Learn Swedish: Journalisten ljög uppenbarligen om händelsen. - The journalist obviously lied about the event.

The suffix -ligen is used to turn certain adjectives into sentence adverbs. Here, it is added to the adjective uppenbar (obvious) to create uppenbarligen (obviously). It is especially common to use this suffix with adjectives ending in the suffix -bar.

Could I use the standard adverb form uppenbart instead of uppenbarligen here?

Yes, but there is a subtle difference in nuance. Uppenbarligen functions specifically as a sentence adverb expressing the speaker's deduction, meaning 'evidently' or 'it is obvious that'. Uppenbart is the standard adverb form meaning 'in an obvious manner'. While they overlap in this context, uppenbarligen is the much more natural choice for commenting on the sentence as a whole.

I sometimes mix up the verb forms for 'to lie'. What are they for ljög?

The verb att ljuga (to tell a lie) is a strong verb. Its principal parts are ljuga (infinitive), ljuger (present), ljög (past tense, used here), and har ljugit (supine/present perfect).

How does the word händelsen relate to the verb att hända?