Timeline for A request for suggestions of advanced topics in representation theory
Current License: CC BY-SA 3.0
11 events
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| Jul 7, 2012 at 5:59 | comment | added | Amitesh Datta | @Emerton Dear Matthew, Thank you very much for these suggestions; I really appreciate it. Could you please suggest a reference for the geometric perspective? If it is not too general a question to ask (but it probably is), then I would also be interested in the possible steps that one could take to get toward research in the representation theory of Lie groups and Lie algebras and related areas (say, after Knapp's book); more precisely, are there other important/essential topics that one should learn before/while reading research papers? Thank you very much and best regards, | |
| Jul 7, 2012 at 1:32 | answer | added | user1504 | timeline score: 2 | |
| Jul 6, 2012 at 23:11 | comment | added | Emerton | ... I think you would enjoy learning (based on my impression of your tastes), and which you would be well-positioned to learn after picking up a little background in the classical aspects of the theory. Regards, Matthew | |
| Jul 6, 2012 at 23:10 | comment | added | Emerton | Dear Amitesh, The theory of Harish-Chandra modules and its relationship to the theory of unitary representations of semisimple Lie groups is probably the natural next large topic following the classification of semisimple Lie groups and their finite-dimensional representations. There are some questions/answers here on MO and on Math.SE that give a quick overview, and there are various books; one that I like is Knapp's "Overview by examples". There is also the geometric perspective of Beilinson and Bernstein (a far-reaching sheaf-theoretic generalization of Borel--Weil--Bott), which ... | |
| Jul 6, 2012 at 18:31 | comment | added | Justin Campbell | I see: if you want to move further in that direction, you might try Humphreys's Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, or for a more geometrical approach Dennis Gaitsgory's notes on geometric representation theory from his 2005 course at Chicago (available on his website). | |
| Jul 6, 2012 at 16:47 | comment | added | Claudio Gorodski | You wrote "I am interested in studying representation theory beyond that which is covered in Daniel Bump's Lie Groups." This reminds me of Knapp's "Lie groups beyond an introduction". | |
| Jul 6, 2012 at 16:18 | history | edited | Amitesh Datta |
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| Jul 6, 2012 at 16:16 | comment | added | Amitesh Datta | @JustinCampbell I think that the coverage of Bump's book is broader than simply the representation theory of $\mathfrak{sl}_{2}$. Of course, chapter 12 is entitled "Representations of $\mathfrak{sl}_{2}(\mathbb{C})$" but there are 50 chapters in Bump's book and many of them discuss a wide range of different topics. For example, general semisimple compact Lie groups are studied in chapter 23. Also, standard topics such as root systems, heighest-weight theory, the Iwasawa and Bruhat decompositions, symmetric spaces etc. are discussed in chapters 1 - 33 (i.e., the first two parts of the book). | |
| Jul 6, 2012 at 16:14 | comment | added | Justin Campbell | I would recommend Serre's book Complex Semisimple Lie Algebras to this end. It is very terse (like most of Serre's writing) so for more details you might refer to Humphreys's Introduction to Lie Algebras and Representation Theory and Dixmier's Universal Enveloping Algebras. | |
| Jul 6, 2012 at 16:09 | comment | added | Justin Campbell | It looks like Bump's book covers the representation theory of $\mathfrak{sl}_2$. From my perspective, understanding the representation theory of general semisimple Lie algebras (over $\mathbb{C}$, say) is a necessity for any serious representation theorist. Certainly it is prerequisite to many areas of current research | |
| Jul 6, 2012 at 15:55 | history | asked | Amitesh Datta | CC BY-SA 3.0 |