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The Matthew Maths Show: Galois Quantum Groups of Finite Fields

http://mebassett.blogspot.com/2013/01/galois-quantum-groups-of-finite-fields.html

The Matthew Maths Show. Maths blog - programming blog. Friday, January 4, 2013. Galois Quantum Groups of Finite Fields. In Chater 4 of Majid's A Quantum Groups Primer. Let $A$ be an algebra over a field $k$. We call $B$ a comeasuring. Of $A$ if it comes with an algebra map $ beta: A rightarrow A otimes B$, this is very similar to a coaction. That is, an object $M(A)$ such that for every other object $B$ in the category, there exists one and only one unique morphism $M(A) rightarrow B$. For $i,j,k=1$ to $...

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The Matthew Maths Show: November 2011

http://mebassett.blogspot.com/2011_11_01_archive.html

The Matthew Maths Show. Maths blog - programming blog. Monday, November 28, 2011. Climbing Out of Black Holes. I published a two questions that had me stuck inside Majid's ` almost commutative black hole' . We'll start today be answering those questions while trying to avoid insulting myself and my readers, and then move on to calculate the Grassmann connection on our Black Hole Algebra. 1 NCG Vector Bundles For the Schwarzschild Solution. 2 Grassmann Connections in NCG. 3 The Grassmann Connection for ou...

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The Matthew Maths Show: What I learned today: Braided (or quasi-triangular) Hopf Algebras

http://mebassett.blogspot.com/2013/01/what-i-learned-today-braided-or-quasi.html

The Matthew Maths Show. Maths blog - programming blog. Saturday, January 5, 2013. What I learned today: Braided (or quasi-triangular) Hopf Algebras. This is a post in my ` psuedo-daily' series ` What I learned Today'. Displaystyle tau circ Delta h = mathcal{R}( Delta h) mathcal{R} {-1}$. Id est, $G = langle x, y : x 2 = y 2 = (xy) 2 = 1 rangle$ and let $kG$ be the group Hopf algebra. For $q in k$ define. Displaystyle ( Delta otimes I) mathcal{R} = mathcal{R} {13} mathcal{R} {23}$. Where $ mathcal{R} {23}...

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The Matthew Maths Show: February 2011

http://mebassett.blogspot.com/2011_02_01_archive.html

The Matthew Maths Show. Maths blog - programming blog. Tuesday, February 8, 2011. Fun and Games with X and f: A talk on Ergodic Theory. Or just read them here:. 1 First Game: Sets and Functions. Let $X$ be an arbitrary set (e.g. points in space, animals in a zoo.) and let $f: X rightarrow X$ be some function on $X$. For some $x in X$, we're gonna look at what happens to the set $ { x, f(x), f(f(x) , ldots }$. We'll call this set the orbit. Displaystyle n mapsto f n(x)$. We're gonna play three games with ...

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The Matthew Maths Show: January 2012

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The Matthew Maths Show. Maths blog - programming blog. Wednesday, January 18, 2012. Don't Censor the Web. Congress is considering two Orwellian-named laws, SOPA and PIPA, that are threatening free speech, internet security, and innovation. This is a reminder to call your representatives in Congress. And/or donate to the EFF. Today to help stop internet censorship. Posted by Matthew Eric Bassett. Saturday, January 7, 2012. Constructing the Grothendieck-Teichmuller Group. So for the past six or seven months.

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The Matthew Maths Show: Bosonisation of Quantum Groups, part II (Braided Categories)

http://mebassett.blogspot.com/2013/01/bosonisation-of-quantum-groups-part-ii.html

The Matthew Maths Show. Maths blog - programming blog. Monday, January 14, 2013. Bosonisation of Quantum Groups, part II (Braided Categories). Let's talk about the category of modules ${ H mathcal{M} $ for a braided Hopf algebra ${H}$ for a minute. If ${V}$ and ${W}$ are ${H}$-modules, then we have an easy action of ${H}$ on ${V otimes W}$ as well, by:. Displaystyle h triangleright(v otimes w) = sum h 1 triangleright v otimes h 2 triangleright w $. Actually a natural isomorphism (one can see already that...

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The Matthew Maths Show: Bosonisation of Quantum Groups, an Introduction

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The Matthew Maths Show. Maths blog - programming blog. Sunday, January 13, 2013. Bosonisation of Quantum Groups, an Introduction. Naturally, $B$ will have it's own modules. But what if we restrict the $B$-modules we want to look at? In fact, let's only look at $B$-modules in $ mathcal{C}$, we'll call these the braided. Does there exist a Hopf algebra $ mathcal{H}$ such that it's category of modules is exactly $ B mathcal{C}$? To do this, I need to study a few things:. That will color all calculations in ...

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The Matthew Maths Show: January 2013

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The Matthew Maths Show. Maths blog - programming blog. Monday, January 21, 2013. Bosonization Part III, Co-Bosonization. A few days ago I sketched out some of the basic category theoretic ideas behind the theory of bosonization of quantum groups. Today I want to talk about the dual version, co-bosonization. So let $(H, mathcal{R})$ be a dually. Quasitriangular Hopf algebra (I called this a co-braided Hopf algebra. In an earlier post on braided categories. Also as before, $ B mathcal{C}$ is the category o...

mebassett.blogspot.com

The Matthew Maths Show: Bosonization Part III, Co-Bosonization

http://mebassett.blogspot.com/2013/01/bosonization-part-iii-co-bosonization.html

The Matthew Maths Show. Maths blog - programming blog. Monday, January 21, 2013. Bosonization Part III, Co-Bosonization. A few days ago I sketched out some of the basic category theoretic ideas behind the theory of bosonization of quantum groups. Today I want to talk about the dual version, co-bosonization. So let $(H, mathcal{R})$ be a dually. Quasitriangular Hopf algebra (I called this a co-braided Hopf algebra. In an earlier post on braided categories. Also as before, $ B mathcal{C}$ is the category o...