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Let g be a finite dimensional Lie algebra over k, and let U be its universal enveloping Lie algebra. Is there a left module M of U which is projective but not free? That is, is the Quillen-Suslin theorem still true for enveloping algebras?

Quillen-Suslin says this there are no non-free projectives for S(g), the associated graded algebra of U. Thus, if the associated graded module of a projective is projective, then it is free (and so the original module was also free). Therefore, this question is equivalent to the question "Is the associated graded module of a projective U-module always projective?"

My guess is no, because the Weyl algebra has non-free projectives, even though it's associated graded algebra is a polynomial algebra. However, the tricks I know that work for the Weyl algebra don't work for Lie algebras. I would love a simple example of a non-free projective U-module.

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In this paper Stafford shows that whenever g is a finite-dimensional non-abelian Lie algebra the enveloping algebra has non-free but stably free (and therefore projective) right ideals. He also shows how to construct them.

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