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:

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

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 $\mathbb {R}$-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 $\mathbb {R} ^{3}$.