Is there any way to fix broken art glass?  

The only possible way
The only possible way is to grind the beaks down to a more pleasing shape.
There is no other way of 'repairing' glass, by it's very nature, any mend would be visible.

Not completely...
true. You can not repair glass without melting it and although it can be done locally for instance on the broken beak, it would require unusual skill...but, one can grind a flat surface and invisibly glue an piece of glass onto it and grind it into the right shape. I was very surprised seeing it the firt time but a french glass artisan who worked here fore a while showed me.

What
sort of adhesive would it be, do you know ?
Forty years ago a couple of friends were throwing pillows in the room where their small art glass collection stood (!) (the old definition of the term: turn-of-the-century post-art nouveau) and a miniature Tiffany vase, transparent golden colored, was smashed. I saw it after repair; only the faintest hint of cracks could be seen. I was amazed. . .

No...
but I could ask..

Not important
to me -- I was just pondering. I'm only aware of the epoxies and the cyanoacrylates; I wonder what that stuff is that is used to fill "star cracks" in auto windshields. . .and what is used to glue the mirror mount to a windshield ?
So many questions; so little time !

oh ys, so little time...
epoxies and the cyanoacrylates are widely used, but in glass silicones are very succesful glues, mostly not by adhesion but by cohesion.