Scales to Play Over Amaj9
Every chord is a subset of several scales: any scale that contains all of a chord's tones fits over it. Pick a chord to see the matching scales from its own root, the major keys the chord belongs to, and full fretboard diagrams for every scale on guitar.
An Amaj9 chord contains the notes A – B – C# – E – G#. Any scale that includes every one of those 5 tones fits over it — from the root A, that means A Major, A Lydian and A Mixolydian Bebop. They differ only in the notes between the chord tones, and that is where each scale gets its color.
Amaj9 is also diatonic to 2 major keys: A major (relative minor F#m), E major (relative minor C#m). In a song from one of those keys you can also solo with the key's own major scale or its relative minor pentatonic across the whole progression, not just this chord.