New York Journal of Mathematics
Volume 7 (2001) 71-86


M. P. de Oliveira

On Commutation Relations for 3-Graded Lie Algebras

Published: September 5, 2001
Keywords: 3-graded Lie algebras, kernel functions
Subject: 17B70, 16W10

We prove some commutation relations for a 3-graded Lie algebra, i.e., a Z-graded Lie algebra whose nonzero homogeneous elements have degrees -1, 0 or 1, over a field K. In particular, we examine the free 3-graded Lie algebra generated by an element of degree -1 and another of degree 1. We show that if K has characteristic zero, such a Lie algebra can be realized as a Lie algebra of matrices over polynomials in one indeterminate. In the end, we apply the results obtained to derive the classical commutation relations for elements in the universal enveloping algebra of sl2(K).

Author information

Department of Mathematics and Statistics, University of Sao Paulo, Brazil