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Transactions of the American Mathematical Society

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Multiloop realization of extended affine Lie algebras and Lie tori

Authors: Bruce Allison, Stephen Berman, John Faulkner and Arturo Pianzola
Journal: Trans. Amer. Math. Soc. 361 (2009), 4807-4842
MSC (2000): Primary 17B65; Secondary 17B67, 17B70
Published electronically: April 21, 2009
MathSciNet review: 2506428
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Abstract: An important theorem in the theory of infinite dimensional Lie algebras states that any affine Kac-Moody algebra can be realized (that is to say constructed explicitly) using loop algebras. In this paper, we consider the corresponding problem for a class of Lie algebras called extended affine Lie algebras (EALAs) that generalize affine algebras. EALAs occur in families that are constructed from centreless Lie tori, so the realization problem for EALAs reduces to the realization problem for centreless Lie tori. We show that all but one family of centreless Lie tori can be realized using multiloop algebras (in place of loop algebras). We also obtain necessary and sufficient conditions for two centreless Lie tori realized in this way to be isotopic, a relation that corresponds to isomorphism of the corresponding families of EALAs.

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  • [AABGP] 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). MR 1376741 (97i:17015)
  • [ABFP] B. Allison, S. Berman, J. Faulkner, and A. Pianzola, Realization of graded-simple algebras as loop algebras, Forum Mathematicum 20 (2008), 395-432. MR 2418198
  • [ABGP] B. Allison, S. Berman, Y. Gao and A. Pianzola, A characterization of affine Kac-Moody Lie algebras, Comm. Math. Phys. 185 (1997), 671-688. MR 1463057 (98h:17026)
  • [ABG] B. Allison, G. Benkart and Y. Gao, Lie algebras graded by the root system BC$ _r$, $ r\ge 2$, Mem. Amer. Math. Soc. 158 #751 (2002). MR 1902499 (2003h:17038)
  • [ABP] B.N. Allison, S. Berman and A. Pianzola, Iterated loop algebras, Pacific J. Math. 227 (2006), 1-42. MR 2247871 (2007g:17022)
  • [AF] B. Allison and J. Faulkner, Isotopy for extended affine Lie algebras and Lie tori, arXiv:0709.1181v3 [math.RA] on
  • [AG] B. Allison and Y. Gao, The root system and the core of an extended affine Lie algebra, Selecta Math. 7 (2001), 149-212. MR 1860013 (2002g:17041)
  • [BSZ] Y.A. Bahturin, I.P. Shestakov and M.V. Zaicev, Gradings on simple Jordan and Lie algebras, J. Algebra 283 (2005), 849-868. MR 2111225 (2005i:17038)
  • [BN] G. Benkart and E. Neher, The centroid of extended affine and root graded Lie algebras, J. Pure Appl. Algebra 205 (2006), 117-145. MR 2193194 (2006i:17043)
  • [BZ] G. Benkart and E. Zelmanov, Lie algebras graded by finite root systems and intersection matrix algebras, Invent. Math. 126 (1996), 1-45. MR 1408554 (97k:17044)
  • [BGK] S. Berman, Y. Gao and Y. Krylyuk, Quantum tori and the structure of elliptic quasi-simple Lie algebras, J. Funct. Anal. 135 (1996), 339-389. MR 1370607 (97b:17007)
  • [BGKN] S. Berman, Y. Gao, Y. Krylyuk and E. Neher, The alternative torus and the structure of elliptic quasi-simple Lie algebras of type $ A_2$, Trans. Amer. Math. Soc. 347 (1995), 4315-4363. MR 1303115 (96b:17009)
  • [BFM] A. Borel, R. Friedman and J.W. Morgan, Almost commuting elements in compact Lie groups, Mem. Amer. Math. Soc. 157 #747 (2002). MR 1895253 (2003k:22006)
