Covariant Bimodule Connections and Noncommutative Geometric Yang-Baxter Solutions

Abstract

We introduce a new class of Yang--Baxter solutions, which we call \emph{noncommutative geometric solutions}, constructed from quantum group noncommutative geometries. These solutions are built from a quantum homogeneous space, endowed with a covariant differential graded algebra and a noncommutative Koszul connection $\nabla$. If $\nabla$ is a bimodule connection, that is, if its generalised flip map $\sigma$ is well defined, then a simple set of sufficient conditions for $\sigma$ to be a Yang--Baxter solution is given. These results are applied to the cominuscule quantum flag manifolds ${\mathcal{O}}_q(G/L_S)$ endowed with their Heckenberger--Kolb calculi. The noncommutative complex geometry of the calculi decomposes their Yang--Baxter operator $\sigma$ into three parts: holomorphic, anti-holomorphic, and cross-type. For the non-exceptional examples, the holomorphic and anti-holomorphic solutions recover, up to scalar multiple, the usual quasitriangular braidings of the associated Drinfeld--Jimbo $U_q({\mathfrak{l}}_S^s)$-modules. However, we get a new extension of these braidings to the direct sum, which remarkably restricts to an involutive operator on the cross terms. As an application, we produce a simple uniform presentation of the degree two relations of the Heckenberger--Kolb calculi as a \emph{quantum exterior algebra} associated to the generalised flip map $\sigma$.

arXiv commentComplete rewrite, including an extension from the quantum Grassmannians to all cominuscule quantum flag manifolds, and a proof that the bimodule map of our bimodule connection is a Yang--Baxter solution