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.


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