Arduino triggering a TRIAC for EL Wire, no optoisolator
Looks like a cool project.

literal answer

The optoisolator is not necessary in this application.

Because you are generating the 100 V, 1000 Hz power to drive the EL from relatively isolated battery power (rather than mains power), there is much less of a safety issue.

Systems without an optoisolator typically connect the A1 pin of the triac is connected to the VCC of the microcontroller (in your case, the +3V supply), using "negative gate current triggering" as recommended.

A digital output pin on your microcontroller is connected with a resistor to the gate of a triac.
When the digital logic pulls the gate pin low (towards the microcontroller GND), the triac is triggered and turns all the way on.
As long as the triac is on, the A1 and A2 pins act like they are shorted together.
Turning the triac off is a little more difficult.

(A few systems without an optoisolator connect the A1 pin of the triac to the GND pin of the microcontroller, using "positive gate current triggering", which is not recommended.
As I recently learned,
hooking the the "GND" pin of the microcontroller to A1 and pulling the gate through a resistor to +3 V or even +5 V doesn't work right with a logic level triac.)

Try to draw your schematic and lay out your parts so it's obvious that:


one end of the inverter output is solidly connected to a harmless-to-the-microcontroller voltage (probably +3V) and pin A1 of the triac
the other end of the inverter output (the "hot side") is not directly or indirectly connected to anything anywhere near the microcontroller -- except for the triac, and even then the hot side is only indirectly connected through the EL wire to pin A2 of the triac.


alternate approach

If you're only going to have one strand of EL wire,
why don't you connect it directly to the inverter output,
and use a FET (rather than a triac) to connect and disconnect the inverter input to the +3 V power?