Hello.
Goal: Identify the italicized word
Sentence: (May) I go to the library?
Kind of sentence: interrogative.
Italicized word: May.
May is a modal verb: a type of verb that is used to indicate modality – that is: likelihood, ability, permission, request, capacity, suggestions, order and obligation, and advice etc. In this case, due to the kind of sentence, may acts like and auxiliary verb.
An auxiliary verb is a verb used in forming the tenses, moods, and voices of other verbs. May, in this case, indicates modality and permission.
Hope it helps you.