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Developments and Trends in Infinite-Dimensional Lie Theory pp 53–126Cite as

Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey

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Part of the book series: Progress in Mathematics ((PM,volume 288))

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This is a survey on extended affine Lie algebras and related types of Lie algebras, which generalize affine Lie algebras.

2000 Mathematics Subject Classifications: Primary 17B65, 17B67. Secondary 16W50, 17B70.

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References

  1. H. Albuquerque, A. Elduque, and J. M. Pérez-Izquierdo, Alternative quasialgebras, Bull. Austral. Math. Soc. 63 (2001), no. 2, 257–268.

    Article  MATH  MathSciNet  Google Scholar 

  2. 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, x+122.

    MathSciNet  Google Scholar 

  3. 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.

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Allison, G. Benkart, and Y. Gao, Lie algebras graded by the root systems BCr, r ≥ 2, Mem. Amer. Math. Soc. 158 (2002), no. 751, x+158.

    MathSciNet  Google Scholar 

  5. B. Allison, S. Berman, J. Faulkner, and A. Pianzola, Realization of gradedsimple algebras as loop algebras, Forum Math. 20 (2008), no. 3, 395–432.

    Article  MATH  MathSciNet  Google Scholar 

  6. B. Allison, S. Berman, J. Faulkner, and A. Pianzola, Multiloop realization of extended affine Lie algebras and Lie tori, Trans. Amer.Math. Soc. 361 (2009), no. 9, 4807–4842.

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Allison, S. Berman, Y. Gao, and A. Pianzola, A characterization of affine Kac-Moody Lie algebras, Comm. Math. Phys. 185 (1997), no. 3, 671–688.

    Article  MATH  MathSciNet  Google Scholar 

  8. B. Allison, S. Berman, and A. Pianzola, Covering algebras. I. Extended affine Lie algebras, J. Algebra 250 (2002), no. 2, 485–516.

    Article  MATH  MathSciNet  Google Scholar 

  9. B. Allison, S. Berman, and A. Pianzola, Covering algebras. II. Isomorphism of loop algebras, J. Reine Angew. Math. 571 (2004), 39–71.

    MATH  MathSciNet  Google Scholar 

  10. B. Allison, S. Berman, and A. Pianzola, Iterated loop algebras, Pacific J. Math. 227 (2006), no. 1, 1–41.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Allison and J. Faulkner, Isotopy for extended affine Lie algebras and Lie tori, pp.3–38 and arXiv:0709.1181 [math.RA].

    Google Scholar 

  12. B. N. Allison and Y. Gao, The root system and the core of an extended affine Lie algebra, Selecta Math. (N.S.) 7 (2001), no. 2, 149–212.

    Article  MATH  MathSciNet  Google Scholar 

  13. S. Azam, Nonreduced extended affine root systems of nullity 3, Comm. Algebra 25 (1997), no. 11, 3617–3654.

    Article  MATH  MathSciNet  Google Scholar 

  14. S. Azam, Construction of extended affine Lie algebras by the twisting process, Comm. Algebra 28 (2000), no. 6, 2753–2781.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Azam, Extended affine root systems, J. Lie Theory 12 (2002), 515–527.

    MATH  MathSciNet  Google Scholar 

  16. S. Azam, Generalized reductive Lie algebras: connections with extended affine Lie algebras and Lie tori, Canad. J. Math. 58 (2006), no. 2, 225–248.

    MATH  MathSciNet  Google Scholar 

  17. S. Azam, Derivations of multi-loop algebras, ForumMath. 19 (2007), no. 6, 1029–1045.

    MATH  MathSciNet  Google Scholar 

  18. S. Azam, S. Berman, and M. Yousofzadeh, Fixed point subalgebras of extended affine Lie algebras, J. Algebra 287 (2005), no. 2, 351–380.

    Article  MATH  MathSciNet  Google Scholar 

  19. S. Azam, V. Khalili, and M. Yousofzadeh, Extended affine root systems of type BC, J. Lie Theory 15 (2005), no. 1, 145–181.

