When I was a very little hare, a big grey wolf told me about the following skew polynomial algebra, which I never understood. My question is whether the following construction is a part of some bigger abstract construction and whether it is been written anywhere.

The skewing depends on a sequence of polynomials $f_i(t)\in K[t]$ satisfying $$f_i(t)f_j(t-j)=f_{i+j}(t-j)$$ for all $i$ and $j$. Something like $f_i(t)=(-1)^i \prod_{k=0}^{i-1} (t+k)$ is an example. I denote $f_i^+=f_i$ and $f_i^-=f_i(t-i)$ and $f^\epsilon_i=1$ if $\epsilon=0$ or $i<1$. Now I introduce symbol $[a,b]=-sgn(ab)min(|a|,|b|)$ for all integers. Given any ${\mathbb Z}$-graded associative $K$-algebra, I can define an algebra structure on $R[t]=R\otimes_K K[t]$ by the formula $$ r_a \otimes F(t) \cdot s_b \otimes G(t) = r_as_b \otimes f^{sgn(b)}_{[a,b]} (t-b) F(t-b) G(t) $$ for homogeneous $r_a\in R_a$ and $s_b\in S_b$. I spare you from the case-by-case proof of associativity that I know but one can clearly see some funny braiding as $F(t)$ and $s_b$ commute. This braiding seems to live on the edge of the usual and tropical worlds and I could never understand where it came from.