[For the entire collection see Zbl 0622.00020.]
The Virasoro algebra is a complex Lie algebra with a base \(e_ i\), \(i\in {\mathbb Z}\), in which \([e_ i,e_ j]=(j-i)e_{i+j}\). Denote by \(L\) a central extension of the Virasoro algebra by a central element \(z\), such that in \(L\)
\[ [e_ i,e_ j]=(j-i)e_{i+j}+(1/12)\delta_{i,-j}(j^ 3- j)z. \]
The authors consider two classes of \({\mathbb Z}\)-graded representations of \(L\)-Verma modules \(V_{h,c}\) and \(L\)-modules \(H_{\lambda,\mu}\) obtained by actions of \(L\) in the generalized Grassmann algebra of an infinite-dimensional vector space. A Verma module \(V_{h,c}\) is completely defined by its generator \(w\) such that \(e_iw=0\), \(i<0\), \(e_0w=hw\), \(zw=cw\), where \(h,c\in\mathbb C\). A criterion is given for \(V_{h,c}\) and \(H_{\lambda,\mu}\) to be irreducible. If \(V_{h,c}\) is reducible then its maximal submodule is isomorphic to \(V_{h+a'a'',c}\) for some \(a',a''\). If \(H_{\lambda,\mu}\) is reducible, then it is a direct sum of some Verma modules.