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On the quiver-theoretical quantum Yang-Baxter equation (with an appendix by M. Takeuchi). (English) Zbl 1113.17007

A quiver, also known as an oriented graph, over a set \(\mathcal{P}\) (of “vertices”) is a set \(\mathcal{A}\) of “arrows” equipped with functions \(\mathfrak{s},\mathfrak{e}:\mathcal{A}\to\mathcal{P}\) into the set \(\mathcal{P}\) specifying the “start/source” and the “end” of each arrow in \(\mathcal{A}\). A groupoid \(\mathcal{G}\) with base space \(\mathcal{G}^{(0)}=\mathcal{P}\) is a special example of quiver over \(\mathcal{P}\). Quivers over the same “base” \(\mathcal{P}\) form the collection of objects of a category Quiv\((\mathcal{P})\) with usual morphisms fixing \(\mathcal{P}\), which becomes a monoidal (or tensor) category when endowed with the operation \(\mathcal{A}\otimes\mathcal{B}:=\mathcal{A}_{\mathfrak{e}}\times_{\mathfrak{s}}\mathcal{B}\) defined by \(\mathcal{A}_{\mathfrak{e}} \times_{\mathfrak{s}}\mathcal{B}:=\{(a,b)\in\mathcal{A}\times\mathcal{B}:\mathfrak{e}(a) =\mathfrak{s}( b)\}\) with \(\mathfrak{s}(a,b):=\mathfrak{s}(a)\) and \(\mathfrak{e}(a,b):=\mathfrak{e}(b)\) for quivers \(\mathcal{A},\mathcal{B}\) over \(\mathcal{P}\).
In this paper, the author studies solutions of the quantum Yang-Baxter equation, or equivalently via the flip \(\tau:\mathcal{A}_{\mathfrak{e}}\times_{\mathfrak{s}}\mathcal{A}\to \mathcal{A}_{\mathfrak{s}}\times_{\mathfrak{e}}\mathcal{A}\), the braid equation, in this monoidal category. More precisely, a pair \((\mathcal{A},\sigma)\) is called a braided quiver if \(\sigma:\mathcal{A}_{\mathfrak{e} }\times_{\mathfrak{s}}\mathcal{A}\to\mathcal{A}_{\mathfrak{e}}\times _{\mathfrak{s}}\mathcal{A}\) is a quiver isomorphism satisfying the braid equation \((\sigma\times\text{id})(\text{id}\times\sigma)(\sigma\times\text{id}) =( \text{id}\times\sigma)(\sigma\times\text{id})(\text{id}\times\sigma)\) on \(\mathcal{A} _{\mathfrak{e}}\times_{\mathfrak{s}}\mathcal{A}_{\mathfrak{e}}\times_{\mathfrak{s}} \mathcal{A}\).
Results of Etingof-Schedler-Soloviev, Lu-Yan-Zhu, and Takeuchi on the set-theoretical quantum Yang-Baxter equation are generalized to the context of quivers, with groupoids playing the role of groups. The concept of matched pair of groups is generalized to groupoids, and a groupoid \(\mathcal{G}\) with left action \(\rightharpoonup\) and right actions \(\leftharpoonup\) on itself is called a braided groupoid if \((\mathcal{G},\mathcal{G})\) with these left and right actions is a matched pair of groupoids. Braided groupoids \((\mathcal{G},\rightharpoonup,\leftharpoonup)\) are braided quivers \((\mathcal{G},\sigma)\) with \(\sigma:\mathcal{G}_{\mathfrak{e}} \times_{\mathfrak{s}}\mathcal{G}\to\mathcal{G}_{\mathfrak{e}}\times_{\mathfrak{s} }\mathcal{G}\) defined by \(\sigma(f,g)=(f\rightharpoonup g,f\leftharpoonup g)\). For a groupoid \(\mathcal{G}\) over \(\mathcal{P} \), the braided structures on \(\mathcal{G}\) are in one-to-one correspondence with bijective 1-cocycles \(\pi:\mathcal{G}\to\mathcal{N}\) where \(\mathcal{N}\) is a group bundle over \(\mathcal{P}\) with a right \(\mathcal{G} \)-action by group bundle automorphisms.
The structure groupoid \(\mathbb{G} _{\mathcal{A}}\) of a non-degenerate braided quiver \((\mathcal{A},\sigma)\), defined as the groupoid generated by elements of \(\mathcal{A}\) with relations \(xy=(x\rightharpoonup y)(x\leftharpoonup y)\) for \((x,y)\in\mathcal{A}_{\mathfrak{e}}\times_{\mathfrak{s}}\mathcal{A}\), is shown to be a braided groupoid. Non-degenerate braided quivers are shown to be in a bijective correspondence with structural pairs which are defined as certain representations of suitable matched pairs of groupoids. By linearization, some star-triangular face models are constructed and realized as modules over quasitriangular quantum groupoids introduced by Aguiar, Natale, and the author. At the end of this paper, there is an appendix by Takeuchi, showing a version of the FRT-construction for matched pairs of groupoids.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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