Vinberg A Course In Algebra Pdf Download ~REPACK~

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I understand from professors who were students at Harvard that MacLane himself would use the book-in all it's versions,from the first mimeographed drafts in the early 1960's to the 3rd edition he used in his last teaching days in the mid-1990's-for both undergraduate and graduate courses in algebra depending on the strength of the students, which would vary enormously from class to class. Usually,for strong undergraduate courses, he'd supplement Algebra with examples drawn from he and Birkoff's classic undergraduate text, A Survey Of Modern Algebra. For a first year graduate course-which at Harvard,usually included a number of undergraduates-he'd use the book straight up and cover most of it.

My question: What level students today can Algebra be used for? I think it would definitely be too difficult for any but the very best undergraduates and I think there are several books that would be better for that purpose: Herstein's Topics in Algebra or Vinberg's A Course in Algebra come to mind. I don't think hitting students with category theory unless they've already had considerable exposure to algebra is a good idea. I doubt they'll be able to fully appreciate the enormous generalization and unification it provides without a giant stock of examples (which is why I love Emily Riehl's Category Theory In Context as the definitive introduction to category theory for mathematics students. She gets this.) But at the same time, since the book contains very little homological algebra and no representation theory beyond the definition,can it really be used as graduate course today?

I do not believe students are significantly less capable today than they were several decades ago, as you seem to suggest. In fact, they may be better when it comes to algebra. Algebra was almost unheard of at the undergraduate level not too long ago, whereas now an undergraduate program would be remiss not to require students take at least an introductory course covering the basic theories of groups, rings, and fields. On the other hand, calculus has been pushed into high school, and introductory courses are substantially easier and less rigorous than in the days of Apostol, Courant, and Spivak. In this way, I do not expect today's students are substantially better than past students, either. Also keep in mind that Dummit & Foote is fairly commonly used at the undergraduate level, and not just at top schools or in honors classes. I would not consider that presentation substantially easier than Birkhoff & Mac Lane, albeit perhaps a bit.

It is unlikely you will cover the entire book, either, even in a one year course. A typical first undergraduate course may cover group theory through the isomorphism theorems and the structure theorem for finite abelian groups,possibly including group actions and the Sylow theorems (c.f. Mac Lane Chapters I, II, VII), basic ring theory (c.f. Chapter III), and some field and vector space theory (c.f. Chapters VI, VII). As aforementioned, some category theory, especially an awareness of universal properties and basic definitions (e.g., categories, functors), can be good, and it seems to be becoming more common, too (c.f. Chapter IV). A second course may cover a bit more group theory (c.f. Chapter VII), more field theory and linear algebra (c.f. Chapters IX-XI), and Galois theory (c.f. Chapter XIII).

The biggest problem is that Birkhoff & Mac Lane use categorical algebra all from Chapter III onward, which reflects the fact that they intended readers to be fairly mature. Although not enough to reconcile this problem, their exposition on categorical notions is quite clear, at least, albeit maybe not as good as some treatments that have since come. They do not avoid using universal properties, and they do not always bother to give students something concrete to hold on. Young students can handle abstraction. Some people mistakenly believe they cannot. Being able to handle abstraction does not mean students should not learn many concrete, basic examples, however, nor does it mean they should learn things at the most abstract possible level and be expected to figure out the less abstract consequences on their own. Maximum generality entailing sophisticated machinery can seem efficient in the abstract, but it rarely works so well in practice. You do not define a group as a groupoid with one object, at least not in an introductory course. (This is not to say Mac Lane and Birkhoff do this!)

Again, I think Gallian's Contemporary Abstract Algebra is better as a primary text. Motivation, examples, clear writing, reasonable exercises, they are all there. Artin (2e, not 1e) is also good and emphasizes linear algebra and geometric intuition, which is good considering how often students will need things from linear algebra and how often they will find themselves ignorant of those things. Vinberg seems similar but more intense, so I imagine it would work well, too. Dummit & Foote is not good enough for ring theory and too encyclopedic to be used as the primary text for an undergraduate course, in my opinion; the group theory is so good, though. Aluffi is really something special, but it is probably a bit too category theory-centric for students new to the subject in the sense that they will not have enough familiarity with the lower-level stuff. Judson is good all around. If you want an older book written by a master expositor and mathematician, then I think Herstein works better than Mac Lane. It has a lot of linear algebra, which is good, and it is not too hard, but it requires some work. It also introduces the student to modules, but it does not insist on working with modules instead of vector spaces whenever possible, which is probably good, because modules often serve to slightly confuse without adding anything more than a bit of generality.

