On the freeness problem for truncated current algebras (Q2042182)

The Takiff algebras of a Lie algebra \(\mathfrak{g}\) are the Lie algebras \(\mathfrak{g} \otimes \mathbb{C}[x]/<x^m>\). This article concerns the structure of the universal enveloping algebras of certain Takiff algebras as modules over their centers. Let \(\mathfrak{g}\) be a finite dimensional complex semisimple Lie algebra. By a classical result of Kostant, \(\mathfrak{U(g)}\) is free over its center. It is known that this also holds if \(\mathfrak{g}\) is replaced by \(\mathfrak{g} \otimes \mathbb{C}[x]/<x^2>\). Here the authors prove that \(\mathfrak{U}(\mathfrak{gl}_3 \otimes \mathbb{C}[x]/<x^m>)\) is free over its center for all \(m\). In addition, they prove that \(\mathfrak{U}(\mathfrak{gl}_2 \otimes \mathbb{C}[x]/<x^m>)\) is free over its Gel'fand-Tsetlin algebra, the commutative subalgebra generated by the centers of the universal enveloping algebras of a nested collection of Levi subalgebras.