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

The positive part $U_q^{+}$ of $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_2)$ 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 ${\mathrm{\square }}_q$ and show how it is related to both $U_q^{+}$ and ${\mathcal{O}}_q$. The algebra ${\mathrm{\square }}_q$ is defined by generators and relations. The generators are $\lbrace x_i \rbrace_{i \in \mathbb Z_4}$where$\mathbb Z_4$isthecyclicgroupoforder4.For$i \in \mathbb Z_4$thegenerators$x_i, x_{i+1}$satisfya$q$-Weylrelation,and$x_i,x_{i+2}$satisfythe$q$-Serrerelations\mathrm{.}Weshowthat$\square_q$ is related to $U_q^{+}$ in the following way. Let ${\mathrm{\square }}_q^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}$ (resp. ${\mathrm{\square }}_q^{\mathrm{o}\mathrm{d}\mathrm{d}}$) denote the subalgebra of ${\mathrm{\square }}_q$ generated by $x_0, x_2$(resp\mathrm{.}$x_1, x_3$)\mathrm{.}Weshowthat(i)thereexistsanalgebraisomorphism$U^+_q \to \square^{\rm even}_q$thatsends$X\mapsto x_0$and$Y\mapsto x_2$; (ii) there exists an algebra isomorphism $U_q^{+}\to {\mathrm{\square }}_q^{\mathrm{o}\mathrm{d}\mathrm{d}}$ that sends $X\mapsto x_1$ and $Y\mapsto x_3$; (iii) the multiplication map ${\mathrm{\square }}_q^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}\otimes {\mathrm{\square }}_q^{\mathrm{o}\mathrm{d}\mathrm{d}}\to {\mathrm{\square }}_q$, $u \otimes v \mapsto uv$isanisomorphismofvectorspaces\mathrm{.}Weshowthat$\square_q$ is related to ${\mathcal{O}}_q$ in the following way. For nonzero scalars $a,b$ there exists an injective algebra homomorphism ${\mathcal{O}}_q\to {\mathrm{\square }}_q$ that sends $A\mapsto a{x}_0+a^{-1}{x}_1$ and $B\mapsto b{x}_2+b^{-1}{x}_3$.