Centralizer construction of the Yangian of the queer Lie superalgebra

Abstract

Consider the complex matrix Lie superalgebra $gl_{N|N}$ with the standard generators $E_{ij}$ where $i,j=-N,...,-1,1,...,N$. Define an involutive automorphism $η$ of $gl_{N|N}$ by $η(E_{ij})=E_{-i,-j}$. The queer Lie superalgebra $q_N$ is the fixed point subalgebra in $gl_{N|N}$ relative to the automorphism $η$. Consider the twisted polynomial current Lie superalgebra $g=\{X(t)\in\gl_{N|N}[t]:η(X(t))=X(-t)\}$. The enveloping algebra $U(g)$ of the Lie superalgebra $g$ has a deformation, called the Yangian of $q_N$. For each $M=1,2,...$ denote by $A_N^M$ the centralizer of $q_M\subset q_{N+M}$ in the superalgebra $U(q_{N+M})$. We describe the projective limit of the sequence of centralizer algebras $A_N^1,A_N^2,...$ in terms of the Yangian of $q_N$.