Q.1.

After that you can look at:

  • P.136, Q.1-3
  • P. 143, Q. 1-4
  • P. 125, Q. 1-4, P.172, Q.1
  • P. 117, Q. 1-4
  • P.112, Q. 1-5
  • Sample Test 2, P.145

If you want to do more again, look at P.113, Q.6-9, P. 118, Q. 5-6, P. 125, Q.5. There is a Weekly Summary for the Chapter 3 Material on P.144.

If you read on there is some information below about solutions to these exercises.

Week 10

For those who were able to make it we had some tutorial time from 18:00-19:00 for parametric, implicit, and related rates differentiation. If you are really interested in understanding how does a curve have an equation, see here.

In class we looked at partial differentiation and error analysis. For those who could not make it, here is video of the partial differentiation material and here are the slides from the error analysis (the video died shortly after we started error analysis).

We only started a revision of Antidifferentiation to start Chapter 4 on (Further) Integration. I have this section completed here.

Week 11

We will have some tutorial time from 18:00-19:00 for further differentiation (i.e. for Test 2, after Easter).

We are under pressure for time but I have made the decision that we will be better off completing our review of antidifferentiation before starting Chapter 4 proper. This might put us under time pressure later on but I believe it is the correct thing to do.

We will look at Integration by Parts, completing the square, and work.

Read the rest of this entry »

Advertisements
%d bloggers like this: