Hopf algebroids and rigid tensor categories associated with dynamical Yang–Baxter maps

Quantum Groups for Restricted SOS Models

“…They are based on a modification of the Yang-Baxter equation, nowadays called the dynamical Yang-Baxter equation, see (1.2) below, which had previously been found by Gervais and Neveu in their study of the exchange relations of vertex operators in the Liouville conformal field theory [26]. The dynamical Yang-Baxter equation reappeared in various contexts since, e.g., [1,9,10,14,15,29,32,42,43]. A recent textbook on elliptic quantum groups is [36].…”

Dynamical Yang-Baxter maps and Hopf algebroids associated with s-sets

“…Table 1: The binary operation on Q 5 [30]. This homogeneous pre-system (S, η) satisfying η(x, y, z) = η(w, η(x, y, w), z) (∀x, y, z, w ∈ S),…”

“…In a similar way, suitable dynamical Yang-Baxter maps [26,28], settheoretic solutions to a version of the quantum dynamical Yang-Baxter equation (see Remark 2.5), produce Hopf algebroids [30,31]. This Hopf algebroid implies two rigid tensor categories, one of which is a category consisting of finite-dimensional L-operators.…”

“…The organization of this paper is as follows. In Section 2, we give a brief exposition of the homogeneous pre-system, the dynamical Yang-Baxter map, the Hopf algebroid, and the rigid tensor category [13,30]. Section 3 discusses the construction of ternary operations by means of the s-sets.…”

“…In this section, we summarize without proofs the relevant material on homogeneous pre-systems, dynamical Yang-Baxter maps, Hopf algebroids, and rigid tensor categories [13,30], to render this paper as self-contained as possible.…”