Ribbon Braided Module Categories, Quantum Symmetric Pairs and Knizhnik–Zamolodchikov Equations

Abstract

Let u be a compact semisimple Lie algebra, and σ be a Lie algebra involution of u. Let Rep q (u) be the ribbon braided tensor C * -category of admissible Uq(u)-representations for 0 < q < 1. We introduce three module C * -categories over Rep q (u) starting from the input data (u, σ). The first construction is based on the theory of 2-cyclotomic KZ-equations. The second construction uses the notion of quantum symmetric pair as developed by G. Letzter. The third construction uses a variation of Drinfeld twisting. In all three cases the module C * -category is ribbon twist-braided in the sense of A. Brochier-this is essentially due to B. Enriquez in the first case, is proved by S. Kolb in the second case, and is closely related to work of J. Donin, P. Kulish, and A. Mudrov in the third case. We formulate a conjecture concerning equivalence of these ribbon twist-braided module C * -categories, and confirm it in the rank one case.

Quantum flag manifolds, quantum symmetric spaces and their associated universal K-matrices

“…We also mention that in a more restricted setting, constructions closely related to the ones in this paper were performed in [15]. Finally, we mention that this paper is a step towards proving part of the conjecture posed as [17,Conjecture 4.1]. However, to prove this part of the conjecture completely, one needs to complement the results of this paper with more refined representationtheoretic results.…”

“…Proof. By [17,Theorem 3.11], we need to check that q 2(ω 0 ,α r +α τ (r) ) c r c τ (r) = q (Θ(α r )−α r ,α τ (r) ) . (4.17)…”

“…Note that this theorem was proven under a different assumption on z mentioned in Remark 4.10, but it is easily verified that for the * -invariance of B the only feature of z which was used was (4.1) and the fact that z r = 1 for α r ⊥ X, which is still valid in the current setup. Following the discussion under [17,Theorem 3.14], it is sufficient to show that (ω 0 , α r ) = 0 for r ∈ X, and…”

Stokes phenomenon and reflection equations

“…For a local picture, the theorem implies that the type B braid group representation, as the monodromy representation ρ(u) of generalized cyclotomic KZ equations (see Section (2)), for different u are equivalent. In a categorical setting, see e.g., [68,69,29,15,22] for various versions, the theorem implies that the τ -braided module category Rep(k) over (the braided monoidal category) Rep(g), constructed from S(u) and K + (u) with different u are equivalent.…”

“…The cyclotomic KZ equation, following Leibman [54], Golubeva-Leksin [43], Enriquez-Etingof [31,Section 4.2], is designed to incorporate various automorphisms on Lie algebras. When the automorphism is simply an involution, the relation between its monodromy and quantum symmetric pairs, has been studied by many authors, see e.g., Enriquez [29], De Commer-Neshveyev-Tuset-Yamashita [22], to some extent, generalizing the works of Drinfeld and Kohno. In this paper, we will consider cyclotomic KZ equations coupled with extra irregular singularities. They can also be seen as cyclotomic analog of the gKZ equations, and thus are called generalized cyclotomic KZ (gcKZ) equation.…”