Timeline for Tensor product of irreducible representations of an algebra
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 18, 2023 at 4:59 | vote | accept | Nanoputian | ||
| Jun 17, 2023 at 14:46 | history | became hot network question | |||
| Jun 17, 2023 at 14:29 | answer | added | Tom Goodwillie | timeline score: 7 | |
| Jun 17, 2023 at 13:52 | answer | added | Benjamin Steinberg | timeline score: 10 | |
| Jun 17, 2023 at 13:18 | comment | added | Benjamin Steinberg | There are lots of semigroup examples. | |
| Jun 17, 2023 at 13:12 | comment | added | Tom Goodwillie | I'm not sure about my example, but I'll think about finding another, if nobody else gives one. | |
| Jun 17, 2023 at 11:58 | comment | added | Nanoputian | @TomGoodwillie Yes that is what I mean. If you write the counterexample as an answer I will be happy to accept it. | |
| Jun 17, 2023 at 11:17 | comment | added | Tom Goodwillie | About Dave Benson's question, presumably you mean that the coproduct of a generator $x$ is $x\otimes x$, in other words a generator operates on $V\otimes W$ by $x(v\otimes w)=xv\otimes xw$. Assuming this, there is a counterexample in which $A$ is free on two generators and $V=W$ is $2$-dimensional. | |
| Jun 17, 2023 at 9:30 | comment | added | Dave Benson | What is your coproduct on the free algebra? | |
| Jun 17, 2023 at 8:02 | comment | added | Nanoputian | @Mare Sorry, I have realised that the tensor product is not in general defined for algebras. So to be more specific, the case I am interested in is when $A$ is the free algebra generated by $n$ elements (even $n = 2$ case is fine). Though, perhaps its too much to ask for such a specific case to have been treated in the literature. | |
| Jun 17, 2023 at 7:58 | history | edited | Nanoputian | CC BY-SA 4.0 |
added 18 characters in body
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| Jun 17, 2023 at 7:44 | comment | added | abx | Indeed in characteristic 0, $V\otimes W$ is semi-simple as a $A\otimes A^{op}$-module. | |
| Jun 17, 2023 at 7:01 | comment | added | Mare | What is the $A$-module strucutre on $V \otimes W$? If it is not a Hopf algebra this might not be so canonical. Or do you view $V \otimes W$ as a module over the enveloping algebra $A \otimes A^{op}$ of $A$? | |
| Jun 17, 2023 at 6:46 | history | asked | Nanoputian | CC BY-SA 4.0 |