Timeline for Does the tensor algebra $T(V)$ of $V$ isomorphic to the symmetric algebra of the free Lie algebra over $V$?
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| Apr 26, 2020 at 5:36 | history | edited | YCor |
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| Apr 26, 2020 at 5:36 | comment | added | YCor | I think the question is a bit ambiguous: here each $K$ of $T$, $S$ and $\mathrm{Lie}$ is considered as a mapping a graded vector space $W$ to a graded algebra $K(W)$ with $K(W)_1=W$ as graded vector space, and the original vector space $V$ is entirely graded in degree 1 (but this should work without this restriction, namely in the "universal" case when $V$ of dimension $d$ is graded in $\mathbf{Z}^d$ with independent weights. (As asked now, another interpretation would be to put $\mathrm{Lie}(V)$ in degree 1, but then the result fails since $T(V)_1$ is finite-dim but not $S(Lie(V))_1$.) | |
| Jan 20, 2017 at 10:08 | comment | added | Adrien | To complete Qiaochu's comment, the key fact is that $T(V) $ is isomorphic as an Hopf algebra to the enveloping algebra of the free Lie algebra on $V$. | |
| Jan 20, 2017 at 10:02 | comment | added | abx | Is there something like "the free algebra over a vector space"? If you consider $V$ just as a set, the answer to your question is obviously no: $T(V)$ depends of course on the vector space structure. | |
| Jan 20, 2017 at 9:09 | comment | added | Qiaochu Yuan | Yes, by PBW, in characteristic zero. | |
| Jan 20, 2017 at 8:17 | history | edited | Jianrong Li | CC BY-SA 3.0 |
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| Jan 20, 2017 at 8:12 | history | edited | Jianrong Li | CC BY-SA 3.0 |
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| Jan 18, 2017 at 7:53 | history | edited | Jianrong Li | CC BY-SA 3.0 |
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| Jan 18, 2017 at 7:10 | history | asked | Jianrong Li | CC BY-SA 3.0 |