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- TL;DR Summary
- I'm trying to grasp Lie algebra with a non-example. Why isn't it a Lie group? The matrices seem invertible but is it a smooth manifold ?

"The group given by ## H = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi i a \theta } } \end{array} \right) | \theta \in \mathbb { R } \right\} \subset \mathbb { T } ^ { 2 } = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi i \phi } } \end{array} \right) | \theta , \phi \in \mathbb { R } \right\} ## with ## a \in \mathbb { P } = \mathbb { R } \backslash \mathbb { Q }

## a fixed irrational number,is a subgroup of the torus ## \mathbb { T}^2## that is not a Lie group when given the subspace topology.

The group
can, however, be given a different topology, in which the distance between two points
is defined as the length of the shortest path
joining
to
. In this topology,
is identified homeomorphically with the real line by identifying each element with the number
in the definition of
. With this topology,
is just the group of real numbers under addition and is therefore a Lie group."

Why isn't it " a Lie group when given the subspace topology" ?

## a fixed irrational number,is a subgroup of the torus ## \mathbb { T}^2## that is not a Lie group when given the subspace topology.

The group

*in the group*Why isn't it " a Lie group when given the subspace topology" ?