Central Extensions of Some Lie Algebras

We consider three Lie algebras: Der C((t)), the Lie algebra of all derivations on the algebra C((t)) of formal Laurent series; the Lie algebra of all differential operators on C((t)); and the Lie algebra of all differential operators on C((t)) ⊗ Cn. We prove that each of these Lie algebras has an essentially unique nontrivial central extension. The Lie algebra of all derivations on the Laurent polynomial algebra C[t, t−1] can also be characterized as the Lie algebra of vector fields on the circle. The analogous object over a field F of characteristic p > 0, Der F [t]/(t), is called the Witt algebra [C], and this name is sometimes applied to Der C[t, t−1] as well. It is known [Bl] that Der F [t]/(t) has an essentially unique nontrivial one-dimensional central extension, and also [GF] that Der C[t, t−1] has an essentially unique nontrivial onedimensional central extension. The proofs of these facts are similar. The nontrivial one-dimensional central extension of Der C[t, t−1] is called the Virasoro algebra. It is one of the fundamental objects in representation theory as well as in theoretical physics. For a positive integer n, the Lie algebra of all differential operators on C[t, t−1]⊗ C has a nontrivial one-dimensional central extension, and the extended Lie algebra is related to the representation theory of affine Lie algebras [KP]. It is proved in [L] that this extension is essentially unique (also see [F]). When n = 1, the Lie algebra of all differential operators on the Laurent polynomial ring C[t, t−1] can also be characterized as the Lie algebra of differential operators on the circle; the corresponding extension is referred to, particularly in the physics literature, as W1+∞. Some representations of W1+∞ have been studied recently (see, e.g., [KR], [FKRW]). In [FKRW], it is shown that some representations of W1+∞ have natural structures of vertex operator algebras (see, e.g., [Bo] and [FLM] for definitions). Each of these constructions involves the Laurent polynomial algebra C[t, t−1]. This algebra is, of course, contained in C((t)), the algebra of formal Laurent series. In this paper, we consider the Lie algebras obtained by replacing C[t, t−1] by C((t)) in each of these constructions. We show that each of the resulting Lie algebras has an essentially unique nontrivial one-dimensional central extension. We work over the field of complex numbers, though all results hold over any field of characteristic zero. Received by the editors September 13, 1996 and, in revised form, February 4, 1997. 1991 Mathematics Subject Classification. Primary 17B65, 17B56; Secondary 17B66.