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Some Factorizations in Universal Enveloping Algebras of Three Dimensional Lie Algebras and Generalizations

Published online by Cambridge University Press:  20 November 2018

Stephen Berman ,
Jun Morita  and
Yoji Yoshii
Stephen Berman
Affiliation:
Department of Mathematics University of Saskatchewan Saskatoon, Saskatchewan S7N 5E6, e-mail: berman@snoopy.usask.ca
Jun Morita
Affiliation:
Institute of Mathematics University of Tsukuba Tsukuba, Ibaraki 305-8571 Japan, e-mail: morita@math.tsukuba.ac.jp
Yoji Yoshii
Affiliation:
Department of Mathematics University of Alberta Edmonton, Alberta T6G 2G1, e-mail: yoshii@math.ualberta.ca
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Abstract

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We introduce the notion of Lie algebras with plus-minus pairs as well as regular plus-minus pairs. These notions deal with certain factorizations in universal enveloping algebras. We show that many important Lie algebras have such pairs and we classify, and give a full treatment of, the three dimensional Lie algebras with plus-minus pairs.

Keywords

17B0517B3517B6717B70
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 4 , 01 December 2002 , pp. 525 - 536
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Allison, B. N., Azam, S., Berman, S., Gao, Y. and Pianzola, A., Extended affine Lie algebras and their root systems. Mem. Amer.Math. Soc. 126, Providence, 1997.Google Scholar
[2] Allison, B. N., Benkart, G. and Gao, Y., Central extensions of Lie algebras graded by finite root systems. Math. Ann. 316 (2000), 499–527.Google Scholar
[3] Berman, S. and Moody, R. V., Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy. Invent.Math. 108 (1992), 323–347.Google Scholar
[4] Borcherds, R. E., Generalized Kac-Moody algebras. J. Algebra 115 (1988), 501–512.Google Scholar
[5] Humphreys, J. E., Introduction to Lie algebras and representation theory. Graduate Texts in Math. 9, Springer-Verlag, New York, 1972.Google Scholar
[6] Jacobson, N., Lie algebras. Interscience, New York, 1962.Google Scholar
[7] Kac, V. G., Simple irreducible graded Lie algebras of finite growth. Math. USSR-Izv. 2 (1968), 211–230.Google Scholar
[8] Kac, V. G., Infinite dimensional Lie algebras. 3rd edition, Cambridge University Press, Cambridge, 1990.Google Scholar
[9] Moody, R. V., A new class of Lie algebras. J. Algebra 10 (1968), 211–230.Google Scholar
[10] Moody, R. V. and Pianzola, A., Lie algebras with triangular decompositions. J. Wiley & Sons, New York, 1995.Google Scholar
[11] Tits, J., Théorie des groupes. Résumé des cours et travaux (1980–1981), 75–87, Collège de France, Paris, 1981.Google Scholar
[12] Wakimoto, M., Infinite-dimensional Lie algebras. Translations of Mathematical Monographs, Iwanami Series of Modern Mathematics, 2001.Google Scholar
[13] Yoshii, Y., Root-graded Lie algebras with compatible grading. Comm. Algebra 29 (2001), 3365–3391.Google Scholar