Realization of locally extended affine Lie algebras of type A_1

Locally extended ane Lie algebras were introduced by Morita and Yoshii as a natural generalization of extended ane Lie algebras. After that, various generalizations of these Lie algebras have been investigated by others. It is known that a locally extended ane Lie algebra can be recovered from its centerless core, i.e., the ideal generated by weight vectors corresponding to nonisotropic roots modulo its centre. In this paper, in order to realize locally extended ane Lie algebras of type A1, using the notion of Tits-Kantor-Koecher construction, we construct some Lie algebras which are isomorphic to the centerless cores of these algebras