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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Associating geometry to the Lie superalgebra $\mathfrak {sl}(1|1)$ and to the color Lie algebra $\mathfrak {sl}^c_2(\Bbbk )$
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by Susan J. Sierra, Špela Špenko, Michaela Vancliff, Padmini Veerapen and Emilie Wiesner PDF
Proc. Amer. Math. Soc. 147 (2019), 4135-4146 Request permission

Abstract:

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra $\mathfrak {sl}_2(\mathbb {C})$, there is a correspondence between Verma modules and certain line modules that associates a pair $(\mathfrak {h}, \phi )$, where $\mathfrak {h}$ is a $2$-dimensional Lie subalgebra of $\mathfrak {sl}_2(\mathbb {C})$ and $\phi \in \mathfrak {h}^*$ satisfies $\phi ([\mathfrak {h}, \mathfrak {h}]) = 0$, to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra $\mathfrak {sl}(1|1)$ and for a color Lie algebra associated to the Lie algebra $\mathfrak {sl}_2$.
References
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Additional Information
  • Susan J. Sierra
  • Affiliation: School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
  • MR Author ID: 860198
  • Email: s.sierra@ed.ac.uk
  • Špela Špenko
  • Affiliation: Departement Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium
  • Email: spela.spenko@vub.be
  • Michaela Vancliff
  • Affiliation: Department of Mathematics, Box 19408, University of Texas at Arlington, Arlington, Texas 76019-0408
  • MR Author ID: 349363
  • Email: vancliff@uta.edu
  • Padmini Veerapen
  • Affiliation: Department of Mathematics, Tennessee Technological University, Cookeville, Tennessee 38505
  • MR Author ID: 353637
  • Email: pveerapen@tntech.edu
  • Emilie Wiesner
  • Affiliation: Department of Mathematics, Ithaca College, Ithaca, New York 14850
  • MR Author ID: 685116
  • Email: ewiesner@ithaca.edu
  • Received by editor(s): August 16, 2018
  • Published electronically: July 1, 2019
  • Additional Notes: The first and second authors were supported in part by EPSRC grant EP/M008460/1
    Some of this work was completed while the second author was a postdoctoral scholar at the University of Edinburgh
    The third author was supported in part by NSF grant DMS-1302050
  • Communicated by: Kailash Misra
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4135-4146
  • MSC (2010): Primary 14A22, 17B70, 17B75
  • DOI: https://doi.org/10.1090/proc/14647
  • MathSciNet review: 4002531