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Let $k$ be a commutative ring and $L$ a $k$-module. The tensor algebra $\otimes L$ is $\mathbb{Z}$-graded and $\mathbb{Z}_2$-graded (an element of $L^{\otimes n}$ has degree $n$ and $\mathbb{Z}_2$-degree $n\mod 2$), hence it is a superalgebra over $k$. This canonically induces a supercommutator $\left[\cdot,\cdot \right]_{\mathrm{s}}$ on $\otimes L,$ which is simply


for any $U\in L^{\otimes n}$ and any $V\in L^{\otimes m}$.

Define a map $T:\otimes L\to \otimes L$ as

$$T\left(u_1\otimes u_2\otimes \ldots\otimes u_k\right) = \sum\limits_{i=1}^{k} \left(-1\right)^i u_i \otimes u_1 \otimes u_2 \otimes \ldots \otimes u_{i-1} \otimes u_{i+1} \otimes \ldots \otimes u_k$$

(this is for the pure tensors; for the rest, just continue this by linearity). It is easy to see that $L^{\otimes 0}\subseteq \mathrm{Ker} T$ and that

$$\mathrm{Ker} T\cdot \mathrm{Ker} T\subseteq \mathrm{Ker} T$$

(where multiplication is the multiplication in the tensor algebra $\otimes L$), so that $\mathrm{Ker} T$ is a subalgebra of $\otimes L$. (Thus, in particular, $\left[\mathrm{Ker} T,\mathrm{Ker} T\right]_{\mathrm{s}}\subseteq \mathrm{Ker} T$.) Also, $\left[L, \mathrm{Ker} T\right]_{\mathrm{s}}\subseteq \mathrm{Ker} T$.

Consequently, by induction, any nontrivial tree of supercommutator brackets decorated by elements of $L$ must evaluate to an element of $\mathrm{Ker} T$, and so must any tensor product of such trees (including empty products). I am wondering: do these generate (over $k$) all of $\mathrm{Ker} T$ or is there more? I am mostly interested in the case when the characteristic of $K$ is zero: if it is $2$, then there is surely more.

Here is a bit of motivation (I said this is a curiousity question, but in fact the original curiousity question was about bilinear forms):

Let $f:L\times L\to k$ be a bilinear form. We define a bilinear map $U:L\times\left( \otimes L\right)\to \otimes L$ by $U\left(u,v_1\otimes v_2\otimes ...\otimes v_k\right)=\sum\limits_{i=1}^k\left(-1\right)^{i-1}f\left(u,v_i\right)\cdot v_1\otimes v_2\otimes ...\otimes \hat{v_i}\otimes \ldots\otimes v_k$, where $\hat{v_i}$ means that the factor $v_i$ is omitted from the product. (Of course, we have just defined $U\left(u,V\right)$ for pure tensors $V$ only, but the rest is clear by linearity.)

This bilinear map $U$ is rather natural; people use to denote the similarly defined map $L\times\left( \wedge L\right)\to \wedge L$ (where all $\otimes$ signs have been replaced by $\wedge$ signs) as the "interior product" (the "exterior product" is just the wedge product) - best known in the context of differential forms.

Now I'm wondering what tensors $V\in \otimes L$ satisfy ($U\left(u,V\right)=0$ for every $u\in L$ and every bilinear form $f$). This is equivalent to $V\in\mathrm{Ker} T$.

On a related matter, how can I make a computer do this kind of algebra for me, say on $L^{\otimes 7}$ ? Are there Haskell/ML libraries for tensor algebras or is there CAS software for this? (I am aware that my questions are reducible to representation theory of $S_n$.) A link (preferrably with a short tutorial) would be very appreciated. Thanks!

share|improve this question
You need to enclose complicated TeX in backticks, just as the sidebar "How to write math" instructs you to do. I've fixed it and made a couple of style/nomenclature changes, I hope it's all right. –  Victor Protsak Aug 12 '10 at 7:46
Thanks a lot! I tried enclosing some signs in backticks, but I didn't think of enclosing just everything. –  darij grinberg Aug 12 '10 at 9:34

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