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I'm a graduate student studying algebraic geometry. Recently, When I studying Hodge theory, I saw sl2-representation is used in Hodge theory. So I think that studying representation theory may be helpful for me. Can you recommand me some good text for studying representation theory, focused on materials helpful for algebraic geometry, and not so difficult to read.

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Should be community wiki – Igor Rivin Jun 13 '12 at 14:40
This question is not exactly identical with yours, but the answers posted there could just as well be posted here. – darij grinberg Jun 13 '12 at 20:15
up vote 2 down vote accepted

I would recommend Introduction to Lie Algebras and Representation Theory by Humphreys. It covers all the basics of Lie algebras and their representations, though mostly in characteristic 0 and over an algebraically closed field. But then, going into any depth with the theory without these assumptions requires a lot of additional work anyway, and knowing the theory for this nice case is a good start.

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Recently, Pavel Etingof published a book about his course on representation theory with his students. This is a description of the book:

"The goal of this book is to give a "holistic" introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints.

The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra."

This is a very enjoyable book to read. A version can be found here: link text. You get a bonus buying the book, where some nice historical remarks are added.

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The bonus also contains two short digression-style chapters on homological algebra (although I think they won't surprise an algebraic geometer). – darij grinberg Jun 13 '12 at 20:18

A nice good book has been written by Fulton and Harris, of course there are many others

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Indeed Fulton and Harris is very appropriate for someone focusing on algebraic geometry. – Claudio Gorodski Jun 13 '12 at 19:13
That's a good collection of examples and exercises, but a pretty bad text for reading. Flawed proofs, lots of missing details, lack of separation between results and proofs make learning much harder than it should be. – darij grinberg Jun 13 '12 at 20:13

I'd also recommend

Erdmann, Karin; Wildon, Mark J. Introduction to Lie algebras.

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I taught a 1-semester class for seniors/1st year grad students out of this book and they all really enjoyed it. It should be very suitable for self study as well. – daveh Jun 14 '12 at 12:38

If you are particularly interested in sl2-representation theory, there is the book by Mazorchuk: Lectures on $\mathfrak{sl}_2(\mathbb{C})$ - modules.

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