A conjecture concerning the q-Onsager algebra

Abstract

The q-Onsager algebra Oq is defined by two generators W0,W1 and two relations called the q-Dolan/Grady relations. Recently Baseilhac and Kolb obtained a PBW basis for Oq with elements denoted{Bnδ+α0}n=0∞,{Bnδ+α1}n=0∞,{Bnδ}n=1∞. In their recent study of a current algebra Aq, Baseilhac and Belliard conjecture that there exist elements{W−k}k=0∞,{Wk+1}k=0∞,{Gk+1}k=0∞,{G˜k+1}k=0∞ in Oq that satisfy the defining relations for Aq. In order to establish this conjecture, it is desirable to know how the elements on the second displayed line above are related to the elements on the first displayed line above. In the present paper, we conjecture the precise relationship and give some supporting evidence. This evidence consists of some computer checks on SageMath due to Travis Scrimshaw, a proof of the analog conjecture for the Onsager algebra O, and a proof of our conjecture for a homomorphic image of Oq called the universal Askey-Wilson algebra.