    MATH  MathSciNet  Google Scholar 

  20. G. Benkart, Derivations and invariant forms of Lie algebras graded by finite root systems, Canad. J. Math. 50 (1998), no. 2, 225–241.

    MATH  MathSciNet  Google Scholar 

  21. G. Benkart and R. Moody, Derivations, central extensions, and affine Lie algebras, Algebras Groups Geom. 3 (1986), no. 4, 456–492.

    MATH  MathSciNet  Google Scholar 

  22. G. Benkart and E. Neher, The centroid of extended affine and root graded Lie algebras, J. Pure Appl. Algebra 205 (2006), no. 1, 117–145.

    Article  MATH  MathSciNet  Google Scholar 

  23. G. Benkart and J. M. Osborn, Derivations and automorphisms of nonassociative matrix algebras, Trans. Amer. Math. Soc. 263 (1981), no. 2, 411–430.

    MATH  MathSciNet  Google Scholar 

  24. G. Benkart and O. Smirnov, Lie algebras graded by the root system BC1, J. Lie Theory 13 (2003), no. 1, 91–132.

    MATH  MathSciNet  Google Scholar 

  25. G. Benkart and Y. Yoshii, Lie G-tori of symplectic type, Q. J. Math. 57 (2006), no. 4, 425–448.

    Article  MATH  MathSciNet  Google Scholar 

  26. G. Benkart and E. Zelmanov, Lie algebras graded by finite root systems and intersection matrix algebras, Invent. Math. 126 (1996), 1–45.

    Article  MATH  MathSciNet  Google Scholar 

  27. 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.

    Article  MATH  MathSciNet  Google Scholar 

  28. S. Berman, Y. Gao, Y. Krylyuk, and E. Neher, The alternative torus and the structure of elliptic quasi-simple Lie algebras of type A2, Trans. Amer. Math. Soc. 347 (1995), 4315–4363.

    Article  MATH  MathSciNet  Google Scholar 

  29. S. Berman and R. Moody, Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy, Invent. Math. 108 (1992), 323–347.

    Article  MATH  MathSciNet  Google Scholar 

  30. S. Bloch, The dilogarithm and extensions of Lie algebras, Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math., vol. 854, Springer, Berlin, 1981, pp. 1–23.

    Google Scholar 

  31. R. Block, Determination of the differentiably simple rings with a minimal ideal., Ann. of Math. (2) 90 (1969), 433–459.

    Article  MathSciNet  Google Scholar 

  32. N. Bourbaki, Groupes et alg`ebres de Lie, chapitres 7–8, Hermann, Paris, 1975.

    Google Scholar 

  33. N. Bourbaki, Groupes et alg`ebres de Lie, chapitres 4–6, Masson, Paris, 1981.

    Google Scholar 

  34. T. S. Erickson, W. S. Martindale, and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49–63.

    MATH  MathSciNet  Google Scholar 

  35. S. Eswara Rao and R. V. Moody, Vertex representations for n-toroidal Lie algebras and a generalization of the Virasoro algebra, Comm. Math. Phys. 159 (1994), no. 2, 239–264.

    Article  MATH  MathSciNet  Google Scholar 

  36. R. Farnsteiner, Derivations and central extensions of finitely generated graded Lie algebras, J. Algebra 118 (1988), 33–45.

    Article  MATH  MathSciNet  Google Scholar 

  37. Y. Gao, Steinberg unitary Lie algebras and skew-dihedral homology, J. Algebra 179 (1996), no. 1, 261–304.

    Article  MATH  MathSciNet  Google Scholar 

  38. Y. Gao, The degeneracy of extended affine Lie algebras, Manuscripta Math. 97 (1998), no. 2, 233–249.

    Article  MATH  MathSciNet  Google Scholar 

  39. Y. Gao and S. Shang, Universal coverings of Steinberg Lie algebras of small characteristic, J. Algebra 311 (2007), no. 1, 216–230.

    Article  MATH  MathSciNet  Google Scholar 

  40. E. García and E. Neher, Tits-Kantor-Koecher superalgebras of Jordan superpairs covered by grids, Comm. Algebra 31 (2003), no. 7, 3335–3375.

