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

TL;DR

This paper proves that the algebraic and geometric constructions of $ extit{i}$-canonical bases in Letzter's coideal subalgebra of quantum $ extit{sl}_2$ are equivalent, unifying two previously distinct approaches.

Contribution

It demonstrates that the geometric basis coincides with the algebraic basis, confirming a conjecture and unifying different constructions of $ extit{i}$-canonical bases.

Findings

The geometric basis admits the same polynomial description as the algebraic basis.

The two bases are shown to coincide, confirming their equivalence.

Provides an explicit formula for basis elements in the geometric approach.

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.