Kempf, George R. The Ext-dual of a Verma module is a Verma module. (English) Zbl 0758.17004 J. Pure Appl. Algebra 75, No. 1, 47-49 (1991). Let \(V\) be a split semi-simple Lie algebra of Chevalley type over a field \(k\), \(\ell\) be the rank of \(V\), and \(\Gamma^ +\) be the set of positive roots of \(V\). Let \(U(V)\) be the universal enveloping algebra of \(V\). Let \(\lambda=(\lambda_ 1,\dots,\lambda_ \ell)\in k^ \ell\) be a weight of \(V\), \(M(\lambda)\) be the left Verma module, and \(M'(\lambda)\) be the right Verma module. By constructing a projective resolution of \(M(\lambda)\) the author shows the following: (a) \(\text{Ext}^ n_{U(V)}(M(\lambda),U(V))=0\) unless \(n=\ell+r\), where \(r=\#\Gamma^ +\); (b) \(\text{Ext}^ n_{U(V)}(M(\lambda),U(V))\) is isomorphic to the Verma module \(M'(\lambda-\sum_{\alpha\in\Gamma^ +}\alpha)\), where \(\lambda-\sum_{\alpha\in\Gamma^ +}\alpha\) is the weight \((\lambda_ i-\sum_{\alpha\in\Gamma^ +}\alpha(h_ i))\). Reviewer: N.Kawamoto (Kure) Cited in 3 Documents MSC: 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B55 Homological methods in Lie (super)algebras 17B20 Simple, semisimple, reductive (super)algebras Keywords:split semi-simple Lie algebra of Chevalley type; Verma module; projective resolution PDFBibTeX XMLCite \textit{G. R. Kempf}, J. Pure Appl. Algebra 75, No. 1, 47--49 (1991; Zbl 0758.17004) Full Text: DOI References: [1] Chevalley, C.; Eilcnberg, S., Trans. AMS, 63, 85-124 (1948) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.