Morita invariants of quasitriangular coideal subalgebras
Monique Müller, Chelsea Walton
TL;DR
This work develops Morita invariants for quasitriangular coideal subalgebras by constructing representations of braid groups of Coxeter types $BC$ and $D$ from braided module categories. The main results show that braided Morita equivalence of quasitriangular left coideal subalgebras $B$ and $B'$ implies the equivalence of their $Br_n^{BC}$- and $Br_n^{D}$-representations for all $n\ge2$, yielding computable invariants; it also provides explicit $K$-matrix classifications in the finite group algebra and Sweedler $H_4$ cases. The approach extends type-$A$ invariants to new Coxeter types, leveraging Andruskiewitsch–Mombelli and Skryabin machinery to relate module-category Morita theory to braiding data. Concrete classifications illustrate how these invariants distinguish braided Morita classes and connect to boundary Yang–Baxter (reflection) equations via $K$-matrices. Overall, the results deepen categorical symmetry phenomena in braided settings and offer tools for distinguishing braided Morita classes in modular and braided fusion contexts, with explicit examples and open directions for further generalization.
Abstract
We use representations of braid groups of Coxeter types BC and D to produce invariants of representation categories of quasitriangular coideal subalgebras. Such categories form a prevalent class of braided module categories. This is analogous to how representations of braid groups of Coxeter type A produce invariants of representation categories of quasitriangular Hopf algebras, a prevalent class of braided monoidal categories. This work also includes concrete examples, and classification results for $K$-matrices of quasitriangular coideal subalgebras.