The Alternating Central Extension of the q-Onsager Algebra

Abstract

The q-Onsager algebra $$O_q$$ is presented by two generators $$W_0$$ , $$W_1$$ and two relations, called the q-Dolan/Grady relations. Recently Baseilhac and Koizumi introduced a current algebra $${\mathcal {A}}_q$$ for $$O_q$$ . Soon afterwards, Baseilhac and Shigechi gave a presentation of $${\mathcal {A}}_q$$ by generators and relations. We show that these generators give a PBW basis for $${\mathcal {A}}_q$$ . Using this PBW basis, we show that the algebra $${\mathcal {A}}_q$$ is isomorphic to $$O_q \otimes {\mathbb {F}} [z_1, z_2, \ldots ]$$ , where $${\mathbb {F}}$$ is the ground field and $$\lbrace z_n \rbrace _{n=1}^\infty $$ are mutually commuting indeterminates. Recall the positive part $$U^+_q$$ of the quantized enveloping algebra $$U_q(\widehat{{\mathfrak {s}}{\mathfrak {l}}}_2)$$ . Our results show that $$O_q$$ is related to $${\mathcal {A}}_q$$ in the same way that $$U^+_q$$ is related to the alternating central extension of $$U^+_q$$ . For this reason, we propose to call $${\mathcal {A}}_q$$ the alternating central extension of $$O_q$$ .