There are a couple quick tests you can try on your lawn tractor. Unfortunately, your Manage my Home profile was not completed so I have no specific model information to provide more detail.
The simplest test is to put a Volt Meter on the battery posts before you start the engine to see what the current charge level is. Then check the voltage at the battery posts while the engine is running at full throttle. You should see a noticeable increase in the overall voltage reading at the battery when the engine is running.
Starting and running the engine draws voltage from the battery and, once the engine is running at full throttle, the charging system compensates by sending voltage back to the battery to charge it. If the voltage does not increase at all, or very little, when you check it with the engine running at full throttle then there may be a problem with the charging system.
If there is increased voltage going to your battery and the battery still keeps running low, you may just need a new battery. Sometimes a battery will lose its ability to "hold" onto the charge voltage it is being sent.
Another possibility is some kind of short or draw in the lawn tractor electrical system. To check, turn off the tractor then remove the ground cable from the battery (normally this is the black cable), then remove the positive battery cable. Reattach and tighten the ground cable to the ground post. Then put an Amp Meter between the positive cable and the positive post of your battery, be careful not to touch any metal parts with the cable or your tester.
The Amp Meter should show no amperage being drawn off the battery at all. If there is an amperage reading then something is drawing power from the battery even when the tractor is turned off and this is slowly killing the battery.