After a semicolon what follows isn't capitalized like at the beginning of a sentence. However, what if it were? Would that be incorrect?
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Proper nouns are always capitalized, even if they follow a semicolon; that is true for adjectives derived from proper nouns, when they are normally capitalized.
The "rules" about capitalizing a word after a semicolon are not different from the "rules" about capitalizing a word after a comma.