  • [B1] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968, 288 pp. MR 0240238 (39:1590)
  • [B2] N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975. 271 pp. MR 0453824 (56:12077)
  • [DM] C. Draper and C. Martin, Gradings on the Albert Algebra and on $ f_4$, arXiv:math/ 0703840v1 [math.RA] on
  • [GP] P. Gille and A. Pianzola, Galois cohomology and forms of algebras over Laurent polynomial rings, Math. Ann. 338 (2007), 497-543. MR 2302073 (2008b:20055)
  • [GOV] V.V. Gorbatsevich, A.L. Onishchik and E.B. Vinberg, Structure of Lie groups and Lie algebras, in Lie Groups and Lie Algebras III, Encyclopaedia of Mathematical Sciences Vol. 41, A.L. Onishchik and E.B. Vinberg, editors, Springer-Verlag, Berlin, 1994. MR 1349140 (96d:22001)
  • [H] J. Humphreys, Linear algebraic groups, Springer-Verlag, New York, 1975. MR 0396773 (53:633)
  • [J] N. Jacobson, Lie algebras, Dover, New York, 1979. MR 559927 (80k:17001)
  • [K] V.G. Kac, Infinite dimensional Lie algebras, Third edition, Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
  • [KS] V.G. Kac and A.V. Smilga, Vacuum structure in supersymmetric Yang-Mills theories with any gauge group. The many faces of the superworld, 185-234, World Sci. Publ., River Edge, NJ, 2000. MR 1885976 (2003k:22031)
  • [KW] V.G. Kac and S.P. Wang, On automorphisms of Kac-Moody algebras and groups, Advances in Math. 92 (1992), 129-195. MR 1155464 (93f:17041)
  • [Mc] K. McCrimmon, A taste of Jordan algebras, Springer, New York, 2004. MR 2014924 (2004i:17001)
  • [N1] E. Neher, Lie tori, C.R. Math. Acad. Sci. Soc. R. Can., 26 (2004), no. 3, pp. 84-89. MR 2083841 (2005d:17030)
  • [N2] E. Neher, Extended affine Lie algebras, C.R. Math. Acad. Sci. Soc. R. Can., 26 (2004), no. 3, pp. 90-96. MR 2083842 (2005f:17024)
  • [P] A. Pianzola, On automorphisms of semisimple Lie algebras, Algebras Groups Geom. 2 (1985), 95-116. MR 901575 (88h:17013)
  • [Sel] G.B. Seligman, Rational methods in Lie algebras, Lect. Notes in Pure and Applied Math. 27, Marcel Dekker, New York, 1976. MR 0427394 (55:428)
  • [vdL] J. van de Leur, Twisted toroidal Lie algebras, preprint, arXiv math/0106119 v1, 2001.
  • [Y1] Y. Yoshii, Coordinate algebras of extended affine Lie algebras of type $ A_1$, J. Algebra 234 (2000), pp. 128-168. MR 1799481 (2001i:17031)
  • [Y2] Y. Yoshii, Lie tori--A simple characterization of extended affine Lie algebras, Publ. Res. Inst. Math. Sci. 42 (2006), 739-762. MR 2266995 (2007h:17027)
  • [Y3] Y. Yoshii, Root systems extended by an abelian group and their Lie algebras, J. Lie Theory 14 (2004), pp. 371-374. MR 2066861 (2005e:17016)

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Additional Information

Bruce Allison
Affiliation: Department of Mathematics and Statistics, University of Victoria, PO Box 3060 STN CSC, Victoria, British Columbia, Canada V8W 3R4

Stephen Berman
Affiliation: Saskatoon, Saskatchewan, Canada

John Faulkner
Affiliation: Department of Mathematics, University of Virginia, Kerchof Hall, P.O. Box 400137, Charlottesville, Virginia 22904-4137

Arturo Pianzola
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Received by editor(s): September 7, 2007
Published electronically: April 21, 2009
Additional Notes: The first and fourth authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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