2025/01/30 by Lee, Jae-Ho
#Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2501.18364
The Onsager Lie algebra O is an infinite-dimensional Lie algebra defined by generators A, 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 \boxtimes, 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, B of O. Finally, we discuss the action of ℤ2 × ℤ2 on these bases and determine some transition matrices among the bases.