2022/10/06 by Hattori, Masamune, Yanagida, Shintarou
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2210.02777
We introduce a family of dynamical Hopf algebroids Uq,p(g,Xl) 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 Xl. If g is set to be certain theta functions, then our family recovers the elliptic algebras Uq,p(\widehat\mathfrakg) for untwisted affine Lie algebras \widehat\mathfrakg studied by Konno (1998, 2009), Jimbo-Konno-Odake-Shiraishi (1999) and Farghly-Konno-Oshima (2014). Also, taking the limit p → 0 in the case Xl=Al, we recover the Hopf algebras Uq(g,Al) of type Al with structure functions g := limp → 0 g, introduced by Ding-Iohara (1998) as a generalization of Drinfeld quantum affine algebras. Thus, our Hopf algebroid Uq,p(g,Xl) 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.