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What are some examples of Lie algebras of the $BC$ root system type please? I am actually interested in the corresponding groups too. I heard that there were Lie algebras over $\mathbb{R}$ having $BC$ root system type. Can someone provide a definition, or at least, some references please?

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    $\begingroup$ @Travis: I think what the OP means is Lie algebras whose (rational = belonging to a maximal split torus) roots form a root system of the non-reduced type usually called $BC$. That can of course not happen for split Lie algebras / over algebraically closed fields. $\endgroup$ – Torsten Schoeneberg Mar 3 at 5:38
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First, here is a quite generic example of simple Lie algebras over certain fields $k$ whose $k$-rational root system is of the non-reduced type $BC$. It works for any field $k$ of characteristic $0$ which has a proper quadratic extension, and gives a quasi-split form of type $A_{n \ge 2}$; I quote from example 3.2.10 (p. 53) of my thesis (for a more geometric interpretation, cf. exercise 16.a to ch. VIII, §13 of Bourbaki's Lie Groups and Lie Algebras):


Let $k$ be any field of characteristic $0$ which has a proper quadratic extension $K = k(y)$ with $y^2\in k$, and let $\sigma \in Gal(K\vert k)$ satisfy $\sigma(y) = -y$; for $k =\Bbb R$, take $y=i$ (the imaginary unit $\in K=\Bbb C$), and $\sigma$ the complex conjugation.

Let $n \ge 2$, $d = n+1$, and consider, inside $\mathfrak{sl}_d(K)$ viewed as a $k$-Lie algebra, those matrices $(x_{i,j})$ which satisfy $x_{i,j} = -\sigma(x_{d+1-j, d+1-i})$; that is, those traceless $d \times d$ matrices over $K$ such that each entry is the negative conjugate of the one mirrored at the secondary diagonal (in particular, the entries on the secondary diagonal are $k$-multiples of $y$); or in yet other words, the traceless $d \times d$ matrices $(x_{i,j})$ over $K$ satisfying $(x_{i,j}) \cdot H + H \cdot \, ^t(\sigma(x_{i,j})) = 0$ where $H$ is the $d \times d$ matrix with entries $1$ on the secondary diagonal and $0$ else. A maximal split toral subalgebra is \begin{align*} \mathfrak{s} := \{ diag(x_{1,1}, ..., x_{d/2, d/2}, \;-x_{d/2, d/2}, ..., -x_{1,1}) : x_{i,i} \in k \} \end{align*} or \begin{align*} \mathfrak{s} := \{ diag(x_{1,1}, ..., x_{n/2, n/2},\; 0, \; -x_{n/2, n/2}, ..., -x_{1,1}) : x_{i,i} \in k \} \end{align*} according to whether $n$ is odd or even. One calculates that for odd $n$, the rational root system $\overline R$ is of type $C_{d/2}$, whereas for even $n$, it is of type $BC_{n/2}$.


Note that over $\Bbb R$, the above Lie algebra (for even $n$) is denoted by $\mathfrak{su}_{\frac{n}{2}, \frac{n}{2}+1}$ e.g. in Onishchik/Vinberg: see Table 9 (p. 312) here. The simplest case $\mathfrak{su}_{1,2}$ is fleshed out a bit here and in example 3.2.9, p. 51-53 of my thesis.

There are more simple Lie algebras over $\Bbb R$ whose real roots form a system of type $BC$. According to and with the notations of the above Onishchik/Vinberg reference, the full list is:

  • $\mathfrak{su}_{p, l+1-p}$ ($l \ge 2$, $1\le p \le l/2$): rational root system $BC_p$ (complexification is of type $A_l$); for even $n=l$ the above was the special case $p=n/2$;
  • $\mathfrak{sp}_{p, l-p}$ ($l \ge 3$ odd, $1 \le p \le (l-1)/2)$: rational root system of type $BC_p$ (complexification is of type $C_l$);
  • $\mathfrak{u}^\ast_{2p+1}(\Bbb H)$: rational root system of type $BC_p$ (complexification is of type $D_{2p+1}$);
  • $EIII$ (called $^2 E^{16'}_{6,2}$ by Tits in the Boulder Proceedings): rational root system of type $BC_2$ (complexification is of type $E_6$);
  • $FII$ (called $F_{4,1}^{21}$ by Tits in the Boulder Proceedings): rational root system of type $BC_1$ (complexification is of type $F_4$).

Note: Over number fields, there might be some more, I am trying to work through Tits' list in the Boulder Proceedings.

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  • $\begingroup$ Interesting. Thank you! I understood the example you explained. What are some standard references on the topic please? You mentioned Onishchik/Vinberg and your thesis. I am slowly making my way into algebraic groups, knowing only the corresponding theory over C only. $\endgroup$ – Malkoun Mar 3 at 12:37
  • $\begingroup$ For semisimple algebraic groups / Lie algebras over $\Bbb R$, see the answers to this recent question: math.stackexchange.com/q/3121110/96384. For general fields, I found parts of Springer's textbook on Linear Algebraic Groups very helpful (in particular the second edition, the first dealt mainly with the alg. closed case). And there's always the big Borel-Tits article numdam.org/item/PMIHES_1965__27__55_0, and the "Boulder Proceedings": www-fourier.ujf-grenoble.fr/~panchish/ETE%20LAMA%202018-AP/… $\endgroup$ – Torsten Schoeneberg Mar 3 at 19:18
  • $\begingroup$ Thank you! Particularly for the Boulder Proceedings, which I could not find online yesterday. I will have a look at the answer you referred to as well. $\endgroup$ – Malkoun Mar 4 at 20:55

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