Abstract
We construct dynamical Yang-Baxter maps, which are set-theoretical solutions to a version of the quantum dynamical Yang-Baxter equation, by means of homogeneous pre-systems, that is, ternary systems encoded in the reductive homogeneous space satisfying suitable conditions. Moreover, a characterization of these dynamical Yang- Baxter maps is presented.
Highlights
The quantum dynamical Yang-Baxter equation (QDYBE for short) [9,10], a generalization of the quantum YangBaxter equation (QYBE for short) [2,3,40,41], has been studied extensively in recent years
A map R(λ) : X × X → X × X (λ ∈ H) is a dynamical Yang-Baxter map associated with H, X and (·), if and only if, for every λ ∈ H, R(λ) satisfies the following equation on X × X × X: R23(λ)R13 λ · X(2) R12(λ) = R12 λ · X(3) R13(λ)R23 λ · X(1)
R12(λ), R12(λ · X(3)), R23(λ · X(1)), and others are the maps from X × X × X to itself defined as follows: for u, v, w ∈ X, R12(λ)(u, v, w) = R(λ)(u, v), w, R12 λ · X(3) (u, v, w) = R12(λ · w)(u, v, w), R23 λ · X(1) (u, v, w) = u, R(λ · u)(v, w)
Summary
The quantum dynamical Yang-Baxter equation (QDYBE for short) [9,10], a generalization of the quantum YangBaxter equation (QYBE for short) [2,3,40,41], has been studied extensively in recent years (see [7] and the references therein). Each triple (L, M, π) consisting of a left quasigroup L = (L, ·) (Definition 1(1)), a ternary system M satisfying (2.2) and (2.3), and a (set-theoretic) bijection π : L → M gives a dynamical Yang-Baxter map R(λ) associated with L, L and (·) (see Section 2 for more details). The aim of this paper is to produce the dynamical Yang-Baxter maps by means of homogeneous pre-systems, which generalize the homogeneous system. We focus on its construction by means of the ternary system This construction yields a category A concerning the ternary systems, which is equivalent to a category D consisting of the dynamical Yang-Baxter maps. Every homogeneous pre-system satisfying (4.1) can produce a dynamical Yang-Baxter map via the ternary system. Our viewpoint sheds some light on the relation between geometry and the dynamical Yang-Baxter map
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