The RLL relation for the elliptic quantum group L-operator

Let L^+(v)End(CN)Uq,p(sl^N)\widehat{L}^+(v)\in\operatorname{End}(\mathbb{C}^N)\otimes U_{q,p}(\widehat{\mathfrak{sl}}_N) be the L-operator constructed from the half currents, and let R+(v,P)R^+(v,P) be the evaluation RR-matrix of the universal RR-matrix of Bq,λ(sl^N){\cal B}_{q,\lambda}(\widehat{\mathfrak{sl}}_N). For tensor factors 11 and 22, write L^+(1)\widehat{L}^{+(1)} and L^+(2)\widehat{L}^{+(2)} for the corresponding copies, and let PP and hh denote the dynamical Cartan variables appearing in the relation. The RLL relation. The L-operator satisfies

R+(12)(v1v2,P+h)L^+(1)(v1)L^+(2)(v2)=L^+(2)(v2)L^+(1)(v1)R+(12)(v1v2,P).R^{+(12)}(v_1-v_2,P+h)\widehat{L}^{+(1)}(v_1)\widehat{L}^{+(2)}(v_2)=\widehat{L}^{+(2)}(v_2)\widehat{L}^{+(1)}(v_1)R^{+*(12)}(v_1-v_2,P).

This relation is conjectured from comparison with the half-current relations and is intended to identify the L-operator realization of the dynamical elliptic quantum group; the source provides no resolution of the conjecture.

Sources & referencesView supporting material

Primary source

Takeo Kojima and Hitoshi Konno, “The elliptic algebra U_q,p(sl_N) and the Drinfeld realization of the elliptic quantum group B_q,λ(sl_N)”, arXiv:math/0210383 (2002).

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