Learn Arabic (Modern Standard): هل يمكن أن أجلس هنا؟

In Arabic, the verb يمكن (it is possible) is usually kept in the 'it' form. Instead of saying 'I can sit', Arabic phrasing is closer to 'Is it possible that I sit?'. You don't change يمكن to match 'I'; you just change the second verb (أجلس) to match who is doing the action.

What is the exact role of أن in this sentence?

The word أن means 'that' or 'to'. When you have an expression like 'it is possible' followed by an action, you need أن to connect them. It works a lot like the English word 'to' in 'I want to sit', but notice that the Arabic verb after it (أجلس) is still fully conjugated for 'I' (I sit).

How do we know it means 'I sit' if the word أنا (I) isn't in the sentence?

The 'I' is built right into the verb! The letter أ (alif) at the very beginning of أجلس is the prefix that tells us the person sitting is 'I'. You don't need to add the separate word أنا unless you want to add special emphasis.

Can I use this same هل يمكن أن structure to ask if someone else can do something?

Yes! هل يمكن أن (Is it possible that...) is a very common and polite way to ask about ability or permission. To ask 'Can we sit here?', you just keep the first part the same and change the second verb to match 'we': هل يمكن أن نجلس هنا؟.