Summary
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Albuquerque, A. Elduque, and J. M. Pérez-Izquierdo, Alternative quasialgebras, Bull. Austral. Math. Soc. 63 (2001), no. 2, 257–268.
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.
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.
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.
B. Allison, S. Berman, J. Faulkner, and A. Pianzola, Realization of gradedsimple algebras as loop algebras, Forum Math. 20 (2008), no. 3, 395–432.
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.
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.
B. Allison, S. Berman, and A. Pianzola, Covering algebras. I. Extended affine Lie algebras, J. Algebra 250 (2002), no. 2, 485–516.
B. Allison, S. Berman, and A. Pianzola, Covering algebras. II. Isomorphism of loop algebras, J. Reine Angew. Math. 571 (2004), 39–71.
B. Allison, S. Berman, and A. Pianzola, Iterated loop algebras, Pacific J. Math. 227 (2006), no. 1, 1–41.
B. Allison and J. Faulkner, Isotopy for extended affine Lie algebras and Lie tori, pp.3–38 and arXiv:0709.1181 [math.RA].
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.
S. Azam, Nonreduced extended affine root systems of nullity 3, Comm. Algebra 25 (1997), no. 11, 3617–3654.
S. Azam, Construction of extended affine Lie algebras by the twisting process, Comm. Algebra 28 (2000), no. 6, 2753–2781.
S. Azam, Extended affine root systems, J. Lie Theory 12 (2002), 515–527.
S. Azam, Generalized reductive Lie algebras: connections with extended affine Lie algebras and Lie tori, Canad. J. Math. 58 (2006), no. 2, 225–248.
S. Azam, Derivations of multi-loop algebras, ForumMath. 19 (2007), no. 6, 1029–1045.
S. Azam, S. Berman, and M. Yousofzadeh, Fixed point subalgebras of extended affine Lie algebras, J. Algebra 287 (2005), no. 2, 351–380.
S. Azam, V. Khalili, and M. Yousofzadeh, Extended affine root systems of type BC, J. Lie Theory 15 (2005), no. 1, 145–181.
G. Benkart, Derivations and invariant forms of Lie algebras graded by finite root systems, Canad. J. Math. 50 (1998), no. 2, 225–241.
G. Benkart and R. Moody, Derivations, central extensions, and affine Lie algebras, Algebras Groups Geom. 3 (1986), no. 4, 456–492.
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.
G. Benkart and J. M. Osborn, Derivations and automorphisms of nonassociative matrix algebras, Trans. Amer. Math. Soc. 263 (1981), no. 2, 411–430.
G. Benkart and O. Smirnov, Lie algebras graded by the root system BC1, J. Lie Theory 13 (2003), no. 1, 91–132.
G. Benkart and Y. Yoshii, Lie G-tori of symplectic type, Q. J. Math. 57 (2006), no. 4, 425–448.
G. Benkart and E. Zelmanov, Lie algebras graded by finite root systems and intersection matrix algebras, Invent. Math. 126 (1996), 1–45.
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.
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.
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.
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.
R. Block, Determination of the differentiably simple rings with a minimal ideal., Ann. of Math. (2) 90 (1969), 433–459.
N. Bourbaki, Groupes et alg`ebres de Lie, chapitres 7–8, Hermann, Paris, 1975.
N. Bourbaki, Groupes et alg`ebres de Lie, chapitres 4–6, Masson, Paris, 1981.
T. S. Erickson, W. S. Martindale, and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49–63.
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.
R. Farnsteiner, Derivations and central extensions of finitely generated graded Lie algebras, J. Algebra 118 (1988), 33–45.
Y. Gao, Steinberg unitary Lie algebras and skew-dihedral homology, J. Algebra 179 (1996), no. 1, 261–304.
Y. Gao, The degeneracy of extended affine Lie algebras, Manuscripta Math. 97 (1998), no. 2, 233–249.
Y. Gao and S. Shang, Universal coverings of Steinberg Lie algebras of small characteristic, J. Algebra 311 (2007), no. 1, 216–230.
