1 CENTRAL EXTENSION OF VIRASORO TYPE SUBALGEBRAS OF THE ZAMOLODCHIKOV-w ∞ LIE ALGEBRA LUIGI ACCARDI Centro Vito Volterra, Universit` a di Roma Tor Vergata via Columbia, 2– 00133 Roma, Italy E-mail: accardi@Volterra.mat.uniroma2.it ANDREAS BOUKAS Centro Vito Volterra, Universit` a di Roma Tor Vergata via Columbia, 2– 00133 Roma, Italy E-mail: andreasboukas@yahoo.com It is known that the centerless Zamolodchikov–w∞ *–Lie algebra of conformal field theory does not admit nontrivial central extensions, but the Witt *–Lie al- gebra, which is a sub–algebra of w∞, admits a nontrivial central extension: the Virasoro algebra. Therefore the following question naturally arises: are there other natural sub–algebras of w∞ which admit nontrivial central extensions other than the Virasoro one? We show that for certain infinite dimensional closed subalgebras of w∞, which are natural generalizations of the Witt alge- bra the answer is negative. Keywords : Witt algebra; Virasoro algebra; Central extensions 1. Introduction The centerless Virasoro (or Witt)-Zamolodchikov-w ∞ ∗–Lie algebra (cf. 3 - 6 ) is the infinite dimensional ∗–Lie algebra, with generators { ˆ B n k : n ∈ N,n ≥ 2,k ∈ Z} (1) commutation relations [ ˆ B n k , ˆ B N K ]=(k (N − 1) − K (n − 1)) ˆ B n+N−2 k+K (2) and involution