I was skimming a paper and got stuck in the middle. As you see in the underlined parts, the authors first assumed that $\mathcal{G}$ is a simple algebraic group. Then $\mathcal{G}$ is defined to be $SL_n(\mathbb{F})$.
But according to the descriptions of this MO question, $SL_n(\mathbb{F})$ is almost-simple but not simple algebraic group. I'm not so familiar with theory of algebraic groups and got so confused. How should I explain this? I would be grateful for any guide.
A algebraic group over a field k is simple if it is non-commutative and has no closed connected normal subgroups other than itself and e. The word "almost simple" is used if we wish to emphasize that the group need not be simple as an abstract group).(see, J. E. Humphreys, Linear algebraic group (1998), pp168).