Subring structure of upper-triangular nilpotent matrix Lie ring:u(3,p)
This article discusses the structure of Lie subrings of the Lie ring upper-triangular nilpotent matrix Lie ring:u(3,p), which is also a Lie algebra over . This is is the Lie ring whose elements are matrices over the prime field , with 0s on and below the diagonal, i.e., matrices of the form:
The addition is defined as matrix addition and the Lie bracket is defined as where the product is matrix multiplication. Explicitly:
|Operation||How it's defined (short version)||How it's defined (formula)|
|Identity for addition||Zero matrix|
|Negative for addition||Negative as matrix|
|Lie bracket||Commutator as matrices, i.e.,|
Note that when is odd, there is a Baer correspondence between U(3,p) and u(3,p), i.e., a correspondence between this and a corresponding group prime-cube order group:U(3,p), which can be defined as the group of upper-triangular unipotent matrices over . The Baer correspondence relates subrings and subgroups, so the subring structure matches the subgroup structure of prime-cube order group:U(3,p).