Subring structure of upper-triangular nilpotent matrix Lie ring:u(3,p)
Template:Lie ring-specific information
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) |
---|---|---|
Addition | Matrix addition | ![]() |
Identity for addition | Zero matrix | ![]() |
Negative for addition | Negative as matrix | ![]() |
Lie bracket | Commutator as matrices, i.e., ![]() |
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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).
When , the Baer correspondence does not work, so the subring structure of
differs from the subgroup structure of
, which is dihedral group:D8 (see subgroup structure of dihedral group:D8).