 |
Post ASqwYgSi6QNZP53qO8 by [email protected] |
 |
More posts by [email protected] |
|
Post #ASkWakmDGj63bikYXA by [email protected] |
|
0 likes, 1 repeats |
|
|
|
A few days ago I managed to watch @johncarlosbaez' talk on the Tenfold Way.… |
|
|
|
Post #ASkWarFVCk5fgIV21I by [email protected] |
|
0 likes, 0 repeats |
|
|
|
The Tenfold Way is a package of 8+2 beautiful structures which appear in a numb… |
|
|
|
Post #ASkWawAitwEqwvImlU by [email protected] |
|
0 likes, 0 repeats |
|
|
|
1. They describe ten kinds of symmetries we can use to make free fermions. John… |
|
|
|
Post #ASkWb1uHZyeWjfejyK by [email protected] |
|
0 likes, 0 repeats |
|
|
|
2. Superalgebras are πβ graded algebras, that is, they consist of an odd a… |
|
|
|
Post #ASkWb6iPhX85eJ8pfM by [email protected] |
|
0 likes, 0 repeats |
|
|
|
What we're interested in has only a short page with some basic definitions … |
|
|
|
Post #ASkWbBdHQ2zgtpmIrI by [email protected] |
|
0 likes, 0 repeats |
|
|
|
John also looks into πβ graded vector spaces, because they give a nice mne… |
|
|
|
Post #ASkWbEqrS0dksp9gwa by [email protected] |
|
0 likes, 0 repeats |
|
|
|
3. The Tenfold Way also appears in conjunction with Clifford algebras, and sinc… |
|
|
|
Post #ASkWbIUfuRGoBAo1uy by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Simple Clifford algebras for π βΏ accomodate vectors, but also scalars at th… |
|
|
|
Post #ASkWbLqlQlioaYADGS by [email protected] |
|
0 likes, 0 repeats |
|
|
|
The real Clifford algebras are popular under the name geometric algebra, maybe … |
|
|
|
Post #ASkWbP6TKp4Mg8XIfI by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Let me just say this much, the rotation groups, which are half of the symmetry … |
|
|
|
Post #ASkWbS6w9YDw0RmDJ2 by [email protected] |
|
0 likes, 0 repeats |
|
|
|
There are also generalizations of Clifford algebra. For one, we can construct a… |
|
|
|
Post #ASkWbUSHRAU7HDmHqq by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Now, Clifford algebras aren't all the same. I mean, of course, if we change… |
|
|
|
Post #ASkWbW1Pc2BI8TDrFY by [email protected] |
|
0 likes, 0 repeats |
|
|
|
John describes Morita equivalence as two structures which have the same categor… |
|
|
|
Post #ASkWbXJApTz07qXYZc by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Plus eight, you ask? Well, yes, the real Clifford algebras sort of repeat every… |
|
|
|
Post #ASkWbYDBTEmmvYkHsO by [email protected] |
|
0 likes, 0 repeats |
|
|
|
You should really watch this fun talk, even if you suspect you won't get ev… |
|
|
|
Post #ASkWbYtiv4oV3UeGBM by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Addendum, I can't resist to add one:There seem to be some interesting avenu… |
|
|
|
Post #ASkWbZjpnKUtf71sPI by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Remember, representation theory is a technique to learn about an algebraic stru… |
|
|
|
Post #ASkWbaNXPiFxeFbaIC by [email protected] |
|
0 likes, 0 repeats |
|
|
|
The point being: all these are associative, but the octonions aren't. I'… |
|
|
|
Post #ASkWbbno7Wrc40u4sS by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@bhaugen asked for this. Hello, I did it :) |
|
|
|
Post #ASkXPWgjPPCMgKOd2e by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - Thanks for your great summary! "John, I don't know i… |
|
|
|
Post #ASkXVoKCkUuV7Or94i by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Allright, I'm just teasing, @johncarlosbaez. I mean, there's a lot of f… |
|
|
|
Post #ASkY896DxT6CYNb2nI by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - when I talk in front of a live audience I wave my hands in en… |
|
|
|
Post #ASkZosZTNxWRhMDaL2 by [email protected] |
|
0 likes, 0 repeats |
|
|
|
> I assume nobody is looking at me.Too bad! If I remember correctly it was … |
|
|
|
Post #ASkjc3iHeBZQ22rg0W by [email protected] |
|
0 likes, 0 repeats |
|
|
|
I suppose you have put up lecture notes somewhere with some more details and le… |
|
|
|
Post #ASkjc3nbKPp8IXLvIO by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Found it. I'm sure others will be interested in this, too:Folks, you can fi… |
|
|
|
Post #ASkjc3sv0e4qZ1qAaG by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - yes, so far the best thing to read is the webpage you pointed… |
|
|
|
Post #ASkokRuC5KGB9MFvAe by [email protected] |
|
0 likes, 0 repeats |
|
|
|
> talk with James DolanYeah, it's tempting. I just snatched that one, th… |
|
|
|
Post #ASkrMktQ8btqx2HtYm by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - I think this particular conversation with James Dolan may be … |
|
|
|
Post #ASohRcibjfcu8Zkhhg by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro wrote: "I'd be rather curious to read about non-associ… |
