Timeline for Do non-associative objects have a natural notion of representation?
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5 events
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| Jun 8, 2022 at 14:11 | comment | added | Vladimir Dotsenko | @LSpice in fact, what you and Qiaochu Yuan are discussing is that in the case of Lie algebras there is a very remarkable coincidence of two different universal enveloping algebras. There is the universal "multiplicative" enveloping algebra that exists for any type of algebras, as in my answer to this question dating from 2010, and the universal enveloping algebra which is the left adjoint to the "forgetful functor" from associative algebras to Lie algebras which only remembers the antisymmetrized operation $AB-BA$; this is certainly something special that does not exist in general. | |
| Feb 25, 2020 at 0:43 | comment | added | LSpice | @QiaochuYuan, I think it only looks like the universal enveloping algebra privileges the binary operation $A B - B A$ if you think of your Lie algebra as coming from an associative algebra; otherwise it privileges just the abstract (non-associative) multiplication of the Lie algebra, which is what a representation should do, shouldn't it? | |
| Apr 12, 2010 at 20:23 | comment | added | Adam Gal | My guess would be then that representations in a vector space in such a case would be meaningless, and you would need to search for some other structure. | |
| Apr 12, 2010 at 20:20 | comment | added | Qiaochu Yuan | The construction of the universal enveloping algebra privileges the bilinear operation AB - BA; my guess is that this operation isn't generic enough to really capture the behavior of an algebra that is very far from being associative, e.g. many interesting non-associative algebras might collapse. | |
| Apr 12, 2010 at 20:18 | history | answered | Adam Gal | CC BY-SA 2.5 |