The $q$-Onsager algebra and the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$

Abstract

The positive part $U_q^{+}$ of $U_q({\widehat{\mathfrak{sl}}}_2)$ has a presentation by two generators $X,Y$ that satisfy the $q$-Serre relations. The $q$-Onsager algebra ${\mathcal{O}}_q$ has a presentation by two generators $A,B$ that satisfy the $q$-Dolan/Grady relations. We give two results that describe how $U_q^{+}$ and ${\mathcal{O}}_q$ are related. First, we consider the filtration of ${\mathcal{O}}_q$ whose $n$th component is spanned by the products of at most $n$ generators. We show that the associated graded algebra is isomorphic to $U_q^{+}$. Second, we introduce an algebra ${\square }_q$ and show how it is related to both $U_q^{+}$ and ${\mathcal{O}}_q$. The algebra ${\square }_q$ is defined by generators and relations. The generators are $\{x_i{\}}_{i\in {\mathbb{Z}}_4}$ where ${\mathbb{Z}}_4$ is the cyclic group of order 4. For $i\in {\mathbb{Z}}_4$ the generators $x_i,x_{i+1}$ satisfy a $q$-Weyl relation, and $x_i,x_{i+2}$ satisfy the $q$-Serre relations. We show that ${\square }_q$ is related to $U_q^{+}$ in the following way. Let ${\square }_q^{\mathrm{even}}$ (resp. ${\square }_q^{\mathrm{odd}}$) denote the subalgebra of ${\square }_q$ generated by $x_0,x_2$ (resp. $x_1,x_3$). We show that (i) there exists an algebra isomorphism $U_q^{+}\to {\square }_q^{\mathrm{even}}$ that sends $X\mapsto x_0$ and $Y\mapsto x_2$; (ii) there exists an algebra isomorphism $U_q^{+}\to {\square }_q^{\mathrm{odd}}$ that sends $X\mapsto x_1$ and $Y\mapsto x_3$; (iii) the multiplication map ${\square }_q^{\mathrm{even}}\otimes {\square }_q^{\mathrm{odd}}\to {\square }_q$, $u\otimes v\mapsto uv$ is an isomorphism of vector spaces. We show that ${\square }_q$ is related to ${\mathcal{O}}_q$ in the following way. For nonzero scalars $a,b$ there exists an injective algebra homomorphism ${\mathcal{O}}_q\to {\square }_q$ that sends $A\mapsto a{x}_0+a^{-1}{x}_1$ and $B\mapsto b{x}_2+b^{-1}{x}_3$.

38 pages

References in corpus (1)

• The q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach

Cited by in corpus (4)

• The alternating central extension of the $q$-Onsager algebra
• The alternating central extension for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• A conjecture concerning the $q$-Onsager algebra
• The $q$-Onsager algebra and its alternating central extension