A dynamical analogue of Ding-Iohara quantum algebras

We introduce a family of dynamical Hopf algebroids $U_{q,p}(g,X_l)$ depending on a complex parameter $q$, a formal parameter $p$, a set $g$ of structure functions satisfying the so-called Ding-Iohara condition, and a finite root system of type $X_l$. If $g$ is set to be certain theta functions, then our family recovers the elliptic algebras $U_{q,p}(\widehat{\mathfrak{g}})$ for untwisted affine Lie algebras $\widehat{\mathfrak{g}}$ studied by Konno (1998, 2009), Jimbo-Konno-Odake-Shiraishi (1999) and Farghly-Konno-Oshima (2014). Also, taking the limit $p \to 0$ in the case $X_l=A_l$, we recover the Hopf algebras $U_q(\overline{g},A_l)$ of type $A_l$ with structure functions $\overline{g} := \lim_{p \to 0} g$, introduced by Ding-Iohara (1998) as a generalization of Drinfeld quantum affine algebras. Thus, our Hopf algebroid $U_{q,p}(g,X_l)$ can be regarded as a dynamical analogue of the Ding-Iohara quantum algebras. As a byproduct, we obtain an extension of the Ding-Iohara quantum algebras to those of non-simply-laced type.