# 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 $$S'^n(V)$$ than write them myself.