E. García and E. Neher, Tits-Kantor-Koecher superalgebras of Jordan superpairs covered by grids, Comm. Algebra 31 (2003), no. 7, 3335–3375.
H. Garland, The arithmetic theory of loop groups, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 5–136.
P. Gille and A. Pianzola, Galois cohomology and forms of algebras over Laurent polynomial rings, Math. Ann., 338 (2007), no. 2, 497–543.
[GiP2] P. Gille and A. Pianzola, Isotriviality and ´etale cohomology of Laurent polynomial rings, J. Pure Appl. Algebra, 212 (2008), no. 4, 780–800.
B. H. Gross and G. Nebe, Globally maximal arithmetic groups, J. Algebra 272 (2004), no. 2, 625–642.
J. T. Hartwig, Locally finite simple weight modules over twisted generalized Weyl algebras, J. Algebra 303 (2006), no. 1, 42–76.
J.-Y. Hée, Syst`emes de racines sur un anneau commutatif totalement ordonn´e, Geom. Dedicata 37 (1991), no. 1, 65–102.
R. Høegh-Krohn and B. Torrésani, Classification and construction of quasisimple Lie algebras, J. Funct. Anal. 89 (1990), no. 1, 106–136.
G. Homann, The abelianizations of Weyl groups of root systems extended by abelian groups J. Algebra 320 (2008), no. 4, 1741-1763.
J. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990.
N. Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (division of John Wiley & Sons), New York-London, 1962.
V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96.
V. G. Kac, Infinite Dimensional Lie Algebras, third ed., Cambridge University Press, 1990.
V. G. Kac, The idea of locality, Physical applications and mathematical aspects of geometry, groups and algebras (Singapure), World Sci., 1997, pp. 16–32.
I. L. Kantor, Classification of irreducible transitive differential groups, Dokl. Akad. Nauk SSSR 158 (1964), 1271–1274.
I. L. Kantor, Non-linear groups of transformations defined by general norms of Jordan algebras, Dokl. Akad. Nauk SSSR 172 (1967), 779–782.
I. L. Kantor, Certain generalizations of Jordan algebras, Trudy Sem. Vektor. Tenzor. Anal. 16 (1972), 407–499.
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.
C. Kassel and J.-L. Loday, Extensions centrales d’alg`ebres de Lie, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 4, 119–142 (1983).
M. Koecher, Imbedding of Jordan algebras into Lie algebras I, Amer. J. Math. 89 (1967), 787–816.
M. Koecher, Imbedding of Jordan algebras into Lie algebras II, Amer. J. Math. 90 (1968), 476–510.
J.-L. Loday, Cyclic Homology, Grundlehren, vol. 301, Springer-Verlag, Berlin Heidelberg, 1992.
O. Loos, Spiegelungsr¨aume und homogene symmetrische R¨aume, Math. Z. 99 (1967), 141–170.
O. Loos and E. Neher, Locally finite root systems, Mem. Amer.Math. Soc. 171 (2004), no. 811, x+214.
,Reflection systems and partial root systems, Jordan Theory Preprint Archives, paper #182.
Y. I. Manin, Topics in Noncommutative Geometry, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1991.
K. McCrimmon, A Taste of Jordan Algebras, Universitext, Springer-Verlag, New York, 2004.
K. McCrimmon and E. Zel’manov, The structure of strongly prime quadratic Jordan algebras, Adv. in Math. 69 (1988), no. 2, 133–222.
R. V. Moody and A. Pianzola, Lie algebras with triangular decompositions, Canad. Math. Soc. series of monographs and advanced texts, John Wiley, 1995.
J. Morita and Y. Yoshii, Locally extended affine Lie algebras, J. Algebra 301 (2006), no. 1, 59–81.
K.-H. Neeb, Integrable roots in split graded Lie algebras, J. Algebra 225 (2000), no. 2, 534–580.
