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Associating geometry to the Lie superalgebra $\mathfrak {sl}(1|1)$ and to the color Lie algebra $\mathfrak {sl}^c_2(\Bbbk )$

Authors: Susan J. Sierra, Špela Špenko, Michaela Vancliff, Padmini Veerapen and Emilie Wiesner
Journal: Proc. Amer. Math. Soc. 147 (2019), 4135-4146
MSC (2010): Primary 14A22, 17B70, 17B75
Published electronically: July 1, 2019
MathSciNet review: 4002531
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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 $\mathfrak {sl}_2$.

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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

Špela Špenko
Affiliation: Departement Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium

Michaela Vancliff
Affiliation: Department of Mathematics, Box 19408, University of Texas at Arlington, Arlington, Texas 76019-0408
MR Author ID: 349363

Padmini Veerapen
Affiliation: Department of Mathematics, Tennessee Technological University, Cookeville, Tennessee 38505
MR Author ID: 353637

Emilie Wiesner
Affiliation: Department of Mathematics, Ithaca College, Ithaca, New York 14850
MR Author ID: 685116

Keywords: Lie algebra, Lie superalgebra, color Lie algebra, line module
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
Article copyright: © Copyright 2019 American Mathematical Society