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ON UNIVERSAL COVERINGS OF LIE TORI

  • Received : 2012.06.28
  • Published : 2012.11.30

Abstract

In this paper we give an introduction to the theory of universal central extensions of perfect Lie algebras. In particular, we will provide a model for the universal coverings of Lie tori and we show that automorphisms and derivations lift to the universal coverings. We also prove that the universal covering of a Lie ${\Lambda}$-torus of type ${\Delta}$ is again a Lie ${\Lambda}$-torus of type ${\Delta}$.

References

  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 (1997), no. 603, 1-122.
  2. B. Allison, G. Benkart, and Y. Gao, Central extensions of Lie algebras graded by finite root systems, Math. Ann. 316 (2000), no. 3, 499-527. https://doi.org/10.1007/s002080050341
  3. B. Allison, G. Benkart, and Y. Gao, Lie algebras graded by the root systems $BC_r$, r ${\geq}$ 2, Mem. Amer. Math. Soc. 158 (2002), no. 751, x+158.
  4. S. Azam, Generalized reductive Lie algebars: coonections with extended affine Lie alge-bars and Lie tori, Canad. J. Math. 58 (2006), no. 2, 225-248. https://doi.org/10.4153/CJM-2006-009-8
  5. S. Azam and V. Khalili, Lie tori and their fixed point subalgebra, Algebra Colloq. 16 (2009), no. 3, 381-396. https://doi.org/10.1142/S1005386709000376
  6. S. Azam, V. Khalili, and M. Yousofzadeh, Extended affine root system of type BC, J. Lie Theory 15 (2005), no. 1, 145-181.
  7. S. Azam, H. Yamane, and M. Yousofzadeh, A finite presentation of universal coverings of Lie tori, Publ. RIMS Kyoto Univ. 46 (2010), 507-548.
  8. G. Benkart and O. Smirnov, Lie algebras graded by the root system $BC_1$, J. Lie Theory 13 (2003), no. 1, 91-132.
  9. S. Berman and R. Moody, Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy, Invent. Math. 108 (1992), no. 2, 323-347. https://doi.org/10.1007/BF02100608
  10. N. Bourbaki, Groupes et algebres de Lie, Chap. 4-6, Hermann, Paris 1968.
  11. Y. Gao, Steinberg unitary Lie algebras and skew-dihedral homology, J. Algebra. 176 (1996), no. 1, 261-304.
  12. H. Garland, The arithmetic theory of loop groups, Inst. Hautes Etudes Sci. Publ. Math. (1980), no. 52, 5-136.
  13. V. Khalili, Extension data and their Lie algebras, Algebra Colloq. 18 (2011), no. 3, 461-474. https://doi.org/10.1142/S1005386711000344
  14. V. Khalili, The core of a locally extended affine Lie algebras, Comm. Algebra 39 (2011), no. 10, 3646-3661. https://doi.org/10.1080/00927872.2010.510812
  15. R. V. Moody and A. Pianzola, Lie Algebras with Triangular Decomposition, John Wiley, New York, 1995.
  16. J. Morita and Y. Yoshii, Locally extended affine Lie algebras, J. Algebra 301, (2006), no. 1, 59-81. https://doi.org/10.1016/j.jalgebra.2005.06.013
  17. K. H. Neeb, Universal central extensions of Lie groups, Acta Appl. Math. 73 (2002). no. 1-2, 175-219. https://doi.org/10.1023/A:1019743224737
  18. E. Neher, Lie algebras graded by 3-graded root systems and Jordan pairs covered by grids, Amer. J. Math. 118 (1996), no. 2, 439-491. https://doi.org/10.1353/ajm.1996.0018
  19. E. Neher, Lie tori, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004), no. 3, 84-89.
  20. E. Neher, Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004), no. 3, 90-96.
  21. E. Neher, An introduction to universal central extensions of Lie superalgebras, Groups, rings, Lie and Hopf algebras (St. John's, NF, 2001), 141-166, Math. Appl., 555, Kluwer Acad. Publ., Dordrecht, 2003.
  22. A. Pianzola, D. Prelet, and J. Sun, Descent constructions for central extensions of infinite dimensional Lie algebras, Manuscripta Math. 122 (2007), no. 2, 137-148. https://doi.org/10.1007/s00229-006-0053-3
  23. W. L. J. van der Kallen, Infinitesimacally Centrals Extension of Chevally Groups, Springer-Verlag, Berlin, 1973, Lecture Notes in Mathematics, Vol. 356, 1973.
  24. C. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994.
  25. Y. Yoshii, Root-graded Lie algebras with compatible grading, Comm. Algebra 20 (2001), no. 8, 3365-3391.
  26. Y. Yoshii, Root systems extended by an abelian group, and their Lie algebras, J. Lie theory, 14 (2004), no. 2, 371-394.
  27. Y. Yoshii, Lie tori-a simple characterization of extended affine Lie algebras, Publ. Res. Inst. Math. Sci. 42 (2006), no. 3, 739-762. https://doi.org/10.2977/prims/1166642158
  28. M. Yousofzadeh, A presentation of Lie tori of type $B_l$, Publ. Res. Inst. Math. Sci. 44 (2008), no. 1, 1-44.