Activation allows (to reduce/ reducing) the temperature of the process.
I found that "reducing" can be used after "allow". Moreover, we can write "to reduce" after "allow", too. But which one is more common?
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I am making this a community wiki since this answer is totally based upon the conclusions of another answer, which is this answer given by @FumbleFingersReinstateMonica on a similar (almost same) question.
Answering your question in simple words, these are the possible and correct variations:
Activation allows us to reduce the temperature of the process.
Activation allows reducing the temperature of the process.
Activation allows reduction of the temperature of the process.
For detailed explanation you can refer to the post I have linked or the post which is linked therein pointing to an answer on ELU.
Without more context it is hard to see how you have ended up trying to use the construction you are attempting to use.
“Activation allows” is an ugly way to start the sentence and leads you into difficulties.
Would “Activation reduces the temperature” meet the needs of the message being communicated?
The option "allows to reduce" needs an object in between to complete the meaning, so it will be "allows us to reduce."
The combination "allow + -ing" is both pragmatically and idiomatically valid in your sentence in that I think it's most similar to the verbs "enables/provides/includes an [additional] option".
Both are common - it's a matter of style and interpretation.