Open AccessThe Koszul dual of the ring of three commuting matrices
Author(s) -
Freyja Hreinsdóttir
Publication year - 2000
Publication title -
mathematica scandinavica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.553
H-Index - 30
eISSN - 1903-1807
pISSN - 0025-5521
DOI - 10.7146/math.scand.a-14304
Subject(s) - mathematics , dimension (graph theory) , ideal (ethics) , lie algebra , ring (chemistry) , nilpotent , field (mathematics) , combinatorics , algebra over a field , pure mathematics , chemistry , philosophy , organic chemistry , epistemology
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.