⧼exchistory⧽
6 exercise(s) shown, 0 hidden
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Prove that we have the free complexification formula
[[math]]
\widetilde{O_N^+}=U_N^+
[[/math]]
directly, without character computations and free probability.
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Prove that we have the free complexification formula
[[math]]
\widetilde{SU_2}=U_2^+
[[/math]]
and then look for analogues of this formula, at arbitrary [math]N\in\mathbb N[/math].
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Prove that we have the formula
[[math]]
PO_N^+=PU_N^+
[[/math]]
directly, without using the free complexification results.
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Prove that we have the formula
[[math]]
PU_2^+=SO_3
[[/math]]
and then look for analogues of this, at arbitrary [math]N\in\mathbb N[/math].
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Try developing a theory of real and complex free projective spaces [math]P^{N-1}_{\mathbb R,+}[/math] and [math]P^{N-1}_{\mathbb C,+}[/math], and explain what fails.
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Find axioms for the projective quadruplets
[[math]]
\xymatrix@R=50pt@C=50pt{
P\ar[r]\ar[d]\ar[dr]&PT\ar[l]\ar[d]\ar[dl]\\
PU\ar[u]\ar[ur]\ar[r]&PK\ar[l]\ar[ul]\ar[u]
}
[[/math]]
covering the real, complex and free quadruplets, constructed above.