|
|
|
Post #ASohkoilKopMreyd9s by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - One could of course simply drop the module axiom(ab)m = a(bm)… |
|
|
|
Post #ASoi1C34N5ANqZHgEC by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - That is, L(A) consists of all linear transformations of A tha… |
|
|
|
Post #ASoiAicwc8QlUgbf3g by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - if you're really interested in these issues I recommend R… |
|
|
|
Post #ASoilBTCr15VNERJHk by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro Thank you very much! I hope you wanted to do it for yourself, t… |
|
|
|
Post #ASoilCK1gdL4139UcC by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@bhaugen said:> I hope you wanted to do it for yourself, too.Yes, of course!… |
|
|
|
Post #ASoilCOzOBJCGRTSLo by [email protected] |
|
0 likes, 0 repeats |
|
|
|
There's certainly a balance between the parts that make sense right away an… |
|
|
|
Post #ASoilCTb72zkUjd8XA by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Oh, and let us know what you make of it in case anything sticks!@bhaugen |
|
|
|
Post #ASoilCXUsY78gpSFc0 by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro I watched about 15 minutes into the video, taking some notes, a… |
|
|
|
Post #ASoilCc6bPngv7bvnM by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Admittedly, the Tenfold way is somewhat obscure and requires a bit more knowled… |
|
|
|
Post #ASoilCgiKHUF9Plbyi by [email protected] |
|
0 likes, 0 repeats |
|
|
|
And thanks for the update, @bhaugen! |
|
|
|
Post #ASoilClK39AnNhvIA4 by [email protected] |
|
0 likes, 0 repeats |
|
|
|
> for the sake of itHm. Maybe that's not quite honest. Years ago I heard… |
|
|
|
Post #ASoilCpZnKZlatugnA by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro wrote: "Maybe that's not quite honest. Years ago I hea… |
|
|
|
Post #ASqutfPgzrvCK9WkRk by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Thanks for the suggestion. I'm going to take a look, it sounds fun!@johncar… |
|
|
|
Post #ASqvCaMjzJXKAgASCO by [email protected] |
|
0 likes, 0 repeats |
|
|
|
I just spent some time listening to you two guys. I love these informal remarks… |
|
|
|
Post #ASqw6t0nqhHODZkjB2 by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - Sorry, I said it in my talk, but probably quite fast: Mβ(X)… |
|
|
|
Post #ASqwK8GoQkuRIyYbc8 by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Now I'm very keen to look into the central simple super division algebras..… |
|
|
|
Post #ASqwK8Lm8IsZYMsZLk by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - people have classified central simple Z/n-graded algebras but… |
|
|
|
Post #ASqwXuWyJHGckOgZMm by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Now that you say it it rings a bell. Thanks, @johncarlosbaez!> Yes, category… |
|
|
|
Post #ASqwYgO6NYh1AmuACm by [email protected] |
|
0 likes, 0 repeats |
|
|
|
I've just flipped through Richard D. Schafer's book. I would have expec… |
|
|
|
Post #ASqwYgSi6QNZP53qO8 by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - no, Schafer's book is not about universal algebra. It… |
|
|
|
Post #ASqwpLe6kYHplZL24O by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - yes, that's why in my official talk I didn't use cate… |
|
|
|
Post #ASqx4iTm8dHGvUpjm4 by [email protected] |
|
0 likes, 0 repeats |
|
|
|
> central simple Z/n-graded algebras[,] So far they have been completely use… |
|
|
|
Post #ASqxPQdPg9uvfDDZ3o by [email protected] |
|
0 likes, 0 repeats |
|
|
|
> But most people will actually enjoy hearing Dyson's 3-fold way and the… |
|
|
|
Post #ASqxX2dkm2dxYx0V7Y by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Yes, the idea was to find out what representation theory for the octonions migh… |
|
|
|
Post #ASqyKJpldWTt4t7jUG by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro wrote: "You did write the category way to do this down I g… |
|
|
|
Post #ASqyiTkwOihoniO1wG by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - the first step would be to define representations of nonassoc… |
|
|
|
Post #ASr1zxzhWgidMsJxU8 by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - I don't think you should hope that there's something … |
|
|
|
Post #ASr6ho4cAjmt9pmNDk by [email protected] |
|
0 likes, 0 repeats |
|
|
|
It's just that the complex numbers work so well for many important things, … |
|
|
|
Post #ASr7J19KWDL2QWjDMW by [email protected] |
|
0 likes, 0 repeats |
|
|
|
Is 'simplical sets' code for generalizations of cohomology? In that cas… |
|
|
|
Post #ASr7J1EIDlJAfv3B68 by [email protected] |
|
0 likes, 0 repeats |
|
|
|
@RefurioAnachro - "simplicial sets" are the foundation of modern homo… |
