Can I power my Arduino board and main circuit with the same DC power supply?

Hello I am a newbie. I wish to let Arduino board drive a bunch of high power LED lights using a transister, and I plan to use 9V DC power supply for both Arduino board and the lights. Can I simply use a single DC adapter and split it into 2 with a barrel connector splitter and power them both at the same time? Can this cause any potential problem?

Based on the information you have given I don't see a problem.

The only possible problem I see is in the ground connections, this is wire size and current dependent defined by the LED load. Use a single point (star) grounding configuration. Led current flowing on the wire that feeds the Arduino can introduce noise into the Arduino giving random problems. To keep things cooler and save parts consider using a n-channel avalanche rated MOSFET. The transistor will have a 0.7 to 1.4 volt drop from emitter to collector. The MOSFET will have very little depending on the FET you chose. This response is to help you get started in solving your problem, not solve it for you.
Good Luck & Have Fun!

Yes It is possible!

Make sure to common all the GNDs.