Timeline for How many Lie and associative algebras over a finite field are there?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Sep 8, 2020 at 20:05 | comment | added | Thiago | Ok, I agree that for now explicit formulas are just a dream. | |
| Sep 8, 2020 at 17:02 | comment | added | Qiaochu Yuan | An explicit formula seems quite hopeless except maybe for very small values of $n$, say $n \le 3$? | |
| Sep 8, 2020 at 12:57 | comment | added | YCor | As regards your edit (you want an explicit formula, not just estimates): it's clearly out of reach at the moment. | |
| Sep 8, 2020 at 11:55 | comment | added | Thiago | Yes, I mean algebra over a field - i.e., just a vector space with a bilinear map. | |
| Sep 8, 2020 at 11:50 | history | edited | Thiago | CC BY-SA 4.0 |
edited tags, replaced $n$ by $q$ in the main questions.
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| Sep 8, 2020 at 11:29 | comment | added | LSpice | @SamHopkins, I shouldn't go on too long since I got it wrong, and now understand, but, to me, 'associative' is always part of 'algebra' (unless one explicitly says 'non-associative'), and 'unital' often is, and of course neither of those properties is satisfied for a random $n^3$-tuple. Since the question mentioned varieties of algebras, I figured it came from the point of view of universal algebra, which is the only place I have encountered this much more general definition. Anyway, it was my mistake, and I understand now. | |
| Sep 8, 2020 at 8:07 | history | became hot network question | |||
| Sep 8, 2020 at 3:59 | comment | added | Sam Hopkins | @LSpice: they meant an algebra over a field (en.wikipedia.org/wiki/Algebra_over_a_field), not necessarily associative. I'm not sure how that's related to "universal algebra"? | |
| Sep 8, 2020 at 2:29 | comment | added | LSpice | Oops, it gradually sunk in that you mean "algebra" in the "universal algebra" sense, so that you genuinely meant all of the $n^3$ tuples to define an algebra. (At first I didn't register the word 'Lie' as an option indicating you didn't just want associative algebras ….) But then there's still the issue of whether to distinguish isomorphic algebras. | |
| Sep 8, 2020 at 1:28 | answer | added | Qiaochu Yuan | timeline score: 16 | |
| Sep 8, 2020 at 1:15 | comment | added | LSpice | An algebra is determined by its $n^3$-tuple, but not every $n^3$-tuple determines an algebra, right? Unless I'm off on that, maybe the first question is how many $n$-dimensional algebras there are, full stop. And then, of course, among the ones that are algebras, there will be isomorphisms. Do you want to distinguish isomorphic algebras? | |
| Sep 8, 2020 at 0:14 | comment | added | Sam Hopkins | There's almost certainly no chance of an exact answer here... so you might want to be more specific about what kind of answer you'd like | |
| Sep 7, 2020 at 23:58 | history | asked | Thiago | CC BY-SA 4.0 |