I wonder what's the way to regenerate O2 in air without using consumable chemicals (where one can use electricity through electrolisis or using UV lamps)?
We can dissolve water into O2 & H2, but it won't remove CO2 from air.
Here is a Babelfish translation from German Wiki on Life support systems. (Including NASA!) (English WIKI on the topic is very bad)
But topic here is some (realistic) method for barsmonster to survive the Moscow summer. This is best done by cooling the air intake, demoist it (which would hopefully remove some of the organic "smog" particles) and one could try some PP microfibre filters. The exhaust air would be used to cool down the air taken in. Any separation of carbon dioxide by ad(ab)sorbents would be much more expensive than this air exchange.
There is no inorganic method of which I am aware that is not "messy." They all consume and/or produced something extraneous to the clean reaction you want ---> CO2 + energy = C + O2.
NASA (these are the people who should know) use electrolysis on the ISS to split 2 H2O into 2 H2 + O2. The H2 is vented overboard, the O2 is vented to the cabin. CO2 is captured by a separate system and also vented overboard. With the cost of upmass being on the order of $10K/lb., you can bet that if there was an easy way to rip the O2 out of CO2, that they'd be doing it.
Using photosynthesis sounds like the easiest way to re-use O2 that is bound in CO2. The NET equation is 6 CO2 + 6 H2O + energy (light) ---> C6H12O6 + 6 O2. This uses half the water per liter of O2 produced as electrolysis. If the sugar produced can be consumed, then that upmass is saved also (reduces the weight of food that needs to be shipped to orbit).
Rather than just splitting water, you want to react it with the CO2 to make O2 and carbohydrates. I think this can be done with an inorganic catalyst (I'd have to Google for the details) but of course the easiest solution is to use photosynthesis.