    Article  MATH  MathSciNet  Google Scholar 

  41. H. Garland, The arithmetic theory of loop groups, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 5–136.

    Google Scholar 

  42. P. Gille and A. Pianzola, Galois cohomology and forms of algebras over Laurent polynomial rings, Math. Ann., 338 (2007), no. 2, 497–543.

    Article  MATH  MathSciNet  Google Scholar 

  43. [GiP2] P. Gille and A. Pianzola, Isotriviality and ´etale cohomology of Laurent polynomial rings, J. Pure Appl. Algebra, 212 (2008), no. 4, 780–800.

    Article  MATH  MathSciNet  Google Scholar 

  44. B. H. Gross and G. Nebe, Globally maximal arithmetic groups, J. Algebra 272 (2004), no. 2, 625–642.

    Article  MATH  MathSciNet  Google Scholar 

  45. J. T. Hartwig, Locally finite simple weight modules over twisted generalized Weyl algebras, J. Algebra 303 (2006), no. 1, 42–76.

    Article  MATH  MathSciNet  Google Scholar 

  46. J.-Y. Hée, Syst`emes de racines sur un anneau commutatif totalement ordonn´e, Geom. Dedicata 37 (1991), no. 1, 65–102.

    Article  MATH  MathSciNet  Google Scholar 

  47. R. Høegh-Krohn and B. Torrésani, Classification and construction of quasisimple Lie algebras, J. Funct. Anal. 89 (1990), no. 1, 106–136.

    Article  MathSciNet  Google Scholar 

  48. G. Homann, The abelianizations of Weyl groups of root systems extended by abelian groups J. Algebra 320 (2008), no. 4, 1741-1763.

    Article  MathSciNet  Google Scholar 

  49. J. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  50. N. Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (division of John Wiley & Sons), New York-London, 1962.

    Google Scholar 

  51. V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96.

    Article  MATH  MathSciNet  Google Scholar 

  52. V. G. Kac, Infinite Dimensional Lie Algebras, third ed., Cambridge University Press, 1990.

    Google Scholar 

  53. V. G. Kac, The idea of locality, Physical applications and mathematical aspects of geometry, groups and algebras (Singapure), World Sci., 1997, pp. 16–32.

    Google Scholar 

  54. I. L. Kantor, Classification of irreducible transitive differential groups, Dokl. Akad. Nauk SSSR 158 (1964), 1271–1274.

    MathSciNet  Google Scholar 

  55. I. L. Kantor, Non-linear groups of transformations defined by general norms of Jordan algebras, Dokl. Akad. Nauk SSSR 172 (1967), 779–782.

    MathSciNet  Google Scholar 

  56. I. L. Kantor, Certain generalizations of Jordan algebras, Trudy Sem. Vektor. Tenzor. Anal. 16 (1972), 407–499.

    MathSciNet  Google Scholar 

  57. C. Kassel, K¨ahler differentials and coverings of complex simple Lie algebras extended over a commutative algebra, Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983), vol. 34, 1984, pp. 265–275.

    MATH  MathSciNet  Google Scholar 

  58. C. Kassel and J.-L. Loday, Extensions centrales d’alg`ebres de Lie, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 4, 119–142 (1983).

    MATH  MathSciNet  Google Scholar 

  59. M. Koecher, Imbedding of Jordan algebras into Lie algebras I, Amer. J. Math. 89 (1967), 787–816.

    Article  MATH  MathSciNet  Google Scholar 

  60. M. Koecher, Imbedding of Jordan algebras into Lie algebras II, Amer. J. Math. 90 (1968), 476–510.

    Article  MATH  MathSciNet  Google Scholar 

  61. J.-L. Loday, Cyclic Homology, Grundlehren, vol. 301, Springer-Verlag, Berlin Heidelberg, 1992.

    Google Scholar 

  62. O. Loos, Spiegelungsr¨aume und homogene symmetrische R¨aume, Math. Z. 99 (1967), 141–170.

    Article  MATH  MathSciNet  Google Scholar 

  63. O. Loos and E. Neher, Locally finite root systems, Mem. Amer.Math. Soc. 171 (2004), no. 811, x+214.

    MathSciNet  Google Scholar 

  64. ,Reflection systems and partial root systems, Jordan Theory Preprint Archives, paper #182.

