Kostant's formula and homology vanishing theorems for generalized Kac-Moody algebras
Let \({\mathfrak g} (A) = {\mathfrak n}^ - + {\mathfrak h} + {\mathfrak n}^ +\) be a generalized Kac-Moody (GKM) algebra associated with a symmetrizable generalized Cartan matrix \(X\), with the Cartan subalgebra \({\mathfrak h}\) and nilpotent part \({\mathfrak n}^ +\) of the Borel subalgebra. In a previous paper [Tohoku Math. J., II. Ser. 44, 567--580 (1992; Zbl 0767.17020)] the author studied the \({\mathfrak h}\)-module structure of the homology \(H_ j ({\mathfrak n}^ -, L(\lambda))\) of \({\mathfrak n}^ -\) and the cohomology \(H_ c^ j ({\mathfrak n}^ +, L(\lambda))\) of \({\mathfrak n}^ +\) with coefficients in the irreducible highest weight \({\mathfrak g} (A)\)-module \(L (\lambda)\). In the present paper, using the idea of \textit{L. Liu} [J. Algebra 149, 155--178 (1992; Zbl 0779.17024)], he extends the study to the case of the nilpotent part of a parabolic subalgebra containing the Borel subalgebra. He proves the Kostant formula as well as some vanishing theorems for the homology of GKM algebras.
scientific article; zbMATH DE number 567072