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I'm starting to learn about Lie algebras in order to decompose a complex 16-dimensional Lie algebra that I have given in terms of the whole multiplication table for the Lie bracket. I want to decompose it into a sum of simple Lie algebras. I should be able to decompose it according to the Levi decomposition and a sum of simple Lie algebras?

Could you please outline the steps and techniques that I will need to do this decomposition? It may help me select the material I need to learn.

Is there a computer package to do that?

Gere
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  • "I want to decompose it into a product of subalgebras." No, I don't think so. You probably want a sum. For groups, one wants a product, e.g., $G=AB$ of subgroups $A$,$B$, and for Lie algebras $L=L_1+L_2$ a vector space sum of subalgebras $L_1$ and $L_2$. Or am I wrong? Onishchik has described such "decompositions". – Dietrich Burde Apr 24 '23 at 12:08
  • The Levi decomposition is a semidirect sum decomposition $L=S\rtimes R$, where $R$ is the solvable radical and $S$ is a Levi subalgebra. There are algorithms (see GAP) to compute it. Your question needs more focus. What exactly do you mean by a decomposition of Lie algebras? – Dietrich Burde Apr 24 '23 at 12:10
  • @DietrichBurde The Lie group should decompose into a product. So maybe I meant that the Lie algebras decompose into a sum. Essentially, there should be parts which one can understand independently of the rest. Are you aware of a software/function which can do that? Or a good references going straight for this recipe to do that? Ideally, I'd not read multiple books for this particular task. – Gere Apr 24 '23 at 12:25
  • Yes, I know how to do decompositions. You only need to say me exactly what kind of decomposition you want. You said " Essentially, there should be parts which one can understand independently of the rest". This is much too vague. – Dietrich Burde Apr 24 '23 at 13:02
  • @DietrichBurde A decomposition into a sum of simple Lie algebras (and whatever is left over which cannot be decomposed further) – Gere Apr 24 '23 at 13:12

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First check the Killing form of $L$. If it is non-degenerate, then $L$ is semisimple, so that $$ L=L_1\oplus \cdots \oplus L_n, $$ with simple ideals $L_i$. Because of $\dim(L)=16$ we only can have $\dim(L_i)\in \{3,8,10\}$, and here we only have the possibilities $16=8+8=10+3+3$, i.e., $$ L\cong \mathfrak{sl}_3(\Bbb C)\oplus \mathfrak{sl}_3(\Bbb C),\quad L\cong \mathfrak{so}_5(\Bbb C)\oplus \mathfrak{sl}_2(\Bbb C) \oplus \mathfrak{sl}_2(\Bbb C). $$

If not, apply the Levi decomposition $L=S\rtimes {\rm rad}(L)$ using GAP and repeat the above decomposition with the Levi subalgebra. The solvable radical ${\rm rad}(L)$ then is left.

Dietrich Burde
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  • Thanks. I still have to learn these topics, so sorry if my description was incomplete. Are there other names for "GAP" - I'm struggling to google that. Do you know a reference where I can learn these topics as quickly as possible to understand just enough to perform these operations? Is the split into simple ideals obvious or are there methods that can help me there? – Gere Apr 24 '23 at 13:42
  • For GAP, see here. It has many packages for Lie algebras. The decomposition into simple ideals is obvious here, because your Lie algebra has dimension $16$ (as an exercise, find the decomposition of a semisimple Lie algebra of dimension $3$). In general, you can use GAP again. – Dietrich Burde Apr 24 '23 at 13:46
  • Oh, I didn't realize GAP is a software. And I need to install plugins? I've checked the PDFs, but I'm a bit lost :/ Could you link me to the function name that I will need to understand in order to do the Levi decomposition? And what is the function name to split into simple ideals? Then I could try to get it running. The Levi decomposition will split off a part. Can I not have that the rest is 3+8? – Gere Apr 24 '23 at 14:01
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    See 64.6 here. – Dietrich Burde Apr 24 '23 at 14:05
  • Does anything get easier if my Lie algebra is derived from a commutator of an associative algebra which is known to me? – Gere Apr 24 '23 at 14:10
  • For me, yes, for you - probably not (forgive me this comment, but it is not so easy to explain how you can use it, and more importantly, it will not be useful for you, as you also don't know it yet). I suppose the easiest way is just to find a Levi subalgebra first, and determine the solvable radical. – Dietrich Burde Apr 24 '23 at 16:19
  • Ok thanks. I understand. My underlying algebra is the Clifford algebra Cl4(C). I'm guessing the GAP functions, but I couldn't find an obvious way to generate this Clifford algebra comfortably. Is there an easy way to create Cl4(C) in GAP (and then use it in my Lie decomposition)? – Gere Apr 24 '23 at 16:34
  • If you have the Clifford algebra of a quadratic form, the quadratic elements of the Clifford algebra give you the Lie algebra of the orthogonal group of that quadratic form. – Dietrich Burde Apr 24 '23 at 17:00
  • Sorry to ask again - feel free to ignore if it's too much. I didn't understand the last comment. Of course, a Clifford algebra Cl4(C) yields a 16-dimensional vector space and a Lie algebra through the commutator. But can I decompose this 16-dimensional Lie algebra into something simpler? – Gere Apr 24 '23 at 17:18
  • Yes, you can decompose it into something simpler (depending on what you understand by "simpler"), see my answer. You can classify the Levi part, and hopefully also the solvable radical (but the known classification for solvable Lie algebras is only in low dimensions). – Dietrich Burde Apr 24 '23 at 17:34