Yes, LEDs work on AC power, although precautions should be taken when doing that. LEDs do not like to have a reverse voltage applied to them, since that can cause them to fail. Some LEDs will fail with as little as 5V reverse voltage applied to them.
LEDs are polarity sensitive, with the Anode (+) leg needing to be at a higher voltage than the Cathode (-) leg for the LED to light. For a typical red LED, the voltage difference between Anode and Cathode must be around 1.7V for lighting to happen at all. A resistor is placed in series with the LED to limit current to around 10-20mA to prevent LED damage.
When using LEDs with AC power, an easy way to avoid excessive reverse voltage across the LED is to wire a regular 1N4007 diode (or another LED if you wish) in parallel with the original, except with the new diode's leads reversed. So, for AC power just connect the 1N4007 diode's Cathode leg to the LED's Anode leg, and connect the 1N4007 diode's Anode leg to the LED's Cathode leg. That way, no more than 0.6V in reverse voltage appears across the LED, and the LED is happy and won't fail.
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