# Simple Subalgebras of Simple Lie Algebras

Given a complex simple Lie algebra $\mathfrak{g}$ of rank $n\in\mathbb{N}$ with $n$ sufficiently large (say $n\ge10$), is there a way to determine whether $\mathfrak{g}$ contains a simple subalgebra of a prescribed type with rank "close" to $n$? For example, if $\mathfrak{g}$ is of type $B_n$ with $n\ge10$, does $\mathfrak{g}$ contain a subalgebra of type $C_{n-2}$?

• Some more tags are natural here, such as rt.representation-theory. Many of us rely on the tags to identify questions which may be interesting. (Also, it's useful to highlight the basic question being asked with > followed by space.) Sep 11 '18 at 0:04

The answer to the question in your example is no in general: $B_n$ does not contain $C_{n-2}$ for large $n$. To see this, observe that $B_n$ has an irreducible orthogonal representation $V$ of dimension $2n+1$. The Weyl dimension formula shows that for any simple Lie algebra, the dimension of an irreducible representation is the smallest for fundamental representations.

[Edited]

For large $n$, the smallest dimensional fundamental representation of $C_{n-2}$ is the standard one, of dimension $2n-4$. Therefore the restriction of $V$ to $C_{n-2}$ can only be some copies of the trivial representation and ONE copy of the standard representation $W$ of dimension $2n-4$. This means that the group of type $C_{n-2}$ must preserve both a symplectic form and a quadratic form on $W$, which is impossible by Schur's lemma. Hence the restriction of $V$ to $C_{n-2}$ can only be a sum of copies of the trivial representation which is impossible.

I had previously given a wrong reason (assuming that the rep of $B_n$ had dimension $n$; thanks to @BS for pointing this out.

• I thought $B_n$ was $so(2n+1)$. What is its $n$-dim irrep ?
– BS.
Sep 16 '18 at 8:16
• Thank you. You are right (I had a brain fade). I have given the correct reason now, I hope. Sep 16 '18 at 8:45
• Schur's lemma usually refers to the statement that the space of equivariant endomorphisms of an irreducible representation is one-dimensional. How does that implies that $C_{n-2}$ cannot preserve two binary forms? Sep 16 '18 at 9:06
• The existence of an invariant Bilinear form on W gives an equivariant map between W and it’s dual. By Schur’s lemme, there is only one such up to scalars Sep 16 '18 at 10:00

I am not sure if this is exactly what you are looking for, but there have been some classic works, developing general methods for such topics:

Simple algebras of rank >8 are classical, so you are asking is there a representation (linear, orthogonal or symplectic) of a given dimension of a prescribed Lie algebra. This amounts to the question what is the minimal dimension of a nontrivial linear, orthogonal or symplectic representation of the Lie algebra. A table in Bourbaki answers this.

• Can you be more specific about the location of the "table in Bourbaki"? Sep 10 '18 at 23:59
• Tables I and II in "Groupes et algèbres de Lie, Chapitres 7 et 8" (I use Russian translation but I hope the tables are in the original text as well and not added by a translator). Sep 11 '18 at 12:08
• Since my copy is the first printing of the French edition, I can confirm that these tables occur at the end of the text of Chapter 8 before the exercises. (There is an obvious misprint $F_6$ for $E_6$ in Table II.) Note however that the tables claim only to provide data for the fundamental representations in each Lie type. Sep 11 '18 at 13:24
• The table in Bourbaki yields the fundamental reps, their dimension, and say if they are orthogonal, symplectic, or none. If $V$ is any rep of dimension $d$, then $V\oplus V$ can be made both orthogonal and symplectic. So how do we conclude? the issue is that possibly, if there's a $d$-dimensional rep, are there non-fundamental irreducibles of dimensions $<2d$? is there a way to list them, and know if they're orthogonal/symplectic?
– YCor
Sep 11 '18 at 16:10
• For the last part of the question, Proposition 12 in Chapter 8 of Bourbaki gives a nice criterion to be orthogonal or symplectic in terms of the coordinates of the highest weight of a representation. The dimension, as mentioned above, can be computed via the Weyl dimension formula. But I don't know an explicit procedure how to list all representations of dimension less than a given bound. Sep 11 '18 at 18:09