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CLASSICAL LIE SUPERALGEBRAS OVER SIMPLE ASSOCIATIVE ALGEBRAS

Published online by Cambridge University Press:  18 April 2006

GEORGIA BENKART ,
XIAOPING XU  and
KAIMING ZHAO
GEORGIA BENKART
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USAbenkart@math.wisc.edu
XIAOPING XU
Affiliation:
Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, P. R. Chinaxiaoping@math.ac.cn
KAIMING ZHAO
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5 and Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P.R. Chinakzhao@wlu.ca
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Abstract

Over arbitrary fields of characteristic not equal to 2, we construct three families of simple Lie algebras and six families of simple Lie superalgebras of matrices with entries chosen from different one-sided ideals of a simple associative algebra. These families correspond to the classical Lie algebras and superalgebras. Our constructions intermix the structure of the associative algebra and the structure of the matrix algebra in an essential, compatible way. Many examples of simple associative algebras without an identity element arise as a by-product. The study of conformal algebras and superalgebras often involves matrix algebras over associative algebras such as Weyl algebras, and for that reason, we illustrate our constructions by taking various one-sided ideals from a Weyl algebra or a quantum torus.

Keywords

17B2017B6517B6717B68
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 92 , Issue 3 , May 2006 , pp. 581 - 600
Copyright
2006 London Mathematical Society

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Footnotes

The research of GB was supported in part by NSF grant #DMS–0245082 and NSA grant #MDA904-03-1-0068 of the USA, that of XX was supported in part by NSF grants #10371121 and #10431040 of China, and that of KZ was supported in part by NSERC and by NSF grants #10371120 and #10431040 of China.