Question about home theater powered subwoofers, inputs on sub amplifier?

It is hooked up the correct way now. You are not losing anything, because all the frequency management is being handled by the receiver. I would use the y adapter (or try it without - if there is a drop in sub level, keep it).

Back in the day when audio electronics was evolving from monophonic sound to stereophonic sound, manufacturers of stereo audio electronics designated the left channel input for monophonic sound (if you only had one input) and, if there was nothing plugged into the right input jack, the monophonic sound was distributed to both left and right inputs. Since mono is pretty rare these days, I don't know if this is still standard on stereo inputs (including powered subwoofers), but you might try it.

If this is not the case (plugging it into the left input just goes through the left amplifier), whether you split the signal or not is basically a wash. A subwoofer has two voice coils on the driver, one powered by the left input and the other powered by the right input. If you just have, say, the left input plugged in, you're only driving the subwoofer with the left amplifier and voice coil. If you split it, you're now driving both voice coils. But, the split itself cuts the signal level in half. So, you have half as much signal input to each amplifier and the output volume of the subwoofer won't be any different than just plugging in one input.

One thing you could do is to get a small, cheap stereo amp or receiver, plug the LFE output into one input, set the amplifier mode to mono, and run the two line outputs to the subwoofer. That would give you two full line level inputs to the subwoofer from the one LFE cable.

Basically you are connecting multiple amplifiers in series. It usually has wattage and impedance mismatch because they are not designed to be fully compatible with each other. You may notice sound change at low volume. But the maximum bass power will be that of the subwoofer. Subwoofers are fast solutions but if you want a wide expansions in the future, buy a high powered receiver amplifier and loud speaker box. Also you may overheat the subwoofer by overdriving it with that receiver. Subwoofers can handle 5 watts or less input. Above the specs may overheat it resulting in distorted sound when pumping the volume up.

The way you have it connected is fine. The receiver is doing the work, sending the low frequency signal to the sub woofer. You should check the frequency cut off on the receiver to she what setting it is set at, common setting should be around the 80HZ and below going to the sub woofer. Hope this will help you out.