bio  website  math.berkeley.edu/~qchu 

location  Berkeley, CA  
age  24  
visits  member for  5 years, 9 months 
seen  11 mins ago  
stats  profile views  62,695 
I am a thirdyear graduate student at UC Berkeley. I'm interested in the interplay between homotopy theory, quantum field theory, manifold topology, and higher category theory.
5m

comment 
A particularly “natural” algebraic structure with three commutative, pairwisedistributive operations
I don't understand why what you've already written isn't an answer to most of these questions. As you already know, sending a sequence $a_1, a_2, \dots$ to the corresponding formal power series $\sum a_i x^i$ turns Cauchy convolution into "pointwise" multiplication, and sending it to the formal Dirichlet series $\sum \frac{a_i}{i^s}$ turns Dirichlet convolution into "pointwise" multiplication. The quotation marks can be removed if you impose growth conditions. In what way is this not an answer to #3, #4, and #8? 
10h

comment 
Can the projective line be provided with a ring structure?
If $K$ doesn't contain a square root of $1$, then your group is the quotient of the multiplicative group of $K[i]$ by the multiplicative group of $K$. The closest ring in sight is $K[i]$, which, being a field, has no nontrivial quotients. 
10h

comment 
Can the projective line be provided with a ring structure?
Your proposed multiplication isn't welldefined if $K$ itself already contains a square root of $1$: in this case, $K[i]$ has zero divisors, so it's possible for the "product" of two points to have both coordinates zero and hence to not be a welldefined point on the projective line. 
11h

answered  Can the projective line be provided with a ring structure? 
1d

comment 
Virasorolike algebras over the quaternions
@Sébastien: this already fails for $n = 1$. The bracket is $[A, B] = AB  BA$ which is not $\mathbb{H}$bilinear in either variable since neither $A$ nor $B$ need be central. Did you understand my point earlier about $\mathbb{H}$modules not being a symmetric monoidal category? 
2d

comment 
SO$(4)$ (& SO$(n)$) characterization?
Added line breaks where your paragraphs were. Hope you don't mind. 
2d

revised 
SO$(4)$ (& SO$(n)$) characterization?
added 4 characters in body 
2d

comment 
Virasorolike algebras over the quaternions
@Chanler: if you just write down the standard definition as applied to a left $\mathbb{H}$module, you get a definition which doesn't apply to $\mathfrak{gl}_n(\mathbb{H})$ (the Lie bracket on $\mathfrak{gl}_n(\mathbb{H})$ fails to be $\mathbb{H}$bilinear in the appropriate way). Are you sure that's what you want? 
2d

comment 
Virasorolike algebras over the quaternions
What is a Lie algebra over $\mathbb{H}$? (The Lie operad is a symmetric operad in, say, abelian groups, so I know how to define a Lie algebra in any symmetric monoidal $\text{Ab}$enriched category. But $\mathbb{H}$modules aren't such a category.) 
2d

answered  SO$(4)$ (& SO$(n)$) characterization? 
Jul 30 
comment 
Why only Normed Linear Spaces?
We do in fact do this. See, for example, en.wikipedia.org/wiki/Absolute_value_(algebra) and en.wikipedia.org/wiki/Banach_algebra. 
Jul 29 
revised 
Does the following characterize local presentability?
added 3 characters in body 
Jul 29 
answered  Does the following characterize local presentability? 
Jul 28 
comment 
Does every Lawvere theory arise in this way?
@David: no, that already fails in the case of groups. The way you recover a Lawvere theory from the free algebra over it on one generator $X$ is by taking the opposite of the full subcategory on the finite coproducts of $X$, not by taking the full subcategory on the finite products of $X$. 
Jul 26 
comment 
Is there a generalization of homotopy groups to fractional dimensions
Is there an $E_{\infty}$ ring spectrum $E$ and an invertible $E$module spectrum $F$ such that $F^{\otimes 2} \cong E[1]$? ($\otimes$ here denotes the $E$module smash product.) This gives a notion of fractional $E$homology and cohomology. 
Jul 24 
comment 
Loop space generalization
$S^1$ has two interesting kinds of extra structure as an object of the homotopy category of based spaces: first, it's a cogroup object, but second, it's a group object. The first kind of extra structure is the one related to loop spaces, while the second kind of extra structure is the one related to its description as $B \mathbb{Z}$. The EilenbergMacLane spaces $B^n \mathbb{Z}, n \ge 2$, on the other hand, are only group objects, and have no cogroup structure (since their cohomology has interesting cup products). 
Jul 22 
awarded  Good Question 
Jul 22 
awarded  Good Question 
Jul 22 
awarded  Nice Question 
Jul 21 
comment 
Three questions about modular forms frequently asked to me
Crossposted: math.stackexchange.com/questions/1369276/… 