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By means of left quasigroups L = (L, ·) and ternary systems, we construct dynamical Yang-Baxter maps associated with L, L, and (·) satisfying an invariance condition that the binary operation (·) of the left quasigroup L defines. Conversely, this construction characterize such dynamical Yang-Baxter maps. The unitary condition of the dynamical Yang-Baxter map is discussed. Moreover, we establish a correspondence between two dynamical Yang-Baxter maps constructed in this paper. This correspondence produces a version of the vertex-IRF correspondence. §1. Introduction Much attention has been directed to the quantum dynamical Yang-Baxter equation (QDYBE), a generalization of the quantum Yang-Baxter equation (QYBE) (for example, see [4]). The dynamical Yang-Baxter map (dynamical YB map) [13] is a set-theoretical solution to a version of the QDYBE.Let H and X be nonempty sets, and φ a map fromis a dynamical YB map associated with H, X, and φ, iff, for every λ ∈ H, R(λ) satisfies the following equation on X × X × X.(1.1) R 23 (λ)R 13 (φ(λ, X (2) ))R 12 (λ) = R 12 (φ(λ, X (3) ))R 13 (λ)R 23 (φ(λ, X (1) )).

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The Gelfand-Tsetlin basis for irreducible c U q (gl(N4-1))-modules of finite dimensions is constructed by means of the lowering operators. § 0. IntroductionIn the previous paper [1], we constructed the Gelfand-Tsetlin basis for irreducible "Ug(^/(]V+l))-modules of finite dimensions, in terms of the lowering operators. In this paper, we will give detailed accounts of our construction of this basis.We give the result when q is a non-zero complex variable and not a root of unity, however from the view of technical arguments in its proof we also have to deal with the case of q to be a transcendental element over C.Let