Subalgebras of the Universal Central Extension of $\mathfrak{sl}(n)[u^{\pm },v]$ from Two Involutions

Abstract

The affine Yangian associated with $\widehat{\mathfrak{sl}}(n)$ is a deformation of the universal enveloping algebra of the universal central extension of $\mathfrak{sl}(n)[u^{\pm },v]$. In this article, we consider two involutions of the universal central extension of $\mathfrak{sl}(n)[u^{\pm },v]$ and give a presentation of the fixed subalgebras of these involutions. By deforming one fixed subalgebra, we can define the 1-parameter twisted affine Yangian $T{Y}_{\hslash }(\widehat{\mathfrak{so}}(n))$, which is a coideal of the affine Yangian associated with $\widehat{\mathfrak{sl}}(n)$. Another fixed subalgebra is expected to define the 2-parameter twisted affine Yangian by deforming the presentation given in this article.