On the cohomology of the Lie algebra \(L_2\)
Let \(W\) be the Lie algebra of polynomial vector fields in \(\mathbb C\). Then \(W\) has a basis \(e_k=x^{k+1}d/dx\), \(k\geq -1\) with multiplication \([e_i,e_j]= (j-i)e_{i+j}\). For \(k\geq 0\), let \(L_k\) denote the subalgebra of \(W\) generated by \(\{e_j\mid j\geq k\}\). The cohomology rings \(H^*(L_k)\), \(k\geq 0\), are known. For \(\lambda,\mu\in \mathbb C\), define the \(W\)-module \({\mathcal F}_{\lambda,\mu}\) to be the complex vector space with basis \(f_j\), \(j\in\mathbb Z\) and \(W\)-action defined by \(e_i\cdot f_j=(j+\mu-(i+1)\lambda)f_{i+j}\). The subspace \(F_{\lambda,\mu}\) with basis \(f_j\), \(j\geq 0\) and the same action is an \(L_0\)-module. The homology of \(L_1\) with coefficients in \({\mathcal F}_{\lambda,\mu}\) and \(F_{\lambda,\mu}\) has been previously computed by \textit{B. L. Feigin} and \textit{D. B. Fuchs} [Funct. Anal. Appl. 14, No. 3, 201--212 (1981); translation from Funkts. Anal. Prilozh. 14, No. 3, 45--60 (1980; Zbl 0482.57010)]. In this paper the author computes the homology in dimensions 0, 1, and 2 for the algebra \(L_2\) with coefficients in the modules \({\mathcal F}_{\lambda,\mu}\) and \(F_{\lambda,\mu}\) and offers conjectures, with supporting arguments, that the higher-dimensional homology is zero for all values of \(\lambda\) and \(\mu\).
scientific article; zbMATH DE number 910265
scientific article