    Google Scholar 

  65. Y. I. Manin, Topics in Noncommutative Geometry, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1991.

    Google Scholar 

  66. K. McCrimmon, A Taste of Jordan Algebras, Universitext, Springer-Verlag, New York, 2004.

    MATH  Google Scholar 

  67. K. McCrimmon and E. Zel’manov, The structure of strongly prime quadratic Jordan algebras, Adv. in Math. 69 (1988), no. 2, 133–222.

    Article  MATH  MathSciNet  Google Scholar 

  68. R. V. Moody and A. Pianzola, Lie algebras with triangular decompositions, Canad. Math. Soc. series of monographs and advanced texts, John Wiley, 1995.

    Google Scholar 

  69. J. Morita and Y. Yoshii, Locally extended affine Lie algebras, J. Algebra 301 (2006), no. 1, 59–81.

    Article  MATH  MathSciNet  Google Scholar 

  70. K.-H. Neeb, Integrable roots in split graded Lie algebras, J. Algebra 225 (2000), no. 2, 534–580.

    Article  MATH  MathSciNet  Google Scholar 

  71. K.-H. Neeb, Derivations of locally simple Lie algebras, J. LieTheory 15 (2005), no. 2, 589–594.

    MATH  MathSciNet  Google Scholar 

  72. K.-H. Neeb, On the classification of rational quantum tori and structure of their automorphism groups, Canadian Math. Bulletin 51 :2 (2008), 261–282.

    Article  MATH  MathSciNet  Google Scholar 

  73. K.-H. Neeb and N. Stumme, The classification of locally finite split simple Lie algebras, J. Reine Angew. Math. 533 (2001), 25–53.

    MATH  MathSciNet  Google Scholar 

  74. E. Neher, Syst`emes de racines 3-gradu´es, C. R. Acad. Sci. Paris Sér. I 310 (1990), 687–690.

    MATH  MathSciNet  Google Scholar 

  75. E. Neher, Generators and relations for 3-graded Lie algebras, J. Algebra 155 (1993), no. 1, 1–35.

    Article  MATH  MathSciNet  Google Scholar 

  76. E. Neher, Lie algebras graded by 3-graded root systems and Jordan pairs covered by a grid, Amer. J. Math 118 (1996), 439–491.

    Article  MATH  MathSciNet  Google Scholar 

  77. E. Neher, An introduction to universal central extensions of Lie superalgebras, Groups, rings, Lie and Hopf algebras (St. John’s, NF, 2001), Math. Appl., vol. 555, Kluwer Acad. Publ., Dordrecht, 2003, pp. 141–166.

    Google Scholar 

  78. E. Neher, Quadratic Jordan superpairs covered by grids, J. Algebra 269 (2003), no. 1, 28–73.

    Article  MATH  MathSciNet  Google Scholar 

  79. E. Neher, Lie tori, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004), no. 3, 84–89.

    MATH  MathSciNet  Google Scholar 

  80. E. Neher, Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004), no. 3, 90–96.

    MATH  MathSciNet  Google Scholar 

  81. E. Neher and M. Tocón, Graded-simple Lie algebras of type B2 and gradedsimple Jordan pairs covered by a triangle, submitted.

    Google Scholar 

  82. E. Neher and Y. Yoshii, Derivations and invariant forms of Jordan and alternative tori, Trans. Amer. Math. Soc. 355 (2003), no. 3, 1079–1108.

    Article  MATH  MathSciNet  Google Scholar 

  83. J. Nervi, Alg`ebres de Lie simples gradu´ees par un syst`eme de racines et sous-alg`ebres C-admissibles, J. Algebra, 223 (2000), no. 1, 307–343.

