Central extensions of some Lie algebras

Li, Wanglai; Wilson, Robert

doi:10.1090/S0002-9939-98-04348-2

Central extensions of some Lie algebras
HTML articles powered by AMS MathViewer

by Wanglai Li and Robert L. Wilson PDF

Proc. Amer. Math. Soc. 126 (1998), 2569-2577 Request permission

Abstract:

We consider three Lie algebras: $Der \mathbb {C}((t))$, the Lie algebra of all derivations on the algebra $\mathbb {C}((t))$ of formal Laurent series; the Lie algebra of all differential operators on $\mathbb {C}((t))$; and the Lie algebra of all differential operators on $\mathbb {C}((t))\otimes \mathbb {C}^n.$ We prove that each of these Lie algebras has an essentially unique nontrivial central extension.

References

Richard E. Block, On the Mills-Seligman axioms for Lie algebras of classical type, Trans. Amer. Math. Soc. 121 (1966), 378–392. MR 188356, DOI 10.1090/S0002-9947-1966-0188356-3
Richard E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068–3071. MR 843307, DOI 10.1073/pnas.83.10.3068
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
B. L. Feĭgin, Lie algebras $\textrm {gl}(\lambda )$ and cohomology of a Lie algebra of differential operators, Uspekhi Mat. Nauk 43 (1988), no. 2(260), 157–158 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 2, 169–170. MR 940679, DOI 10.1070/RM1988v043n02ABEH001720
Edward Frenkel, Victor Kac, Andrey Radul, and Weiqiang Wang, $\scr W_{1+\infty }$ and $\scr W(\mathfrak {g}\mathfrak {l}_N)$ with central charge $N$, Comm. Math. Phys. 170 (1995), no. 2, 337–357. MR 1334399, DOI 10.1007/BF02108332
Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR 996026
I. M. Gel′fand and D. B. Fuks, Cohomologies of the Lie algebra of vector fields on the circle, Funkcional. Anal. i Priložen. 2 (1968), no. 4, 92–93 (Russian). MR 0245035
Victor G. Kac and Dale H. Peterson, Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 6, 3308–3312. MR 619827, DOI 10.1073/pnas.78.6.3308
Victor Kac and Andrey Radul, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Comm. Math. Phys. 157 (1993), no. 3, 429–457. MR 1243706, DOI 10.1007/BF02096878
Wang Lai Li, $2$-cocyles on the algebra of differential operators, J. Algebra 122 (1989), no. 1, 64–80. MR 994935, DOI 10.1016/0021-8693(89)90237-8