Enright's completions and injectively copresented modules (Q2782660)

For a finite-dimensional simple Lie algebra \(L\) over the complex numbers there is a BGG-category \(\mathcal O\) which decomposes into blocks. It is well known that each block of \(\mathcal O\) is equivalent to the module category of a finite-dimensional associative algebra. In the paper under review the authors study the completion of \(L\)-modules introduced by \textit{T. J. Enright} [Ann. Math. (2) 110, 1-82 (1979; Zbl 0417.17005)] and give a very natural intepretation of the relative and absolute completions in terms of finite-dimensional algebras related to blocks of the category \(\mathcal O\). NEWLINENEWLINENEWLINEThe main tool in this paper is formulated in the first part, in which Auslander's homological method plays a role. The key observation is that the complete modules can be copresented injectively by certain injective modules. Thus the abstract framework in the first part of the paper works since the category of complete modules, roughly speaking, is equivalent to the module category of an algebra of the form \(eAe\) with \(A\) a quasi-hereditary algebra and \(e\) an idempotent. NEWLINENEWLINENEWLINEThis leads naturally to the study of the structure of the algebras of the form \(eAe\) in the second part of the paper, where the authors provide a sufficient condition for an algebra \(eAe\) to be standardly stratified in case \(A\) itself is standardly stratified. Furthermore, following the approach in [\textit{S. König, I. H. Slungard} and \textit{C. C. Xi}, J. Algebra 240, No. 1, 393-412 (2001; Zbl 0980.17003)], the authors also discuss the existence of double centralizer properties and re-prove or generalize many results in the literature. The results in this paper show again that the representation theory of finite-dimensional algebras is powerful for many questions in algebraic Lie theory.