For any \(k\)-linear category \(\mathcal C\) with zero object, we write \(\mathrm{Ch}^b(\mathcal C)\) for the \(k\)-linear category of bounded (on both sides) chain complexes in \(\mathcal C\), with chain maps as morphisms. If \(\mathcal C\) is additive (and thus has a zero object) or equipped with a \(\mathbb{Z}\)-action, then so is \(\mathrm{Ch}^b(\mathcal C)\). If \(\mathcal C\) is additive and equipped with a monoidal structure compatible with \(\oplus\), then this is inherited by \(\mathrm{Ch}^b(\mathcal C)\). There is a natural notion of homotopy between chain maps and the nullhomotopic chain maps form a \(k\)-linear (monoidal) ideal.
The quotient of \(\mathrm{Ch}^b(\mathcal C)\) by the nullhomotopic chain maps is the chain homotopy category\(\mathrm{K}^b(\mathcal C)\). An isomorphism between objects of \(\mathrm{K}^b(\mathcal C)\) is called a chain homotopy equivalence. 13
For \(n\geq 2\) we denote by \(\operatorname{Br}_n\) the braid group with (Artin) generators \(\sigma_i\), \(1\leq i\leq n-1\). Given a generator or its inverse, we will consider the following complexes in \(\mathrm{Ch}^b(\mathrm{Sbim}_n)\): Here the part is in homological degree zero, \(m\) is induced by the multiplication map \(B_i=R \otimes_{R^{s_i}} R \langle
-1 \rangle \rightarrow R\langle-1\rangle\), and \(\Delta\) is the bimodule map determined by \(1 \mapsto x_i\otimes 1 - 1\otimes x_{i+1}\). An expression \(\underline{\beta}=\sigma_{i_1}^{\epsilon_1}\cdots\sigma_{i_r}^{\epsilon_r}\) with \(\epsilon_j\in\{\pm\}\) is called a braid word with corresponding braid element \(\beta\in \operatorname{Br}_n\). The word is positive if \(\epsilon_j=1\) for \(1\leq j\leq r\). Given \(\underline{\beta}\) define \[
F(\underline{\beta})
:=
F(\sigma_{i_1}^{\epsilon_1}) \circ_1\cdots \circ_1F(\sigma_{i_r}^{\epsilon_r})\] where we make use of the horizontal composition \(\mathrm{Ch}^b(\mathrm{Sbim}_n)\) (given by the obvious extension of \(\circ_1=\otimes_R\)). By convention, the empty braid word gives \(F(\emptyset)=R\).
The complexes \(F(\underline{\beta})\) are called Rouquier complexes. They were first thoroughly studied by Rouquier who proved in [Rou06] that, up to canonical homotopy equivalence, these complexes are independent of the chosen braid word representing \(\beta\). More precisely the following holds:
Let \(\underline{\beta}_1\) and \(\underline{\beta}_2\) be braid words representing the same braid \(\beta\), then there exist homotopy equivalences \[\psi_{\underline{\beta}_1,\,\underline{\beta}_2}\colon F(\underline{\beta}_1) \rightarrow F(\underline{\beta}_2)\] which form a transitive system, i.e. if \(\underline{\beta}_3\) is a third braid word representing the same braid, then \[\psi_{\underline{\beta}_2,\,\underline{\beta}_3}\circ_2\psi_{\underline{\beta}_1,\,\underline{\beta}_2} \sim\psi_{\underline{\beta}_1,\,\underline{\beta}_3}.\] If moreover \(\underline{\beta}_1'\), \(\underline{\beta}_2'\) are braid words representing another braid \(\beta'\) then we have \[
\psi_{\underline{\beta}_1\underline{\beta}_1',\underline{\beta}_2\underline{\beta}_2'} \sim\psi_{\underline{\beta}_1,\underline{\beta}_2}\circ_1\psi_{\underline{\beta}_1',\underline{\beta}_2'}.\]
This rephrasing of Rouquier’s results from [Rou06] is a slight strengthening of [EH17, Prop. 2.19]. As a consequence we may abuse notation and write \(F(\beta)\) instead of \(F(\underline{\beta})\).
For \(n \geq 2\) and \(R=R_n\) as above and \(w\in S_n\) we let \(R_{\circlearrowleft w}\) denote the graded \(R\)-bimodule which is isomorphic to \(R\) as left \(R\)-module and with right-action twisted by \(w\): i.e. \(r\in R\) acts on \(R_w\) from the right as multiplication by \(w(r)\). We emphasize that for non-trivial \(w\), this \(R_n\)-bimodule \(R_{\circlearrowleft w}\) is not an object of \(\mathrm{Sbim}_n\).
However, for \(1\leq i\leq n-1\), the bimodule morphism \(R_{\circlearrowleft s_i}\langle
1\rangle \rightarrow B_i\) determined by \(1\mapsto x_i\otimes 1 - 1\otimes x_i\) induces a quasi-isomorphism \(R_{\circlearrowleft s_i}\langle 1\rangle \rightarrow F(\sigma_i)\). Likewise, the multiplication map \(B_i \rightarrow R_{\circlearrowleft s_i}\langle -1 \rangle\) determined by \(1\otimes 1 \mapsto 1\) induces a quasi-isomorphism \(F(\sigma^{-1})\rightarrow R_{\circlearrowleft s_i}\langle -1\rangle\).
Up to a grading shift, the generating Rouquier complexes, and more generally, the Rouquier complexes of positive resp. negative permutation braids, can hence be identified with permutation bimodules upon proceeding to the derived category \(\mathrm{D}^b({}_R\mathrm{grbmod}_{R})\) of graded \(R\)-\(R\)-bimodules. To obtain an interesting (non-symmetric) braiding, it is thus essential to work up-to-chain-homotopy, rather than up-to-quasi-isomorphism. Nevertheless, the comparison with permutation bimodules is important in this paper and the grading shifts in the following definition are motivated by it.
For \(m,n\geq 0\), we define the braiding complexes: \[\begin{aligned}
X_{m,n}&:=F( (\sigma_{n}\cdots\sigma_{1})\cdots (\sigma_{i+n-1}\cdots\sigma_{i}) \cdots (\sigma_{m+n-1}\cdots\sigma_{m}) )\langle -m n \rangle\\
X'_{m,n}&:=F( (\sigma^{-1}_{n}\cdots\sigma^{-1}_{m+n-1}) \cdots (\sigma^{-1}_{i}\cdots\sigma^{-1}_{i+m-1}) \cdots (\sigma^{-1}_{1}\cdots\sigma^{-1}_{m}) )\langle m n \rangle
\end{aligned}\] The braiding complexes \(X_{m,n}\) (resp. \(X'_{m,n}\)) will be called positive (resp. negative) cabled crossings complexes or just cabled crossings, since the underlying braids are cabled crossings.
The braids appearing in the special cabled crossings \(X_{m,1}\), \(X'_{m,1}\), \(X_{1,n}\), and \(X'_{1,n}\) will be called Coxeter braids, since they are braid versions of Coxeter words, see the illustrations in Figure [0016] (as for functions, we compose functors and read diagrams from right to left. The Artin generator \(\sigma_{i}\) acts on the \(i\)-th and \((i+1)\)-th strand from the bottom.).
Cabled crossings \(X_{2,3}\), \(X'_{3,2}\), and Coxeter braids \(X_{1,4}\), \(X'_{1,4}\).[0017]
