On canonical bases of Letzter algebra $\mathbf U^{\imath}(\mathfrak{sl}_2)$

Abstract

Let $\mathbf U^{\imath}\equiv\mathbf U^{\imath} (\mathfrak{sl}_2)$ be Letzter's coideal subalgebra of quantum $\mathfrak{sl}_2$ corresponding to the symmetric pair $(\mathfrak{sl}_2(\mathbb C),\mathbb C)$. As a subalgebra of quantum $\mathfrak{sl}_2$, $\mathbf U^{\imath}$ 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 $\imath$canonical bases, are constructed inside the modified form of $\mathbf U^{\imath}$ via algebraic and geometric approaches respectively. The modified form of $\mathbf U^{\imath}$ 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.