K.-H. Neeb, Derivations of locally simple Lie algebras, J. LieTheory 15 (2005), no. 2, 589–594.
K.-H. Neeb, On the classification of rational quantum tori and structure of their automorphism groups, Canadian Math. Bulletin 51 :2 (2008), 261–282.
K.-H. Neeb and N. Stumme, The classification of locally finite split simple Lie algebras, J. Reine Angew. Math. 533 (2001), 25–53.
E. Neher, Syst`emes de racines 3-gradu´es, C. R. Acad. Sci. Paris Sér. I 310 (1990), 687–690.
E. Neher, Generators and relations for 3-graded Lie algebras, J. Algebra 155 (1993), no. 1, 1–35.
E. Neher, Lie algebras graded by 3-graded root systems and Jordan pairs covered by a grid, Amer. J. Math 118 (1996), 439–491.
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.
E. Neher, Quadratic Jordan superpairs covered by grids, J. Algebra 269 (2003), no. 1, 28–73.
E. Neher, Lie tori, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004), no. 3, 84–89.
E. Neher, Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004), no. 3, 90–96.
E. Neher and M. Tocón, Graded-simple Lie algebras of type B2 and gradedsimple Jordan pairs covered by a triangle, submitted.
E. Neher and Y. Yoshii, Derivations and invariant forms of Jordan and alternative tori, Trans. Amer. Math. Soc. 355 (2003), no. 3, 1079–1108.
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.
J. Nervi, Affine Kac-Moody algebras graded by affine root systems, J. Algebra, 253 (2002), no. 1, 50–99.
J. M. Osborn and D. S. Passman, Derivations of skew polynomial rings, J. Algebra 176 (1995), no. 2, 417–448.
D. S. Passman, Infinite crossed products, Pure and Applied Mathematics, vol. 135, Academic Press Inc., Boston, MA, 1989.
K. Saito, Extended affine root systems. I. Coxeter transformations, Publ. Res. Inst. Math. Sci. 21 (1985), no. 1, 75–179.
G. B. Seligman, Rational methods in Lie algebras, Marcel Dekker Inc.,New York, Lecture Notes in Pure and Applied Mathematics, Vol. 17, 1976.
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.
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.
N. Stumme, The structure of locally finite split Lie algebras, J. Algebra 220 (1999), no. 2, 664–693.
J. Tits, Une classe d’alg`ebres de Lie en relation avec les alg`ebres de Jordan, Indag. Math. 24 (1962), 530–535.
W. L. J. van der Kallen, Infinitesimally central extensions of Chevalley groups, Lecture Notes in Mathematics, Vol. 356, Springer-Verlag, Berlin, 1973.
C. Weibel, An introduction to homological algebra, Cambridge studies in advanced mathematics, vol. 38, Cambridge University Press, 1994.
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.
Y. Yoshii, Root-graded Lie algebras with compatible grading, Comm. Algebra 29 (2001), no. 8, 3365–3391.
Y. Yoshii, Classification of division Zn-graded alternative algebras, J. Algebra 256 (2002), no. 1, 28–50.
Y. Yoshii, Classification of quantum tori with involution, Canad. Math. Bull. 45 (2002), no. 4, 711–731.
Y. Yoshii, Root systems extended by an abelian group and their Lie algebras, J. Lie Theory 14 (2004), no. 2, 371–394.
Y. Yoshii, Lie tori—a simple characterization of extended affine Lie algebras, Publ. Res. Inst. Math. Sci. 42 (2006), no. 3, 739–762.
Y. Yoshii, Locally extended affine root systems, preprint, April 2008.
E. I. Zel’manov, Prime Jordan algebras. II, Sibirsk. Mat. Zh. 24 (1983),no. 1, 89–104, 192.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4741-4_3
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4740-7
Online ISBN: 978-0-8176-4741-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)