The universal central extension of the three-point 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,

**[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 -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 loop algebra,`arXiv:math-ph/0511004`, J. Algebra (in press).**[ITW]**T. Ito, P. Terwilliger, C. Weng, The quantum algebra 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]**Christian Kassel,*Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra*, Proceedings of the Luminy conference on algebraic 𝐾-theory (Luminy, 1983), 1984, pp. 265–275. MR**772062**, https://doi.org/10.1016/0022-4049(84)90040-9**[KL]**C. Kassel and J.-L. Loday,*Extensions centrales d’algèbres de Lie*, Ann. Inst. Fourier (Grenoble)**32**(1982), no. 4, 119–142 (1983) (French, with English summary). MR**694130****[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.