11
$\begingroup$

I need a reference to a complete list of all faithful real 4-dimensional irreducible representations of real Lie algebras.

The list itself is not very hard to obtain. Using the Levi decomposition, it's possible to see that the Lie algebra $\mathfrak g \subset\mathfrak{sl}(4, {\Bbb R})$ which does not fix a proper subspace has to be reductive. Then (up to a center acting by constants and/or by rotations with constant angle), $\mathfrak g$ is a semisimple subalgebra of $\mathfrak{sl}(4, {\Bbb R})=\mathfrak{so}(3,3)$. The list of such subalgebras (I think) is $\mathfrak{so}(1,2)=\mathfrak{sl}(2, {\Bbb R})$, $\mathfrak{so}(3)$, $\mathfrak{so}(2,2)= \mathfrak{so}(1,2)\times \mathfrak{so}(1,2)$, $\mathfrak{so}(4)=\mathfrak{so}(3)\times \mathfrak{so}(3)$, $\mathfrak{so}(1,3)=\mathfrak{sl}(2, {\Bbb C})$, $\mathfrak{so}(2,3)=\mathfrak{sp}(4, {\Bbb R})$, $\mathfrak{sl}(4, {\Bbb R})=\mathfrak{so}(3,3)$. It can be (probably) obtained by removing some vertices of the Dynkin diagram of $A_3$ and using the arcane technique of coloring the remaining vertices to obtain the real forms. All in all, it's a pain.

I am sure there is a reference somewhere to this list (also I might have missed some algebras). I would be much grateful for all pointers.

$\endgroup$
1
  • 1
    $\begingroup$ I think it's recommendable in such a list to separate the 3 cases: centralizer is $\mathbf{R}$ (= absolutely irreducible case), centralizer is $\mathbf{C}$, centralizer is $\mathbf{H}$. $\endgroup$
    – YCor
    Jun 24, 2019 at 16:35

2 Answers 2

5
+50
$\begingroup$

A complete list of semisimple subalgebras of $\mathfrak{sl}(4,\mathbb{R})$ seems to be available here: http://downloads.hindawi.com/journals/jmath/2016/2570147.pdf

$\endgroup$
1
  • $\begingroup$ Million thanks! $\endgroup$ Jun 25, 2019 at 21:10
2
$\begingroup$

Your list of semisimple subalgebras is not quite complete. You are missing (at least) $\mathfrak{sl}(3,\mathbb R)$ and $\mathfrak{sl}(2,\mathbb R)\times \mathfrak{sl}(2,\mathbb R)=\mathfrak{so}(2,2)=\mathfrak{so}(2,1)\times \mathfrak{so}(1,2)$. See Marcel Berger
Les espaces symétriques noncompacts, 1957, MR0104763.

$\endgroup$
3
  • 1
    $\begingroup$ $sl_3$ does not act irreducibly in dimension 4 (but indeed $so(2,2)$ does). $\endgroup$
    – YCor
    Jun 20, 2019 at 8:33
  • $\begingroup$ many thanks! I would modify the question $\endgroup$ Jun 24, 2019 at 16:28
  • 2
    $\begingroup$ I have read (cursorily) Berger's paper, and did not find anything which was helpful, but it's 94 pages long. Could you be so kind to point it more precisely. The paper is here: numdam.org/item/?id=ASENS_1957_3_74_2_85_0 Many thanks! $\endgroup$ Jun 24, 2019 at 16:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.