The Koszul dual of the ring of three commuting matrices

Abstract

Let $X=(x_{ij}), Y=(y_{ij})$ and $Z=(z_{ij})$ be generic $n$ by $n$ matrices. Let $k$ be a field with char $k\neq 2, 3, S=k[x_{11}, \dots , x_{nn}, y_{11}, \dots , y_{nn}, z_{11}, \dots , z_{nn}] $ and let $I$ be the ideal generated by the entries of the matrices $XY-YX, XZ-ZX$ and $YZ-ZY$. We study the Koszul dual of the ring $R=S/I$ and show that for $n\geq 3$ this is the enveloping algebra of a nilpotent Lie algebra. We also give the dimension of the Lie algebra in each degree.