What is...diagrammatic algebra?

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Күн бұрын

Пікірлер: 21

@WhenMathFlows-k7r Ай бұрын

Funny, I wrote up an expository intro to string diagrams for 2-categories a while ago, junior year. There the cup and cap products can be interpreted as the unit and co-unit of an adjunction, and straitening the strand is equivalent to the axioms of an adjunction. String diagrams used this way give such an nice and quick proof that the composition of an adjoint pair gives a monad

@VisualMath Ай бұрын

Excellent, the students probably had a lot of fun 😁 Yes, in my experience these diagrams are great to get started!

@crimsnblade8555 Ай бұрын

It's like a Metal band member teaching me maths, I love it

@VisualMath Ай бұрын

Excellent, playing in a metal band was one of my two alternatives instead of a math path 🤣 I am glad that you like the videos!

@Jaylooker Ай бұрын

I think the diagrams of diagrammatic algebra are knots as closed braids following Alexander’s theorem. The equalities you mentioned throughout with the different swaps are Reidemeister moves or just isotopy with the pairing, copairing, and straightening out at 9:02. Similar string diagrams credited to Penrose are used with monoidal categories. I think there is a connection from an algebra to its monodial category through Drinfeld associators. Drinfeld associators provide a way to construct braided monodial categories out Lie algebras.

@VisualMath Ай бұрын

I like to think of diagrammatic algebra as a generalization of knot diagrams. That is similar, but not quite what you are getting to 😀

@Jaylooker Ай бұрын

@@VisualMath I agree 👍

@astonishinghypothesis Ай бұрын

That was amazing to watch. One interesting piece of information after another while never losing sight of the big picture. Best introduction to diagram algebras on KZbin.

@VisualMath Ай бұрын

Thanks for the feedback; very much appreciated. I am glad that you like the video/topic ☺

@eternaldoorman5228 Ай бұрын

There is some sense of process in all these examples. The diagrams seem to give a handle on thinking about constructions involving building up complex processes from some simple set of fundamental elements.

@VisualMath Ай бұрын

Good observation, that is exactly right 😄 This often goes under the slogan generator and relations and is a general idea that is powerfully reflected in diagrammatics.

@asainpopiu6033 Ай бұрын

I just found your channel, I like the kind of math you present here! I wish I had more time to study that! Thanks for uploading that stuff.

@asainpopiu6033 Ай бұрын

unfortunately your website gives me a 504 bad gateway error... :/

@VisualMath Ай бұрын

@@asainpopiu6033 Welcome, I hope you will enjoy the ride ☺ My provider seems to have server issues; hopefully this will be resolved soon. Sorry for that!

@mathephilia Ай бұрын

Very interesting ! since this is not my forte, I'd be curious to see more concrete examples (like, take a pair of mathematical domains, and show how some concepts / proofs are expressed in both domains in regular algebraic notation, but also explain how these can be expressed in diagrammatic notation). For example, it seemed to me like these diagrams are read top to bottom ? But I'm not even sure. I got that the first part represented commutativity, but I was confused about why the cups and caps of the second part represented the dot product or what seemed to be something mapping 1 in ℂ to the identity map in ℂⁿ ? But then why would the dot product create 2 children, and the diagonal-ish map delete two elements ?

@asainpopiu6033 Ай бұрын

"For example, it seemed to me like these diagrams are read top to bottom ?" In the video they would be read bottom to top, based on the slide at 9:30 "But then why would the dot product create 2 children, and the diagonal-ish map delete two elements ?" No it's the other way around, the dot product deletes things (it produces a scalar) and the diagonal-ish thing creates 2 children. "why the cups and caps of the second part represented the dot product or what seemed to be something mapping 1 in ℂ to the identity map in ℂⁿ ? " Hm, what is the identity map in ℂ^n ? If V is a ℂ-vector space of finite dimension with an inner product, the cap is the inner product V⊗V→ℂ. And the cup goes from ℂ to V⊗V, it sends 1 to the sum of e_i⊗e_i where the e_i form an orthonormal basis of V.

@VisualMath Ай бұрын

Thanks, you two! Yes, I read bottom to top. I agree with asainpopiu6033's other comments and hope they help!

@mathephilia Ай бұрын

@@asainpopiu6033 Oh thanks, that makes a lot more sense. As for the identity map, I meant the identity matrix, but admittedly here you're right that it's used as a dot product, not the identity. Though I suppose that this "diagonal-ish map" would generate from a complex number z the symmetric matrix with only z's in the diagonal ? And so in the general case would return a dot product scaled by z ?

@asainpopiu6033 Ай бұрын

@@mathephilia Ah, I understand, you meant the identity map *on* ℂ^n (or of ℂ^n). Sorry. So yes this is how this diagonal-ish thing acts! And then the inner product sends a matrix to its trace, if we think in terms of matrices.