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Let g be a Lie algebra over an algebraically closed field of characteristic p>0 and let U be the universal enveloping algebra of g. We prove in this paper for g=gl_n and g=sl_n that the centre Z of U is a unique factorisation domain and that its field of fractions is rational. For g=sl_n our argument requires the assumption that p does not divide n while for g=gl_n it works for any p. It turned out that our two main results are closely related to each other. The first one confirms in type A a recent conjecture of A. Braun and C. Hajarnavis while the second answers a question of J. Ale

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Year: 2005

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oai:eprints.soton.ac.uk:43534

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