Four bases for the Onsager Lie algebra related by a Z2×Z2 action

Abstract

The Onsager Lie algebra O is an infinite-dimensional Lie algebra defined by generators A and B and relations [A,[A,[A,B]]]=4[A,B], and [B,[B,[B,A]]]=4[B,A]. Using an embedding of O into the tetrahedron Lie algebra ⊠, we obtain four direct sum decompositions of the vector space O, each consisting of three summands. As we will show, there is a natural action of Z2×Z2 on these decompositions. For each decomposition, we provide a basis for each summand. Moreover, we describe the Lie bracket action on these bases and show how they are recursively constructed from the generators A and B of O. Finally, we discuss the action of Z2×Z2 on these bases and determine some transition matrices among the bases.