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I searched examples for the verb "ageru - raise" and I found it written with two different kanji:
挙げる
上げる
Mostly their meaning is "raise". I found them with the following examples:
棚に箱を上げる。Put a box on the shelves.
学校で生徒は手を挙げる。Put one's hands up
旗を揚げる。to raise a flag (a third kanji for ageru! :) )
I would like to ask if there are strict rules to use these kanji in certain cases just like in the examples, or I can use them in a transposed way?
My dictionary lists five uses:
上げる - moving something to a higher position / status
荷物を棚に上げる Put the luggage on the shelf 利益を上げる Raise profits
上げる - give or supply something
歓声を上げる Give a cheer プレゼントを上げる Give a present
挙げる - indicate, or cause something to happen
手を挙げる Raise your hand 式を挙げる Hold a ceremony
揚げる - suspend in space, fry
国旗を揚げる Raise the national flag てんぷらを揚げる Fry tempura
あげる - do something for somebody
本を貸してあげる Lend a book
To address your actual question: it depends on the meaning you want to convey. Obviously for "fry" you would only use 揚げる, which is a bit disconnected from the other meanings. The difference between 手を揚げる and 手を上げる is a bit more subtle: the latter might refer to the physical motion of raising your hand, the former to indicate you wished to speak. In the last case, the verb ending 〜てあげる is listed as all-hiragana, but I've seen 〜て上げる in the wild.
