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Enright's completions and injectively copresented modules

Authors: Steffen König and Volodymyr Mazorchuk
Journal: Trans. Amer. Math. Soc. 354 (2002), 2725-2743
MSC (2000): Primary 17B10, 16G10
Published electronically: March 11, 2002
MathSciNet review: 1895200
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Abstract: Let $\ensuremath{\mathfrak{A}} $be a finite-dimensional simple Lie algebra over the complex numbers. It is shown that a module is complete (or relatively complete) in the sense of Enright if and only if it is injectively copresented by certain injective modules in the BGG-category ${\mathcal O}$. Let $A$ be the finite-dimensional algebra associated to a block of ${\mathcal O}$. Then the corresponding block of the category of complete modules is equivalent to the category of $eAe$-modules for a suitable choice of the idempotent $e$. Using this equivalence, a very easy proof is given for Deodhar's theorem (also proved by Bouaziz) that completion functors satisfy braid relations. The algebra $eAe$ is left properly and standardly stratified. It satisfies a double centralizer property similar to Soergel's ``combinatorial description'' of ${\mathcal O}$. Its simple objects, their characters and their multiplicities in projective or standard objects are determined.

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Additional Information

Steffen König
Affiliation: Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester, LE1 7RH, England

Volodymyr Mazorchuk
Affiliation: Department of Mathematics, Uppsala University, Box 480, SE-75106, Uppsala, Sweden

Received by editor(s): July 11, 2000
Received by editor(s) in revised form: October 3, 2001
Published electronically: March 11, 2002
Additional Notes: The first author was partially supported by the EC TMR network “Algebraic Lie Representations” grant no ERB FMRX-CT97-0100.
The second author was an Alexander von Humboldt fellow at Bielefeld University.
Article copyright: © Copyright 2002 American Mathematical Society