# The "semi-symmetric" algebra of a vector space

If $$V$$ is a vector space over a field $$K$$, then the symmetric algebra $$S(V)$$ is defined as the tensor algebra $$T(V)$$ factorized by the two-sided ideal generated by $$x\otimes y-y\otimes x$$, with $$x,y\in V$$. The homogeneous component of degree $$n$$ of $$S(V)$$ is $$S^n(V)=T^n(V)/I_n$$, where $$I_n$$ is the subspace of $$T^n(V)$$ generated by $$x_1\otimes\cdots\otimes x_n-x_{\sigma (1)}\otimes\cdots\otimes x_{\sigma (n)}$$, where $$x_1,\ldots,x_n\in V$$ and $$\sigma\in S_n$$.

What I'm interested are the spaces $$S'^n(V):=T^n(V)/I'_n$$, where $$I'_n$$ is generated only by expressions $$x_1\otimes\cdots\otimes x_n-x_{\sigma (1)}\otimes\cdots\otimes x_{\sigma (n)}$$ with $$\sigma\in A_n$$. Alternatively, we may regard $$S'^n(V)$$ as the homogeneous component of degree $$n$$ of the algebra $$S'(V)=T(V)/I'$$, where $$I'$$ is the two-sided ideal of $$T(V)$$ generated by $$x\otimes y\otimes z-y\otimes z\otimes x$$, with $$x,y,z\in V$$. (It is because $$A_n$$ is generated by the cyclic permutations $$(i,i+1,i+2)$$ with $$1\leq i\leq n-2$$.) We may call $$S'(V)$$ the "semi-symmetric algebra of $$V$$".

My question is, is this object already known? Maybe it was introduced by somebody else under other name or other notation. I need it in a paper I'm writing and, if possible, I'd rather quote the definition and the properties of $$S'(V)$$ than write them myself.

• The same question on Mathematics: The “semi-symmetric” algebra of a vector space. You can find some advice on cross-posting, for example, in this answer. May 3, 2019 at 22:23
• Sorry, I'm new to this and I didn't know that posting the same thing on math.stackexchange and mathoverflow is considered cross-posting. I thought they are separate things for different audiences (math.stackexchange for general public, mathoverflow forresearchers). May 4, 2019 at 13:34
• As far as I can tell, it is not really that problematic (especially if you get no response on one of the site for some time).However, as you can see in the linked post on MathOverflow Meta and on Mathematics Meta it is recommended to link each post to the other copy. (I wish I was able to add also something about the problem rather than just technicalities like this - let's hope you get the answer from somebody more knowledgeable.) May 4, 2019 at 13:44

I posted a short article on arxiv. In the second section I speak about the algebra S'(V). However, the main result is in the first section, and it deals with the kernel of the projection map $$T(V)\to S(V)$$. I posted a question about that on mathoverflow here: A description of the kernel of projection map from the tensor algebra to the symmetric algebra $T(V)\to S(V)$

Thea article on arxiv can be found here: https://arxiv.org/abs/1912.03515

Again, if anybody saw any of these results or something similar, then please let me know.