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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Enright’s completions and injectively copresented modules
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by Steffen König and Volodymyr Mazorchuk PDF
Trans. Amer. Math. Soc. 354 (2002), 2725-2743 Request permission

Abstract:

Let $\mathfrak {A}$ 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
  • MR Author ID: 263193
  • Email: sck5@mcs.le.ac.uk
  • Volodymyr Mazorchuk
  • Affiliation: Department of Mathematics, Uppsala University, Box 480, SE-75106, Uppsala, Sweden
  • MR Author ID: 353912
  • Email: mazor@math.uu.se
  • 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.
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 2725-2743
  • MSC (2000): Primary 17B10, 16G10
  • DOI: https://doi.org/10.1090/S0002-9947-02-02958-6
  • MathSciNet review: 1895200