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The main point of the question is: I would like to know whether there are only finitely many, countable infinitely many or even uncountable many isomorphism classes of $3$-dimensional real lie algebras...

For $2$ dimensions, I know that there are exactly $2$ real lie algebras but for 3 dimensions I could not find any reference in the literature...

And even more interesting: Let $U$ be an open ball in $3$-dimensional Euclidean space. Then $U$ is a 3dimensional manifold. How many non-isomorphic Lie-Group strucutres exist on $U$? This question is of course related to the Lie algebra question because every Lie group structure on $U$ gives rise to a Lie algebra, but we dont get all Lie algebras since there are $3$-dimensional real Lie algebras whose simply connected real Lie group is compact and hence not diffeomorphic to $U$ (for example $\frak su (2)$ )

I would be very grateful, if someone could help me out here.

Thanks.

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5  
To address your last question: In fact, the ${\frak{su}}(2)\simeq{\frak{so}}(3)$ case is the only $3$-dimensional Lie algebra whose corresponding connected, simply-connected Lie group is not diffeomorphic to $\mathbb{R}^3$. –  Robert Bryant May 9 at 15:38
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Three-dimensional real Lie algebras were classified by Bianchi in 1898. It’s classical, but I would not suggest reading the original paper even if you read Italian. But googling “Bianchi classification” might help you find references. For completeness, the original reference is Luigi Bianchi, "Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti", Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, Serie Terza, Tomo XI (1898), 267–352. –  José Figueroa-O'Farrill May 9 at 22:22
    

3 Answers 3

There are already uncountably many isomorphism classes of $3$-dimensional real Lie algebras. In fact, there are $1$-parameter families of $3$-dimensional solvable Lie algebras. The classification has been done over an arbitrary field. For references see the paper of Willem A. de Graaf, and the book of Jacobson on Lie algebras, chapter $1$, section $4$.

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See Chapter 3 of "Introduction to Lie Algebras" by Erdmann and Wildon. They do the classification for complex Lie algebras, but most of the stuff there works for real algebras too.

Let $L$ be a $3$-dimensional real Lie algebra. In the case $[L,L] = L$ you have exactly two possibilities, $\mathbb{R}^3$ with the cross product and $\mathfrak{sl}(2,\mathbb{R})$. When $\dim [L,L] = 1$, you have the heisenberg algebra and $\mathbb{R} \oplus M$, where $M$ is the nonabelian $2$-dimensional algebra. That leaves the case $\dim [L,L] = 2$, and in this case you have uncountably many nonisomorphic Lie algebras.

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Exercise: let $K$ be a field and $t\in K$. Then the Lie algebras $\mathfrak{g}_t$ with basis $(T,X,Y)$ with bracket $[T,X]=X$, $[T,Y]=tY$, $[X,Y]=0$ are pairwise non-isomorphic, with the exception $\mathfrak{g}_t\simeq\mathfrak{g}_u$ if $ut=1$. In particular the real Lie algebras $\mathfrak{g}_t$ for $-1\le t\le 1$ are pairwise non-isomorphic.

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