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

Abstract

Let ${\mathbf{U}}^{ı}\equiv {\mathbf{U}}^{ı}({\mathfrak{sl}}_2)$ 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 ${\mathfrak{sl}}_2$, ${\mathbf{U}}^{ı}$ is generated by the sum $\mathbf{E}+v\mathbf{KF}+\mathbf{K}$ of standard generators, and hence can be identified with the polynomial ring $\mathbb{Q}(v)[t]$. In [BW13] and [LW18], two distinguished bases, called $ı$canonical bases, are constructed inside the modified form of ${\mathbf{U}}^{ı}$ via algebraic and geometric approaches respectively. The modified form of ${\mathbf{U}}^{ı}$ can be identified with a direct sum of two copies of ${\mathbf{U}}^{ı}\cong \mathbb{Q}(v)[t]$ 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.

3 pages

References in corpus (2)

• A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs
• Positivity of Canonical Bases under Comultiplication

Cited by in corpus (1)

• Quasi-split symmetric pairs of $U({\mathfrak{gl}}_N)$ and their Schur algebras