Recall that the (first) Weyl algebra is the algebra generated by $x,y$ with the relation $xy-yx=1$. It can be realized as the algebra of differential operators on $k[x]$, where one generator acts as multiplication by $x$ and the other acts as $\frac{\partial}{\partial x}$.
According to I. A. Dynnikov, V. A. Shastin, “On independence of some pseudocharacters on braid groups”, Algebra i Analiz, 24:6 (2012), 21–41, some notable operators in the spaces of quasimorphisms of braid groups satisfy the identity $xy-yx=1$. I followed their computations and noticed that the arguments rely only on some elementary relations concerning the strand deletions and the standard embeddings between the braid groups. As a result, I came up with the following version of this relation.
Recall that a simplicial group is a collection of groups $(G_n)$ and group homomorphisms $d_0,\ldots,d_n\colon G_n \to G_{n-1}$ and $s_0,\ldots,s_n\colon G_n\to G_{n+1}$ satisfying the so-called simplicial identities. Among them there are $d_{i+1}\circ s_0 = s_0\circ d_i$ (for $i>0$) and $d_1\circ s_0={\rm id}$. They imply the identity $$(d_1+...+d_{n+1})\circ s_0 - s_0 \circ (d_1+....+d_n) = {\rm id}$$ in the following sense. Define linear maps $R\colon k[G_n] \to k[G_{n+1}]$ and $T\colon k[G_n] \to k[G_{n-1}]$ of the group rings by $R(x):=s_0(x)$ and $T(x) := d_1(x)+\ldots+d_n(x)$. Then $T\circ R -R \circ T={\rm id}$.
My question is whether such an interplay is well known and whether there are studies that elaborate on this idea that simplicial identities imply the Weyl algebra ones.
