Suggestion of a definition of the twisted affine Yangian of type $D$

Abstract

We suggest a definition of the 1-parameter twisted affine Yangian $TY_{\hbar}(\widehat{\mathfrak{so}}(n))$ and the 2-parameter twisted affine Yangian $TY_{\hbar,\ve}(\widehat{\mathfrak{so}}(2n))$. These twisted affine Yangians are deformations of subalgebras of the universal central extension of $\mathfrak{sl}(2n)[u^{\pm1},v]$. As for the 1-parameter twisted affine Yangian, we show that $TY_{\hbar}(\widehat{\mathfrak{so}}(n))$ is a coideal of the affine Yangian associated with $\widehat{\mathfrak{sl}}(n)$. As for the two parameter twisted affine Yangian, we construct a homomorphism from $TY_{\hbar,\ve}(\widehat{\mathfrak{so}}(8))$ to the universal enveloping algebra of the rectangular $W$-algebra $\mathcal{W}^k(\mathfrak{sp}(16),(2^8))$.