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The centroid of extended affine and root graded Lie algebras. (English) Zbl 1163.17306
Summary: We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.

MSC:
17B70 Graded Lie (super)algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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[1] B.N. Allison, S. Azam, S. Berman, Y. Gao, A. Pianzola, Extended affine Lie algebras and their root systems, Memoirs of the American Mathematical Society 126 vol. 603, Providence, R.I., 1997. · Zbl 0879.17012
[2] Allison, B.N.; Benkart, G.; Gao, Y., Central extensions of Lie algebras graded by finite root systems, Math. ann., 316, 499-527, (2000) · Zbl 0989.17004
[3] B.N. Allison, G. Benkart, Y. Gao, Lie Algebras graded by the root systems \(\operatorname{BC}_r\), \(r \geqslant 2\), Memoirs of the American Mathematical Society 158 vol. 751, Providence, R.I., 2002.
[4] Allison, B.N.; Berman, S.; Gao, Y.; Pianzola, A., A characterization of affine Kac-Moody Lie algebras, Comm. math. phys., 185, 671-688, (1997) · Zbl 0879.17013
[5] Allison, B.N.; Berman, S.; Pianzola, A., Covering algebras II: isomorphism of loop algebras, J. reine angew. math., 571, 39-71, (2004) · Zbl 1056.17018
[6] B.N. Allison, S. Berman, A. Pianzola, Iterated loop algebras, preprint February 2005, to appear in Pacific J. Math.
[7] Benkart, G.; Moody, R.V., Derivations, central extensions, and affine Lie algebras, Algebras, groups and geom., 3, 456-492, (1986) · Zbl 0619.17014
[8] Benkart, G.; Osborn, J.M., Derivations and automorphisms of nonassociative matrix algebras, Trans. amer. math. soc., 263, 411-430, (1981) · Zbl 0453.16020
[9] Benkart, G.; Smirnov, O., Lie algebras graded by the root system \(\operatorname{BC}_1\), J. Lie theory, 13, 91-132, (2003) · Zbl 1015.17028
[10] Benkart, G.; Zelmanov, E., Lie algebras graded by finite root systems and intersection matrix algebras, Invent. math., 126, 1-45, (1996) · Zbl 0871.17024
[11] Berman, S.; Gao, Y.; Krylyuk, Y.; Neher, E., The alternative torus and the structure of elliptic quasi-simple Lie algebras of type \(A_2\), Trans. amer. math. soc., 347, 4315-4363, (1995) · Zbl 0847.17010
[12] Berman, S.; Moody, R.V., Lie algebras graded by finite root systems and the intersection matrix algebras of slodowy, Invent. math., 108, 323-347, (1992) · Zbl 0778.17018
[13] N. Bourbaki, Algèbre, Diffusion C.C.L.C, Paris, 1970 (Chapter II).
[14] Bourbaki, N., Groupes et algèbres de Lie, (1971), Hermann Paris, (Chapter I) · Zbl 0213.04103
[15] Bourbaki, N., Groupes et algèbres de Lie, (1981), Masson Paris, (Chapter VI) · Zbl 0483.22001
[16] Farnsteiner, R., Derivations and central extensions of finitely generated graded Lie algebras, J. algebra, 118, 33-45, (1988) · Zbl 0658.17013
[17] Jacobson, N., Lie algebras, (1962), Interscience New York · JFM 61.1044.02
[18] Kac, V.G., Infinite dimensional Lie algebras, (1990), Cambridge University Press Cambridge · Zbl 0574.17002
[19] Lam, T.Y., A first course in noncommutative rings, () · Zbl 0728.16001
[20] McCrimmon, K., A taste of Jordan algebras, universitext, (2004), Springer New York · Zbl 1044.17001
[21] Melville, D., Centroids of nilpotent Lie algebras, Comm. algebra, 20, 12, 3649-3682, (1992) · Zbl 0954.17502
[22] R.V. Moody, A. Pianzola, Lie Algebras With Triangular Decompositions, Canad. Math. Soc. Series of Monographs and Advanced Texts, Wiley, New York, 1995. · Zbl 0874.17026
[23] Neher, E., Lie algebras graded by 3-graded root systems, Amer. J. math., 118, 439-491, (1996) · Zbl 0857.17019
[24] Neher, E., Lie tori, C. R. math. acad. sci. soc. R. can., 26, 3, 84-89, (2004) · Zbl 1106.17027
[25] Neher, E., Extended affine Lie algebras, C. R. math. acad. sci. soc. R. can., 26, 3, 90-96, (2004) · Zbl 1072.17012
[26] Passman, D., Infinite crossed products, () · Zbl 0662.16001
[27] Pianzola, A., Automorphisms of toroidal Lie algebras and their central quotients, J. algebra appl., 1, 113-121, (2002) · Zbl 1036.17017
[28] Ponomarëv, K., Invariant Lie algebras and Lie algebras with a small centroid, Algebra logic, 40, 365-377, (2001) · Zbl 1033.17014
[29] Saito, K., Extended affine root systems 1 (Coxeter transformations), Publ. RIMS, Kyoto univ., 21, 75-179, (1985) · Zbl 0573.17012
[30] Saito, K., Extended affine root systems 2 (flat invariants), Publ. RIMS, Kyoto university, 26, 15-78, (1990) · Zbl 0713.17014
[31] Seligman, G.B., Rational methods in Lie algebras, () · Zbl 0189.03201
[32] Slodowy, P., A character approach to Looijenga’s invariant theory for generalized root systems, Compositio math., 55, 3-32, (1985) · Zbl 0609.20024
[33] Tits, J., Une classe d’algébres de Lie en relation avec LES algébres de Jordan, Indag. math., 24, 530-535, (1962) · Zbl 0104.26002
[34] Waterhouse, W.C., Introduction to affine group schemes, () · Zbl 0212.25602
[35] Yoshii, Y., Root-graded Lie algebras with compatible grading, Comm. algebra, 29, 3365-3391, (2001) · Zbl 0992.17016
[36] Yoshii, Y., Root systems extended by an abelian group and their Lie algebras, J. Lie theory, 14, 371-394, (2004) · Zbl 1087.17008
[37] Y. Yoshii, Lie tori—A simple characterization of extended affine Lie algebras, preprint 2003. · Zbl 1148.17017
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