Perhaps not every course or book needs to cover all of these topics in detail, though, as much as the books by Lang and Dummit & Foote would have one believe. I am not sure. Certainly, algebraists expect everyone has seen some homological algebra and representation theory by the time they get their Ph.D., but plenty of people do not need either, both topics are often taught in separate courses, and so forth.

I think you could cover the entire book, minus perhaps Chapter XIV on lattices, which are not typically emphasized and are partly historical, and possibly Chapter VIII, which is a bit odd, in my opinion. Supplementing this with another book to get some coverage of representation theory and homological algebra is probably ideal. Dummit & Foote or Lang are the obvious places to go, but I do not think those books' best parts cover these topics. Unfortunately, I am not sure what the best supplements are. I quite like Etingof's Introduction to representation theory, which is available at his site, but it takes two chapters before it gets to the case of finite groups, which is usually what algebra professors focus on in a first graduate course. I do not think using algebras is a problem, but three whole chapters is probably a bit much, so some would have to be cut. I think taking parts from Rotman's An Introduction to Homological Algebra may work in a similar vein, albeit with much more cutting. The advantage of not falling back on a general reference as a supplement is you can use the really great parts of these excellent books, and the students know where to go for much more; plus, both of these are written at a reasonably low level. Adding in some applications may be good, too; I do not remember many being in Mac Lane. Robert Ash's Algebra: The Basic Graduate Year includes some good applications to algebraic geometry and algebraic number theory, for example.

I am less sure what makes a really excellent graduate course in terms of extant texts. I love Lang, especially for things like Galois theory, but it is too hard, too fast, too big, too encyclopedic, and, dare I say, too modern for most graduate students. Certainly, one can survive it, but it is probably suboptimal for most. Lots of people cannot stand Lang's writing, too. Dummit and Foote is great for group theory, but it suffers in other parts and may also too big and encyclopedic. I often found it dry, too. Similarly, books like Cohn, Grillet, and Jacobson can be too advanced or too focused on being references. Rowen has been talked about a good bit, which is deserving for its extensive presentation on algebras and many applications, but I am not sure starting with modules is a good idea, for example.

I think graduate courses should use category theory pretty openly. This is when students should come face-to-face with having to understand universality, or else. Yet, I also think a course should start with basic material. Perhaps a lecture reviewing elementary set theoretical notions, then cover some linear algebra (introducing basic category theory after seeing direct sums), then cover some ring theory, then plenty of group theory, then modules and advanced linear algebra, followed by field and Galois theory, representation theory (using algebras and specializing quickly to groups), commutative algebra (including some applications to algebraic geometry and the like), and finally homological algebra, with some advanced or extra topics at the end, if possible (e.g., a word on universal algebra or further category theory). Linear algebra is first, because students have the best intuition there; ring theory is next, because the examples and applications are nicer there than in groups and the quotient construction is easier. Logic suggests the standard groups, rings, fields, modules, vector spaces, etc. sequence, but it does not demand it.

Several of the books mentioned in other answers are devoted mostly or entirely to Lie algebras and their representations, rather than Lie groups. Here are more comments on the Lie group books that I am familiar with. If you aren't put off by a bit archaic notation and language, vol 1 of Chevalley's Lie groups is still good. I've taught a course using the 1st edition of Rossmann's book, and while I like his explicit approach, it was a real nightmare to use due to an unconscionable number of errors. In stark contrast with Complex semisimple Lie algebras by Serre, his Lie groups, just like Bourbaki's, is ultra dry. Knapp's Lie groups: beyond the introduction contains a wealth of material about semisimple groups, but it's definitely not a first course ("The main prerequisite is some degree of familiarity with elementary Lie theory", xvii), and unlike Procesi or Chevalley, the writing style is not crisp. An earlier and more focused book with similar goals is Goto and Grosshans, Semisimple Lie algebras (don't be fooled by the title, there are groups in there!). 2b1af7f3a8