Noncommutative differential geometry related to the Yang-Baxter equation. (Q1808231)

Let \(V\) be a vector space over a (not necessarily commutative) ring \(k\), and consider the ``symmetry'' operator \(S\colon V^{\otimes 2}\to V^{\otimes 2}\), \(S(e_i\otimes e_j)=S^{kl}_{ij} e_k\otimes e_l\), \(e_i\in V\) (with summation over repeated indices), satisfying the Yang-Baxter equation \(S^{12}S^{23}S^{12}=S^{23}S^{12}S^{23}\). If there is a ``column-wise'' inverse \(T\) to \(S\), such that \(S^{kl}_{ij}T^{jq}_{lp}=\delta^q_i\delta^k_p\), then \(S\) is said to be closed. The authors first show how to extend \(S\) to the rigid tensor category \(\mathfrak A(S)\) of vector spaces with symmetries. They then discuss the classification of symmetries (especially so-called Hecke symmetries, defined by \((S-q\cdot\text{id})(S+\text{id})=0\)). Suppose \(V\) is a (noncommutative) ring \(A\) over a field \(k\), with multiplication \(\circ\colon A^{\otimes 2}\to A\) and a symmetry \(S\colon A^{\otimes 2}\to A^{\otimes 2}\) with respect to which \(\circ\) is \(S\)-invariant. \(A\) is said to be \(S\)-commutative if \(\circ\,S=\circ\) (i.e., \(f\circ g=\circ\,S(f\otimes g)\) for \(f,g\in A\)). Define the space of left derivations \(\text{Der}(A)\) as the elements of \(\text{Hom}_k(A,A)\) that satisfy the obvious \(S\)-Leibniz rule; this space is \(S\)-invariant and is a left \(A\)-module. If \(A\) is an associative \(S\)-algebra, the operation \([X,Y]=X\circ Y-\circ\,S(X\otimes Y)\) turns it into an \(S\)-Lie algebra. Requiring that \(S\) be a unitary symmetry, the authors discuss exterior products on an \(S\)-commutative ring \(A\) (over a field \(k\)) and prove many of the properties of this product. At the end they give some examples, including an example of a sheaf of rings in which local solutions of the quantum Yang-Baxter equation are defined but which does not possess a global section.