Unitriangular matrix group:UT(3,R)

Definition
This group is defined as the member of family::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 $$\R^3$$.