Timeline for What's the name of an algebra? is it isomorphic to $w_\infty \times w_\infty$?
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| Dec 20, 2010 at 3:49 | comment | added | José Figueroa-O'Farrill | Thanks for this. I think that my initial objection to your question was misguided and I've deleted my comment. Sorry about the noise! I now think that indeed what you wrote down is a Lie algebra. It's the Lie algebra obtained from the associative algebra in Scott's answer via the commutator. | |
| Dec 20, 2010 at 1:26 | comment | added | Jack Cheng | Usually, I think $w_\infty$ is an Lie algebra, though it seems that $w_\infty$ can be viewed as an associative algebra. The algebra spanned by {$z^{\alpha_1}z^{\alpha_2}\partial_{z}^{\beta_1}\partial_{y}^{\beta_2}$} should be Lie algebra. "span" means as a linear basis. | |
| Dec 19, 2010 at 15:34 | comment | added | José Figueroa-O'Farrill | I'm very confused by this question and this answer. In which sense is $w_\infty$ an algebra in this context? I have always understood it as a Lie algebra. It may simply come down to what "span" means in this context. Is it meant as a generating set? or as a linear basis? From your answer it seems you take it to mean the former. Is this standard? | |
| Dec 19, 2010 at 12:25 | vote | accept | Jack Cheng | ||
| Dec 19, 2010 at 8:43 | history | answered | S. Carnahan♦ | CC BY-SA 2.5 |