On canonical bases of Letzter algebra ${\mathbf{U}}^{\mathrm{ı}}({\mathfrak{s}\mathfrak{l}}_2)$

Let ${\mathbf{U}}^{\mathrm{ı}}\equiv {\mathbf{U}}^{\mathrm{ı}}({\mathfrak{s}\mathfrak{l}}_2)$ be Letzter’s coideal subalgebra of quantum ${\mathfrak{s}\mathfrak{l}}_2$ corresponding to the symmetric pair $({\mathfrak{s}\mathfrak{l}}_2(\mathbb{C}),\mathbb{C})$. As a subalgebra of quantum ${\mathfrak{s}\mathfrak{l}}_2$, ${\mathbf{U}}^{\mathrm{ı}}$ is generated by the sum $\mathbf{E}+v\mathbf{K}\mathbf{F}+\mathbf{K}$ of standard generators, and hence can be identified with the polynomial ring $\mathbb{Q}(v)[t]$. In [BW13] and [LW18], two distinguished bases, called $\mathrm{ı}$canonical bases, are constructed inside the modified form of ${\mathbf{U}}^{\mathrm{ı}}$ via algebraic and geometric approaches respectively. The modified form of ${\mathbf{U}}^{\mathrm{ı}}$ can be identified with a direct sum of two copies of $\mathbf U^{\imath}\cong\mathbb Q(v)[t]$ itself. An explicit and elegant formula, as a polynomial in $t$, of algebraic basis elements is conjectured in [BW13] and proved in [BeW18]. The purpose of this short paper is to show that the geometric basis in [LW18] admits the same description and, consequently, that the two bases coincide.