I've been asked this immediately after been asked to show that the structure constants $c_{ijk}$ are totally antisymmetric, so I suppose there must be a connection, although I can't figure out where.
I'm really confused about this. A quick search took me to this:
Example of two-dimensional non-abelian Lie algebra?
and to this: Geometry and Quantum Field Theory, from which I quote:
An example of a Lie group of dimension 2 with a non-abelian Lie algebra is the matrix Lie group
$$G=\left\{\begin{pmatrix}a&b\\0&1\end{pmatrix}\bigg|\,a\in\mathbb{R}^+,\,b\in\mathbb{R}\right\}$$ In fact, it is not hard to show that, up to isomorphism, this is the only connected non-abelian Lie group of dimension 2 (…).
The author refers first to the algebra and then to the group, so that makes my confusion worse.
So are there, in fact, non-abelian Lie groups of dimension 2? Or is it the algebra that can be non-abelian? If so, how should I argue that, however, the group must be abelian?
Best Answer
The Lie Group $$ G = \left\{\pmatrix{a&b\\0&1} : a \in \Bbb R^+,b \in \Bbb R\right\} $$ is indeed non-abelian. In particular, note that
$$ \pmatrix{1&1\\0&1}\pmatrix{2&0\\0&1} \neq \pmatrix{2&0\\0&1}\pmatrix{1&1\\0&1} $$ The Lie algebra of this group is also non-abelian (i.e. non-trivial).
I'm not sure where you're getting the idea that there's no such Lie group.