Simple double Lie algebras on the Laurent polynomial space

Abstract

We construct a simple $λ$-double Lie algebra on $k[t,t^{-1}]$ for every nonzero $λ\in k$. We then show that the analogous two-sided construction of weight zero is also simple. Both products admit natural coefficient realizations via row-and-column-finite operators associated with the finitary $\mathbb Z\times\mathbb Z$ matrix algebra.