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I have had little exposure to algebra during my undergraduate degree, covering essentially only the basics of group theory with an emphasis on the symmetries of Euclidean space, and a course on Galois theory. I hugely enjoyed these subjects and I'd love to learn more algebra.

However I need to do this in self - study mode, and also I'd need to narrow the topics a little. I am doing a PhD in topological analysis so the background material I study these days is broadly differential geometry, functional analysis, algebraic topology. I do not confine my interests to these (interesting!) areas in general but I have to restrain myself due to time constraints.

Question 1:

What are the most suitable topics in algebra that I could study on the side? I already have some ideas, such as multilinear algebra, module theory, Clifford algebras (I need to understand Dirac operators) or homological algebra, but I am not sure these are actually the "optimal" choices (subject to the constraint that I need to focus on finishing my PhD). Another way to go would be Lie theory and Representation theory (closely related and very useful in "my" area, though not purely algebraic). But as I said there may be better choices, also I am aware that my suggestions differ widely with regards to their generality.

Another topic that I dearly wish to become familiar with is Category theory! If I find the time to spend more time with Algebra I'd prefer to do it from the point of view of Category theory.

Question 2:

What are the recommendable graduate level books on Algebra that are well suited for self - study, focus on the appropriate topics (as suggested above, for example) and take Category theory into account? (By graduate level I mean that I have some mathematical maturity, though my Algebra background is weak.)

Many thanks!

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    $\begingroup$ Although the text doesn't address most of the specific topics you've listed above, Dummit and Foote's Abstract Algebra covers most all of the basics and is probably perfect for your level. For Lie Theory, Brian Hall's "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction" is a nice intro to that stuff. For category theory, I rather like the text by Steve Awodey. $\endgroup$ – Bill Cook Oct 23 '13 at 22:14
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    $\begingroup$ +1 Bill for the suggestion of Brian Hall's book. Really learned a lot from there. $\endgroup$ – user38268 Oct 23 '13 at 22:32
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    $\begingroup$ How many times has this question been asked on math.SE? $\endgroup$ – Martin Brandenburg Oct 23 '13 at 23:19
  • $\begingroup$ @MartinBrandenburg, unfortunately I don't have enough enough score to access to data in MSE but I am sure it would be an element of $\mathbb{N}$ :-) $\endgroup$ – user231343 Nov 2 '18 at 20:55
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Paolo Aluffi's "Algebra : Chapter 0" is a kind of introduction to algebra in the categorical way. However I would recommend Isaac's "Algebra A graduate Course" despite of it not deals with category theory.

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clifford & preston 1961 , the algebraic theory of semigroup' it is a very interesting book for algebra , a very interesting manner all the concepts covered in this book.

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