Realization of locally extended affine Lie algebras of type $A_1$

Document Type: Research Paper


Department of Mathematics, University of Isfahan, Isfahan, Iran, P.O.Box: 81745-163 and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746.


Locally extended affine Lie algebras were introduced by Morita and Yoshii in [J. Algebra 301(1) (2006), 59-81] as a natural generalization of extended affine Lie algebras. After that, various generalizations of these Lie algebras have been investigated by others. It is known that a locally extended affine 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 affine Lie algebras of type $A_1$, using the notion of Tits-Kantor-Koecher construction, we construct some Lie algebras which are isomorphic to the centerless cores of these algebras.


[1] B. Allison, S. Azam, S. Berman, Y. Gao, and A. Pianzola, Extended affine Lie algebras and their root systems, Mem. Amer. Math. Soc. 126(603), 1997.

[2] S. Azam, Construction of extended affine Lie algebras by the twisting process, Comm. Algebra 28(6) (2000), 2753-2781.
[3] S. Azam, G. Behboodi and M. Yousofzadeh, Direct unions of Lie tori (realization of locally extended ane Lie algebras), Comm. Algebra 44(12) (2016), 5309-5341.
[4] R. Hegh-Krohn and B. Torresani, Classifi cation and construction of quasisimple Lie algebras, J. Funct. Anal. 89(1) (1990), 106-136.
[5] N. Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Publ. 39, 1968.
[6] O. Loos, Spiegelungsraume und homogene symmetrische Raume, Math. Z. 99 (1967), 141-170.
[7] O. Loos and E. Neher, Locally finite root systems, Mem. Amer. Math. Soc. 171(811), 2004.
[8] J. Morita and Y. Yoshii, Locally extended affine Lie algebras, J. Algebra 301(1) (2006), 59-81.
[9] E. Neher, Lie tori, C. R. Math. Acad. Sci. Soc. R. Can. 26(3)(2004), 84-89.
[10] E. Neher, Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can. 26(3) (2004), 90-96.
[11] E. Neher, Extended ane Lie algebras and other generalizations of affine Lie algebras -a survey, in: Developments and trends in infinite-dimensional Lie theory, 53-126, Progr. Math., 288, Birkhauser Boston, Inc., Boston, MA (2011).
[12] Y. Yoshii, Locally extended affine root systems, in: Contemporary Math. 506 (2010), 285-302.