Examples of Lie algebras of the $BC$ root system type

Question

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?

Answer 1

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):

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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 $BC_{n/2}$.

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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.