A description of the kernel of projection map from the tensor algebra to the symmetric algebra $T(V)\to S(V)$

Question

Let $V$ be a vector space over some arbitrary field. Let $T(V)$ and $S(V)$ be the tensor and symmetric algebras over $V$. We have the projection map $T(V)\to S(V)$, given by $x_1\otimes\cdots\otimes x_n\mapsto x_1\cdots x_n$.

I'm interested in the kernel of this map. More precisely, I want an explicit object $M(V)$, in terms of generators and relations, such that we have an exact sequence $$0\to M(V)\to T(V)\to S(V)\to 0.$$

I did find such an object, a $T(V)$-bilagebra which is an explicit quotient of the bialgebra generated by $\Lambda^2(V)$, i.e. $T(V)\otimes\Lambda^2(V)\otimes T(V)$.

I wonder if somebody saw this somewhere. (With a different notation, not necessarily $M(V)$.) After all, it is a natural question, somebody might have done it before.

I need this result in a paper I'm writing and I'd rather quote it than write the proof myself. The proof is not that obvious and it's not short either. (I guess it's two pages or more.)

The precise result is the following:

$\bf Definition$ We define the $T(V)$-bimodule $M(V)=(T(V)\otimes\Lambda^2(V)\otimes T(V))/W(V)$, where $W(V)$ is the subbimodule of $T(V)\otimes\Lambda^2(V)\otimes T(V)$ generated by $f(x,y,z)$, with $x,y,z\in V$, where $$f(x,y,z)=x\wedge y\otimes z+y\wedge z\otimes x+z\wedge x\otimes y-x\otimes y\wedge z-y\otimes z\wedge x-z\otimes x\wedge y,$$ and the expressions $$[x,y]\otimes\xi\otimes z\wedge t-x\wedge y\otimes\xi\otimes [z,t],$$ with $x,y,z,t\in V$ and $\xi\in T(V)$.

$\bf Theorem$ We have an exact sequence $$0\to M(V)\to T(V)\to S(V)\to 0,$$ where the left map is the morphism of $T(V)$-bimodules given on generators by $x\wedge y\mapsto [x,y]$

(By $[x,y]$ we mean the commutator, $[x,y]=x\otimes y-y\otimes x$.)

Answer 1

The proof turned out to be longer than expected, so I decided to write it in a separate paper. You may find it on arxiv here: https://arxiv.org/abs/1912.03515

The second section deals with the "semi-symmetric" algebra $S'(V)$. I asked a question about it on May 3. See here: The "semi-symmetric" algebra of a vector space

I need both these results as a prerequisite for a future paper. Again, if anybody saw any of these results, or something similar, then please let me know.

Answer 2

I suspect that the best way to organise the information that you need will be via the Poincaré-Birkhoff-Witt theorem: the tensor algebra $TV$ is the universal enveloping algebra $ULV$ of the free Lie algebra $LV$ generated by $V$, there is a natural filtration of $TV$ whose associated graded ring is the symmetric algebra $SLV$. We can split $LV$ as $V\oplus \lambda^2 V \oplus L_{\geq 3}V$ and then $SLV=SV\otimes S(\lambda^2 V)\otimes S(L_{\geq 3}V)$.