Centralizer construction of the Yangian of the queer Lie superalgebra

Summary

Consider the complex matrix Lie superalgebra $\mathfrak{g}\mathfrak{l}_{\left. N \right|N}$ with the standard generators E _ij where i, j = ±1, . . . , ± N. Define an involutive automorphism η of $\mathfrak{g}\mathfrak{l}_{\left. N \right|N}$ by η(E _ij) = E _−i,−j . The queer Lie superalgebra q_N is the fixed point subalgebra in $\mathfrak{g}\mathfrak{l}_{\left. N \right|N}$ relative to η. Consider the twisted polynomial current Lie superalgebra

$$ \mathfrak{g} = \left\{ {X\left( t \right) \in \mathfrak{g}\mathfrak{l}_{\left. N \right|N} \left[ t \right]:\eta \left( {X\left( t \right)} \right) = X\left( { - t} \right)} \right\} $$

. The enveloping algebra U($\mathfrak{g}$ ) of the Lie superalgebra g has a deformation, called the Yangian of q_N. For each M = 1,2, . . . , denote by A ^M _N the centralizer of q_M ⊂ q _N+M in the associative superalgebra U(q _N+M). In this article we construct a sequence of surjective homomorphisms U(q_N) ← A ¹ _N ← A ² _N ← . . . . We describe the inverse limit of the sequence of centralizer algebras A ¹ _N , A ² _N , . . . in terms of the Yangian of q_N.

To Professor Anthony Joseph on the occasion of his 60th birthday