\(U(\mathfrak{h})\)-free modules over the block algebra \(\mathcal{B}(q)\)
The Block Lie algebra \(\mathcal{B}(q)\) for \(q \in \mathbb{C}\) is the Lie algebra over \(\mathbb{C}\) with basis \(\{L_n \mid n=(n_1,n_2) \in \mathbb{Z}^2\}\) subject to the relation \([L_m, L_n] = (n_1(m_2 + q)-m_1(n_2 + q))L_{m+n}\). It has an extension \(\mathcal{B}'(q)\) by the derivations \(\partial_1\) and \(\partial_2\) with \(\partial_i(L_n)=n_i L_{n_i}\). The extended Lie algebra has the Cartan subalgebra \(\mathfrak{h}=\mathbb{C} \partial_1 + \mathbb{C} \partial_2\), i.e., a maximal abelian subalgebra with semisimple adjoint action on the whole algebra. The paper under review introduces a new class of modules of \(\mathcal{B}(q)\) and \(\mathcal{B}'(q)\), and describes all the simple objects among them. It is shown that these modules exhaust all \(U(\mathfrak{h})\)-free modules of rank one.
scientific article; zbMATH DE number 7396040
scientific article