    Article  MATH  MathSciNet  Google Scholar 

  84. J. Nervi, Affine Kac-Moody algebras graded by affine root systems, J. Algebra, 253 (2002), no. 1, 50–99.

    Article  MATH  MathSciNet  Google Scholar 

  85. J. M. Osborn and D. S. Passman, Derivations of skew polynomial rings, J. Algebra 176 (1995), no. 2, 417–448.

    Article  MATH  MathSciNet  Google Scholar 

  86. D. S. Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press Inc., Boston, MA, 1989.

    Google Scholar 

  87. K. Saito, Extended affine root systems. I. Coxeter transformations, Publ. Res. Inst. Math. Sci. 21 (1985), no. 1, 75–179.

    Article  MATH  MathSciNet  Google Scholar 

  88. G. B. Seligman, Rational methods in Lie algebras, Marcel Dekker Inc.,New York, Lecture Notes in Pure and Applied Mathematics, Vol. 17, 1976.

    Google Scholar 

  89. T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167–266.

    Google Scholar 

  90. R. Steinberg, G´en´erateurs, relations et revˆetements de groupes alg´ebriques, Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain, 1962, pp. 113–127.

    Google Scholar 

  91. N. Stumme, The structure of locally finite split Lie algebras, J. Algebra 220 (1999), no. 2, 664–693.

    Article  MATH  MathSciNet  Google Scholar 

  92. J. Tits, Une classe d’alg`ebres de Lie en relation avec les alg`ebres de Jordan, Indag. Math. 24 (1962), 530–535.

    MathSciNet  Google Scholar 

  93. W. L. J. van der Kallen, Infinitesimally central extensions of Chevalley groups, Lecture Notes in Mathematics, Vol. 356, Springer-Verlag, Berlin, 1973.

    Google Scholar 

  94. C. Weibel, An introduction to homological algebra, Cambridge studies in advanced mathematics, vol. 38, Cambridge University Press, 1994.

    Google Scholar 

  95. R. L. Wilson, Euclidean Lie algebras are universal central extensions, Lie algebras and related topics (New Brunswick, N.J., 1981), Lecture Notes in Math., vol. 933, Springer, Berlin, 1982, pp. 210–213.

    Google Scholar 

  96. Y. Yoshii, Root-graded Lie algebras with compatible grading, Comm. Algebra 29 (2001), no. 8, 3365–3391.

    Article  MATH  MathSciNet  Google Scholar 

  97. Y. Yoshii, Classification of division Zn-graded alternative algebras, J. Algebra 256 (2002), no. 1, 28–50.

    Article  MATH  MathSciNet  Google Scholar 

  98. Y. Yoshii, Classification of quantum tori with involution, Canad. Math. Bull. 45 (2002), no. 4, 711–731.

    MATH  MathSciNet  Google Scholar 

  99. Y. Yoshii, Root systems extended by an abelian group and their Lie algebras, J. Lie Theory 14 (2004), no. 2, 371–394.

    MATH  MathSciNet  Google Scholar 

  100. Y. Yoshii, Lie tori—a simple characterization of extended affine Lie algebras, Publ. Res. Inst. Math. Sci. 42 (2006), no. 3, 739–762.

    Article  MATH  MathSciNet  Google Scholar 

  101. Y. Yoshii, Locally extended affine root systems, preprint, April 2008.

    Google Scholar 

  102. E. I. Zel’manov, Prime Jordan algebras. II, Sibirsk. Mat. Zh. 24 (1983),no. 1, 89–104, 192.

    MathSciNet  Google Scholar 

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  1. Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada

    Erhard Neher

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  1. Friedrich-Alexander-Universität Erlangen, Department of Mathematics, Bismarckstrasse 1 1/2, Erlangen, 91054, Germany

    Karl-Hermann Neeb

  2. University of Alberta, Department of Mathematical Sciences, Edmonton, T6G 2G1, Alberta, Canada

    Arturo Pianzola

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Neher, E. (2011). Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey. In: Neeb, KH., Pianzola, A. (eds) Developments and Trends in Infinite-Dimensional Lie Theory. Progress in Mathematics, vol 288. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4741-4_3

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