¡Aliviánate! is a colloquial way of saying: relax, chill, calm down, simmer down, you get the idea.
Alivianar is to lighten, so when you think about it, it completely makes sense that alivianarse means to chill, relax, calm down (what we are doing is lightening our load, isn't it?).
Necesito aliviniarme - I need to calm down.
You can describe a person as being alivianado/a:
Juan es muy alivianado -Juan is very easy going.
Lupe no es nada alivianada -Lupe is not easy going at all.
Aliviane is used in some expressions:
¡Qué aliviane! What a relief!
Esto es un aliviane - This is a relief.
You will not be using this expression in business meetings or formal conversations, but it is a great way of showing your friends how much Mexican you speak.
An important note: Some people may use aliviane for when they are high (from drugs), but you will know this by the context of the conversation.