To any vertex algebra one can attach invariants of different nature: its automorphism group, its character (a formal series), its associated variety (a Poisson variety), etc. In this talk, I will explain how to exploite the connection between these invariants to obtain nontrivial isomorphisms between W-algebras at admissible levels. To study a more general setting, one can use totally different technics developed more recently. This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.