Central extensions of nonsymmetrizable Kac-Moody algebras over commutative algebras

Gao, Yun

doi:10.1090/S0002-9939-1994-1185261-3

Central extensions of nonsymmetrizable Kac-Moody algebras over commutative algebras
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Proc. Amer. Math. Soc. 121 (1994), 67-76 Request permission

Abstract:

For a commutative algebra R over a field k of characteristic zero and a nonsymmetrizable Kac-Moody algebra $g(A)$, we prove that the Lie algebra ${g_R}(A) = R{ \otimes _k}g(A)$ is centrally closed. Consequently, we get a characterization of the symmetrizability of $g(A)$ by the second homology group of the Kac-Moody algebra over Laurent polynomials. Also a presentation of ${g_R}(A)$ is given when A is of nonaffine type.

References

G. M. Benkart and R. V. Moody, Derivations, central extensions, and affine Lie algebras, Algebras Groups Geom. 3 (1986), no. 4, 456–492. MR 901810
S. Berman, Isomorphisms and automorphisms of universal Heffalump Lie algebras, Proc. Amer. Math. Soc. 65 (1977), no. 1, 29–34. MR 486024, DOI 10.1090/S0002-9939-1977-0486024-4
S. Berman and R. V. Moody, Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy, Invent. Math. 108 (1992), no. 2, 323–347. MR 1161095, DOI 10.1007/BF02100608
Howard Garland, The arithmetic theory of loop groups, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 5–136. MR 601519
Aziz Haddi, Détermination des extensions centrales des algèbres de Kac-Moody, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 16, 691–694 (French, with English summary). MR 944412
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Christian Kassel, Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra, Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), 1984, pp. 265–275. MR 772062, DOI 10.1016/0022-4049(84)90040-9
Robert V. Moody and Arturo Pianzola, Lie algebras with triangular decompositions, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1323858
Robert V. Moody, Senapathi Eswara Rao, and Takeo Yokonuma, Toroidal Lie algebras and vertex representations, Geom. Dedicata 35 (1990), no. 1-3, 283–307. MR 1066569, DOI 10.1007/BF00147350
Robert Lee 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-New York, 1982, pp. 210–213. MR 675118