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

Document Type : Research Paper

Author

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.

Abstract

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.

Keywords

#### References

[1] B. Allison, S. Azam, S. Berman, Y. Gao, and A. Pianzola, Extended affine Lie algebras and their root systems, Mem. Amer. Math. Soc. 126(603), 1997.
[2] S. Azam, Construction of extended affine 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 ane Lie algebras), Comm. Algebra 44(12) (2016), 5309-5341.
[4] R. Hegh-Krohn and B. Torresani, 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 affine 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 affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can. 26(3) (2004), 90-96.
[11] E. Neher, Extended ane Lie algebras and other generalizations of affine Lie algebras -a survey, in: Developments and trends in infinite-dimensional Lie theory, 53-126, Progr. Math., 288, Birkhauser Boston, Inc., Boston, MA (2011).
[12] Y. Yoshii, Locally extended affine root systems, in: Contemporary Math. 506 (2010), 285-302.