Let $V$ be a $\mathbb{Z}$-graded vector space over an algebraically closed field $k$ of characteristic zero.

Let $\overline{TV}= \bigoplus_{n=1}^{\infty} V^{\otimes n}$ be the reduced tensor algebra. The degree of an element $v_1\otimes \dots \otimes v_k$ is defined as $\sum_i \operatorname{deg}(v_i)$.

Suppose that $d$ is a differential of degree $-1$ on $\overline{TV}$ which respects the (graded) Leibniz rule.

Let $\tau: \overline{TV} \to \overline{TV}$ be the cyclic permutation operator defined on pure tensors by $$\tau(v_1\otimes \dots \otimes v_k)= (-1)^{\operatorname{deg}(v_k)(\sum_1^{k-1} \operatorname{deg}v_i)} v_k \otimes v_1 \otimes \dots \otimes v_{k-1}.$$

We can consider the module of co-invariants $\overline{TV}^{\tau}:= \overline{TV}/ (1- \tau)$, and observe that $d$ passes to the quotient.

Question: What does the homology of $(\overline{TV}^{\tau}, d)$ compute? If $d=0$, this seems to agree with cyclic homology of $\overline{TV}$, but I don't see what it is in general. I assume that this is something well known.


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