Mathematics - 2024-04-27

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Jakobian

12:13 AM

I still don't even know why I'm learning Clifford algebras. What if this is all pointless

Lukas Heger

@Jakobian Clifford algebras are related to K-theory

you can prove Bott periodicity with them

not sure if K-theory is enough to solve an existential crisis, though...

Thorgott

12:34 AM

sounds like an awkward approach to Bott periodicity

I think Balarka tried telling me something along these lines once upon a time

Jakobian

12:45 AM

@LukasHeger nah I'm trying to help my friend so its going to be something applied

I've boiled down how to derive all Clifford algebras, at least the ones over $\mathbb{R}$

$\mathbb{R}^{p, q}$ is the quadratic space (i.e. vector space with quadratic form) I've been talking about above

and $\mathbb{R}_{p, q}$ is the corresponding Clifford algebra

Then one just has to know $\mathbb{R}_{1, 0}$ and $\mathbb{R}_{0, q}$ for $0\leq q \leq 7$ as well as three formulas

1) $\mathbb{R}_{p, q}\cong \mathbb{R}_{q+1, p-1}$ for $p > 0$

2) $\mathbb{R}_{p+1, q+1}\cong \mathbb{R}_{p, q}\otimes \mathbb{R}(2)$

3) $\mathbb{R}_{p, q+8}\cong \mathbb{R}_{p, q}\otimes \mathbb{R}(2^4)$

if you know those, you can write $\mathbb{R}_{p, q}$ as an algebra of matrices over one of the $9$ algebras I've mentioned above

here $\mathbb{R}_{1, 0} = \mathbb{R}^2$, $\mathbb{R}_{0, q}$ is $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{H}^2, \mathbb{H}(2), \mathbb{C}(4), \mathbb{R}(8)$ and $\mathbb{R}^2(8)$

oh I guess one actually only has to know $\mathbb{R}_{1, 0}$ and $\mathbb{R}_{0, q}$ for $0\leq q\leq 3$ if one apples formula 1) above

but yeah everything is an algebra of matrices over either $\mathbb{R}, \mathbb{R}^2, \mathbb{C}, \mathbb{H}$ or $\mathbb{H}^2$

@Thorgott Bott periodicity seems to be related to 3) ?

Thorgott

1:02 AM

yeah, that should be it

real K-theory is 8-periodic

Jakobian

1:16 AM

@Thorgott is complex K-theory 2-periodic?

this would be what I'd guess from how complex Clifford algebras only change from matrices over $\mathbb{C}$ to matrices over $\mathbb{C}^2$ and conversely

leslie townes

yes, it is

Jakobian

oh, are you knowledgeable in the topic leslie?

leslie townes

not really, i took a class that covered it 20 years ago

Jakobian

More knowledgeable than me at least

Magnus Alexander

1:43 AM

Hello everyone! Some easy PT task that I can't solve, maybe someone might give me a hint...
Given $\{X_i\}$ -- uniformly distributed independent random variables. Let $X_{n} = max{X_1, X_2, ..., X_n}$. Find the limit by distribution of $n\cdot (1-X_{(n)})$

Magnus Alexander

2:27 AM

Solved it, but thanks for anyone who considered it !