Fock space representations of the Virasoro algebra. Intertwining operators (Q1083519)

The Virasoro algebra \({\mathcal L}\) is the Lie algebra over the complex numbers with generators \(e'_ 0\), \(e_ n\) (where \(n\) ranges over the integers) and the relations \(e'_ 0\) commutes with everything, \([e_ n,e_ m]=(m-n)e_{n+m}+\delta_{n+m}((m^ 3-m)/12)e'_ 0.\) It appears in string models of elementary particle physics. Quite recently it was used to analyze the critical phenomena in the two dimensional statistical physics. Kac studied the left \({\mathcal L}\)-module \(M(h,c)\) parametrized by two complex numbers h and c, called Verma module. In this paper, another kind of representations \({\mathcal F}(w,\lambda)\) of \({\mathcal L}\) parametrized by two complex numbers \(w\) and \(\lambda\) and intertwining operators between them are constructed. They are called the Fock space representations. The \({\mathcal L}\)-modules \({\mathcal F}(w,\lambda)\) and \(M(h,c)\) and their relationship are investigated.