# Unitriangular matrix group:UT(3,R)

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## Definition

This group is defined as the unitriangular matrix group of degree three over the field of real numbers. Explicitly, it is the following group of matrices under multiplication:

$\left \{ \begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\\end{pmatrix} \mid a,b,c \in \R \right \}$

This group is also sometimes called the continuous Heisenberg group or real Heisenberg group.

### Structures

• The group has the natural structure of an algebraic group over the field of real numbers. (Alternatively, we can think of it as the set of $\R$-points of unitriangular matrix group:UT(3,C), which is the corresponding algebraic group over the field of complex numbers). As an algebraic group, it is a unipotent algebraic group.
• The group has the structure of a real Lie group (and hence also a topological group). The underlying manifold is diffeomorphic to $\R^3$.