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 $O_q\otimes \mathbb{F}[z_1,z_2,\dots ]$, where $\mathbb{F}$ is the ground field and $\{z_n{\}}_{n=1}^{\infty }$ are mutually commuting indeterminates. Recall the positive part $U_q^{+}$ of the quantized enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$. 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 $O_q$.

arXiv comment58 pages

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Cited by in corpus (6)

• The alternating presentation of $U_q(\widehat{g{l}_2})$ from Freidel-Maillet algebras
• Generalised Onsager Algebra in Quantum Lattice Models
• The $q$-Onsager algebra and its alternating central extension
• A compact presentation for the alternating central extension of the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• Using a $q$-shuffle algebra to describe the basic module $V({\Lambda }_0)$ for the quantized enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$
• Freidel-Maillet type presentations of $U_q(s{l}_2)$