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The universal central extension of the three-point $ \mathfrak{sl}_2$ loop algebra


Authors: Georgia Benkart and Paul Terwilliger
Journal: Proc. Amer. Math. Soc. 135 (2007), 1659-1668
MSC (2000): Primary 17B37
DOI: https://doi.org/10.1090/S0002-9939-07-08765-5
Published electronically: January 8, 2007
MathSciNet review: 2286073
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the three-point loop algebra,

$\displaystyle L= \mathfrak{sl}_2\otimes \mathbb{K} \lbrack t, t^{-1}, (t-1)^{-1}\rbrack,$

where $ \mathbb{K}$ denotes a field of characteristic 0 and $ t$ is an indeterminate. The universal central extension $ \widehat L$ of $ L$ was determined by Bremner. In this note, we give a presentation for $ \widehat L$ via generators and relations, which highlights a certain symmetry over the alternating group $ A_4$. To obtain our presentation of $ \widehat L$, we use the realization of $ L$ as the tetrahedron Lie algebra.


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  • [ABG] Bruce Allison, Georgia Benkart, and Yun Gao, Central extensions of Lie algebras graded by finite root systems, Math. Ann. 316 (2000), no. 3, 499–527. MR 1752782, https://doi.org/10.1007/s002080050341
  • [BK] Stephen Berman and Yaroslav Krylyuk, Universal central extensions of twisted and untwisted Lie algebras extended over commutative rings, J. Algebra 173 (1995), no. 2, 302–347. MR 1325778, https://doi.org/10.1006/jabr.1995.1090
  • [Br] Murray Bremner, Generalized affine Kac-Moody Lie algebras over localizations of the polynomial ring in one variable, Canad. Math. Bull. 37 (1994), no. 1, 21–28. MR 1261553, https://doi.org/10.4153/CMB-1994-004-8
  • [EO] A. Elduque and S. Okubo,
    Lie algebras with $ S_4$-action and structurable algebras, arXiv:math.RA/0508558, J. Algebra (in press).
  • [HT] B. Hartwig and P. Terwilliger, The tetrahedron algebra, the Onsager algebra, and the $ \mathfrak{sl}_2$ loop algebra, arXiv:math-ph/0511004, J. Algebra (in press).
  • [ITW] T. Ito, P. Terwilliger, C. Weng, The quantum algebra $ U_q(\mathfrak{sl}_2)$ and its equitable presentation, J. Algebra, 298 (2006), 284-301.
  • [Ka] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219
  • [K] C. Kassel, Kähler differentials and coverings of complex simple Lie algebras extended over a commutative ring, J. Pure and Applied Alg. 34 (1984), 265-275. MR 0772062 (86h:17013)
  • [KL] C. Kassel and J.-L. Loday, Extensions centrales d'algèbres de Lie, Ann. Inst. Fourier, Grenoble 32 (1982), 119-142. MR 0694130 (85g:17004)
  • [MP] Robert V. Moody and Arturo Pianzola, Lie algebras with triangular decompositions, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1323858
  • [O] Lars Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. (2) 65 (1944), 117–149. MR 0010315
  • [P] Jacques H. H. Perk, Star-triangle equations, quantum Lax pairs, and higher genus curves, Theta functions—Bowdoin 1987, Part 1 (Brunswick, ME, 1987) Proc. Sympos. Pure Math., vol. 49, Amer. Math. Soc., Providence, RI, 1989, pp. 341–354. MR 1013140

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Additional Information

Georgia Benkart
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: benkart@math.wisc.edu

Paul Terwilliger
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: terwilli@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08765-5
Keywords: ${\mathfrak{sl}}_2$ loop algebra, universal central extension, tetrahedron Lie algebra, Onsager Lie algebra.
Received by editor(s): December 17, 2005
Received by editor(s) in revised form: February 24, 2006
Published electronically: January 8, 2007
Additional Notes: The first author’s support from NSF grant #DMS–0245082 is gratefully acknowledged.
Communicated by: Dan M. Barbasch
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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