bio | website | mit.edu/~darij/www |
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location | Karlsruhe (home), Munich (until summer 2011), Cambridge/MA (2011-) | |
age | 25 | |
visits | member for | 4 years, 8 months |
seen | 6 hours ago | |
stats | profile views | 11,092 |
I'm just here for asking stupid questions.
Aug 16 |
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Reference request: Book of Linear algebra from categorical point of view
I don't remember right out of my hat whether Paolo Aluffi's "Algebra 0" does much linear algebra (and I'm not at the computer which has it as PDF), but it certainly looks like a step in the right direction. Also, Kostrikin/Manin "Linear Algebra and Geometry", while not using the categorical approach right away, does introduce categories at some point (as well as tons of other interesting things). But now I'm seeing that these are exactly the first two suggestions on the Reddit thread... |
Aug 11 |
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Can one live without actual infinity?
I always found the distinction between potential and actual infinity rather nebulous: what does "actual" mean in mathematics? Is the $2$ in $1+1=2$ a potential $2$ or an actual $2$ ? What about the $1$ or the $+$ or, gods forbid, the $=$ ? Doesn't the abstractness of mathematics preclude its actuality? The only good distinction I am able to make is that between considering infinite sets as sets and considering them as classes / types. Constructivism, as I understand it, does the latter. |
Aug 9 |
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Examples of unexpected mathematical images
What is your $n$ ? |
Aug 7 |
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Prove that the determinants are equal
+1 because this thing is nice and I am wondering whether it generalizes. |
Aug 5 |
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Is antipode unique for bialgebras in arbitrary monoidal categories?
Ah -- I guess the correct way to argue my proof would involve introducing an associator $\operatorname{Ass}_V : \left(V \otimes V\right) \otimes V \to V \otimes \left(V \otimes V\right)$ (functorial in the object $V$), which is going to be pushed past the $\left(f \otimes g\right) \otimes h$, transforming it into $f \otimes \left(g \otimes h\right)$. Alternatively, you can use one of the coherence theorems that show that you can pretend that the associators are trivial. I don't know the meaning of Jenkins' pictures myself, but don't count me surprised if they depend upon such coherence. |
Aug 5 |
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Is antipode unique for bialgebras in arbitrary monoidal categories?
for any three linear maps $f$, $g$ and $h$ from the coalgebra to the algebra. It remains to compare the two equalities and use the coassociativity axiom $\left( \Delta\otimes\operatorname{id}\right) \circ\Delta=\left( \operatorname{id}\otimes\Delta\right) \circ\Delta$ and the associativity axiom $m\circ\left( m\otimes\operatorname{id}\right) =m\circ\left( \operatorname{id}\otimes m\right) $. |
Aug 5 |
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Is antipode unique for bialgebras in arbitrary monoidal categories?
... $=m\circ\left( m\otimes\operatorname{id}\right) \circ\left( \left( f\otimes g\right) \otimes h\right) \circ\left( \Delta\otimes \operatorname{id}\right) \circ\Delta$ $=m\circ\left( m\otimes\operatorname{id}\right) \circ\left( f\otimes g\otimes h\right) \circ\left( \Delta\otimes\operatorname{id}\right) \circ\Delta$ and similarly $f\ast\left( g\ast h\right) =m\circ\left( \operatorname*{id}\otimes m\right) \circ\left( f\otimes g\otimes h\right) \circ\left( \operatorname*{id}\otimes\Delta\right) \circ\Delta$ ... |
Aug 5 |
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Is antipode unique for bialgebras in arbitrary monoidal categories?
... $\left( f\ast g\right) \ast h=m\circ\left( \left( \underbrace{\left( f\ast g\right) }_{=m\circ\left( f\otimes g\right) \circ\Delta}\right) \otimes h\right) \circ\Delta$ $=m\circ\left( \underbrace{\left( m\circ\left( f\otimes g\right) \circ\Delta\right) \otimes h}_{=\left( m\otimes\operatorname{id}\right) \circ\left( \left( f\otimes g\right) \otimes h\right) \circ\left( \Delta\otimes\operatorname{id}\right) }\right) \circ\Delta$ ... |
Aug 5 |
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Is antipode unique for bialgebras in arbitrary monoidal categories?
Denoting the multiplication of the algebra by $m$ and the comultiplication of the coalgebra by $\Delta$, we have ... |
Aug 5 |
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Is antipode unique for bialgebras in arbitrary monoidal categories?
Oh -- but it is very easy to prove the associativity of convolution without Sweedler's notation, just using the axioms of a coalgebra and of an algebra. |
Aug 4 |
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Is antipode unique for bialgebras in arbitrary monoidal categories?
This is really obvious if you look at the standard proof of the uniqueness of the antipode. The antipode is defined as the $*$-inverse of the identity in the convolution algebra. The convolution algebra is a honest algebra, not an algebra-over-a-category; so inverses are unique if they exist. |
Aug 3 |
awarded | Notable Question |
Aug 1 |
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Diagonalization via the Toda flow
Now I am wondering: what happens if we regard the matrix as a formal power series rather than a function? We can then resolve the ODE uniquely (when the ground field has characteristic $0$). Do the analytical statements about the $t \to \infty$ limit have some formal shadows? How do the $t^N$-coefficients of the matrix entries of $X$ look like for big $N$ ? Is there a reasonable way in which the diagonal gets more action than the rest of the matrix? Is there a formal-power-series analogue of the spectral theorem? |
Aug 1 |
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Diagonalization via the Toda flow
Thanks for this exposition, David! I only (briefly) missed an explanation of why $X$ stays symmetric (this is used tacitly in your formula for $\frac{dX_{kk}}{dt}$). |
Aug 1 |
revised |
Diagonalization via the Toda flow
trivial typos |
Aug 1 |
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Factor a sum of products of cofactors
Is your $S_n(k)$ divisible by $\sum_{q=1}^n C_{q,1}$ if you don't require $M$ to have its first column filled with $1$'s? |
Aug 1 |
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Coideals of Hopf algebra coming from right (left) coideals K->K^+
(Note: It is rather easy to prove that $\left(\eta \varepsilon - \operatorname{id}\right)\left(K\right)$ is a coideal of $H$, because every $k \in K$ satisfies $\Delta\left(k - \varepsilon\left(k\right)\right) = \sum_{(k)} \left(k_{(1)} - \varepsilon\left(k_{(1)}\right)\right) \otimes k_{(2)} + 1 \otimes \left(k - \varepsilon\left(k\right)\right)$, using Sweedler's notation.) |
Aug 1 |
revised |
Coideals of Hopf algebra coming from right (left) coideals K->K^+
language (and bump this up) |
Jul 31 |
reviewed | Leave Open Isomorphic Dual and Conjugate Representations of a Lie Algebra |
Jul 29 |
revised |
Why does this antisymmetric product factor out a determinant?
added 8994 characters in body |