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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 $\lbrace x_i \rbrace_{i \in \mathbb Z_4}$ 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^{\rm even}_q$ (resp. $ \square^{\rm odd}_q$) denote the subalgebra of $ \square_q$ generated by $x_0, x_2$ (resp. $x_1, x_3$). We show that (i) there exi

Author

Terwilliger, Paul

Published

2015

Language

EN