The short answer is yes.
Unfortunately it is quite difficult. For racing, a total loss battery system is normal: you get a limited time running then you need to recharge the battery (one would have a few to swap in between heats during a race meet).
There's also the magneto approach, but like the total loss battery you don't have lights (at all with a magneto, and not for long with total loss) so street use is discouraged.
You can take the stock alternator off and install a permanent magnet type. This still presents starting problems, the engine has to be spinning pretty fast to make enough power for the ignition. Many two stroke engines (Yamaha RD400 for example) use a magneto system for ignition and a permanent magnet alternator for lighting, you can see their lights dim and flicker at idle when the alternator output is weak and the AC frequency gets low.
The simplest approach is to use a small gel-cell or AGM battery to provide ignition power for starting and to provide ignition and lighting power at low rpm where the alternator output is below the electrical load power used, plus the alternator regulator needs a battery type load attached to operate properly. You may still get in trouble if you ride in a city a lot, stop and go driving will discharge the battery so you need enough capacity to last you.
