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I am looking for a holistic book on lie algebra representation, including the infinite dimensional cases. I heard Humphrey book is standard but it does not seem to include the infinite dimensional Lie algebras

Could anyone please recommend me a book that would fit my interests?

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Anthony W. Knapp has a book, namely Representation Theory of Semisimple Groups, which includes infinite-dimensional representations of Lie algebras and Lie groups. There are many other books, e.g., Unitary Representations of Reductive Lie Groups by D. Vogan. Which one fits your interests most is of course up to you.

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VS Varadarajan's Lie Groups, Lie Algebras, and their Representations may suit your purposes.

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