2-cocycles on the algebra of differential operators
Let \({\mathbb{C}}[t,t^{-1}]\) be the Laurent polynomial ring over the complex field \({\mathbb{C}}\), and \({\mathfrak a}={\mathbb{C}}[t,t^{-1},D]\) the algebra of differential operators on \({\mathbb{C}}[t,t^{-1}],\) \({\mathfrak A}={\mathfrak a}\otimes gl_ n({\mathbb{C}}).\) We know that \[ \alpha (t^{\ell +m} D^{\ell}, t^{\ell '+m'} D^{\ell '})=\delta_{m,-m'} (-1)^{\ell} \ell ! \ell '! \left( \begin{matrix} m+\ell \\ \ell +\ell '\quad +1\end{matrix} \right) \] is a nontrivial 2-cocycle on \({\mathfrak a}\) and \[ \psi (t^{\ell +m} D^{\ell}\otimes A, t^{\ell '+m'} D^{\ell '}\otimes \quad A')=\delta_{m,-m'} (-1)^{\ell} \ell ! \ell '! \left( \begin{matrix} m+\ell \\ \ell +\ell '+1\end{matrix} \right)\quad tr AA' \] is a nontrivial 2-cocycle on \({\mathfrak A}\). In this paper it is proved that every 2-cocycle on \({\mathfrak a}\) is equivalent to a multiple of \(\alpha\) and every 2- cocycle on \({\mathfrak A}\) is equivalent to a multiple of \(\psi\), solving problems proposed by V. G. Kac.
scientific article; zbMATH DE number 4097594