Rainbow Boomerang Graphs
Shunsuke Hirota
TL;DR
Rainbow Boomerang Graphs develops a unified, graph-theoretic framework for the exchange property underpinning Coxeter groups by introducing rainbow boomerang graphs and embedding the Weyl groupoid formalism of Heckenberger–Yamane. It demonstrates that the exchange property for odd reflections in Lie superalgebras and for Nichols algebras of diagonal type are instances of the same structural principle realized via path subgroupoids and rainbow colorings. The work extends odd Verma's theorem beyond basic Lie superalgebras to regular symmetrizable Kac–Moody Lie superalgebras and to diagonal Nichols algebras, leveraging Lusztig automorphisms of small quantum groups. It also provides a precise combinatorial characterization of the exchange property and connects it to representation-theoretic constructions in braided Hopf algebras and PBW-type bases, offering a cohesive view that unifies several prior, case-specific results.
Abstract
We generalize the well known exchange property of Coxeter groups to the setting of edge-colored graphs. This work aims to unify and extend the results of our companion article, "odd Verma's theorem", which were originally established for basic Lie superalgebras, to the broader setting of regular symmetrizable Kac-Moody Lie superalgebras and Nichols algebras of diagonal type, via the theory of Weyl groupoids in the sense of Heckenberger and Yamane. In particular, we show that the exchange property of odd reflections arises as a special case of the exchange property of Weyl groupoids. To study the exchange property itself, we analyze a class of edge-colored graphs introduced here, called rainbow boomerang graphs, which form an independently natural family of combinatorial objects. We also elaborate on odd Verma theorem in the specific setting of Nichols algebras of diagonal type.