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2 votes
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Wedderburn theorem for finite-dimensional algebras over the complex numbers

[Elaborating as an answer b/c the OP asked for more details that didn't fit into a comment] Suppose $\mathcal{A} \subseteq M_d(F)$ with $F$ algebraically closed and $\mathcal{A}$ semisimple. To keep ...
Joshua Grochow's user avatar
5 votes

Semi-simple algebras over operads

Partial answer to (1). As far as I'm aware, the only definition of a finite-dimensional semisimple algebra over a linear operad is provided by Etingof. Here is an attempt to generalize it. Let $O$ be ...
Cory Gillette's user avatar
3 votes
Accepted

Minimal ideals and subalgebras of semisimple algebras

For a not necessarily unital ring $R$, a left $R$-module $S$ is simple if $RS\neq 0$ and $S$ has no proper submodule. A simple right module is defined dually. For a not necessarily unital ring the ...
Benjamin Steinberg's user avatar
6 votes
Accepted

Formal smoothness of path algebras and connections

It helps to realize that a quiver algebra is in fact a tensor algebra. (I'm assuming that $Q$ has finitely many vertices and arrows.) Let $S \subseteq kQ$ be the span of the vertices, and let $V \...
Manny Reyes's user avatar
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3 votes

Minimal ideals and subalgebras of semisimple algebras

In this answer, I was assuming the naive definition of a simple module $M$ for a nonunital ring $R$ as one with no proper nonzero submodules. It seems to be common to also insist that $RM\neq0$, in ...
Jeremy Rickard's user avatar
3 votes
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Is there a notion of point in noncommutative geometry?

A caveat: everything correct and/or good in this is stolen. I should have probably written this more carefully but given some comments above I felt too much time may have been a bad idea. I pause for ...
JP McCarthy's user avatar
7 votes

Is there a notion of point in noncommutative geometry?

In the early days of noncommutative topology (the 60s and 70s) the idea that pure states are "points" was common. But I think Yemon's comments about there being too many pure states are well ...
Nik Weaver's user avatar
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