The universal central extension of the three-point 𝔰𝔩₂ loop algebra

Abstract

We consider the three-point loop algebra, \[ L = s l 2 ⊗ K [ t , t − 1 , ( t − 1 ) − 1 ] , L= \mathfrak {sl}_2\otimes \mathbb {K} \lbrack t, t^{-1}, (t-1)^{-1}\rbrack , \] where K \mathbb {K} denotes a field of characteristic 0 0 and t t is an indeterminate. The universal central extension L ^ \widehat L of L L was determined by Bremner. In this note, we give a presentation for L ^ \widehat L via generators and relations, which highlights a certain symmetry over the alternating group A 4 A_4 . To obtain our presentation of L ^ \widehat L , we use the realization of L L as the tetrahedron Lie algebra.