Compounds with aluminum and nitrogen atoms directly bonded to each other as part of an aromatic ring?
Again no.  The aluminum-nitrogen bond is much stronger as a sigma bond than as a pi bond.

Borazine, with boron rather than aluminum in the six-membered ring, is known as an aromatic ring and even called "inorganic benzene".  Like its benzene counterpart, the borazine ring can be fused to give polymeric structures ultimately leading to hexagonal boron nitride.  But with aluminum nitride, where pi overlap is weaker and the bonding more polar, such a layered phase is not known, this nitride most commonly having a wurtzite structure.  In effect, the Al-N pi bonds are broken up by cross-linking with other nitrogen atoms even where the pi bonded structure is ostensibly aromatic.

Take a cue from this answer.  You are on firmest ground with aromaticity when the elements involved are a subset of boron, carbon, nitrogen, oxygen, sulfur.