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Feb
3
comment Concept of Facets in the structure of reductive algebraic groups
@Sam: This may cause some confusion, but it's not my idea. It apparently originated in work of Tits on buildings, then was taken over by Bourbaki and others in the context of root systems and Weyl groups. (Conflicting uses of terminology are probably inevitable in mathematics, such as the term "block".)
Feb
1
comment product of root multiplicities in Kac Moody Algebras
This question looks very open-ended, so it would be better motivated if you could point to some specific example where there is an interesting interpretation of the product. (By the way, I added one tag, since the usual finite dimensional semisimple Lie algebras are examples of Kac-Moody algebras. Of course, in those cases $k$ is always 1.)
Feb
1
revised product of root multiplicities in Kac Moody Algebras
edited tags
Feb
1
comment Center of $U_q(sl_3)$ and $U_q(sl_4)$
@Alexander: I was thinking of more general Lie types than type A, but I also don't know all of the possible literature here.
Jan
30
comment Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?
A small point of confusion is the switch to the index $n$ in your summation $(*)$.
Jan
29
awarded  Nice Question
Jan
28
comment Complete reducibility of representations of reductive algebraic groups
The more difficult question is to get complete reducibility in char 0 for (say connected) reductive groups from the Borel-Tits definition. It seems to take a lot of work to show that such a group is the almost-direct product of a torus (for which all rational representations are completely reducible in any characteristic) and a semisimple group (possibly trivial). The root structure gets one close to the full classification. Maybe there's a shortcut, but in prime characteristic complete reducibility usually fails for semisimple groups.
Jan
28
comment Complete reducibility of representations of reductive algebraic groups
As zeno points out, there are explicit statements. Actually, my early Chapter V on characteristic 0 theory does treat the case of a semisimple group with few preliminaries other than Weyl's complete reducibility theorem: see 13.2 and 14.3. [to be continued]
Jan
27
comment Complete reducibility of representations of reductive algebraic groups
@Jason: Presumably my British ancesters were mostly illiterate, so I'm used to all the variant spellings. Anyway, the question being asked is probably too elementary for this site. The concept of "reductive" algebraic group originates in the work of Borel-Tits, but related ideas in characteristic 0 are older in the theory of Lie groups. There is an identification (from the Chevalley classification) of semisimple Lie groups and semisimple algebraic groups over $\mathbb{C}$. Then the question reduces to "reductive" Lie algebras and Weyl's complete reducibility theorem.
Jan
26
revised Describing Levi factors and unipotent radicals of parabolic subgroups in classical groups
edited tags
Jan
26
answered Describing Levi factors and unipotent radicals of parabolic subgroups in classical groups
Jan
26
comment Who gave the generalized Stone-Weierstrass Theorem?
To add an amusing anecdote to a serious question, I recall Marshall Stone himself remarking decades ago that he found it comical when some of his students asked him about his "collaboration" with Weierstrass (who was to Stone an entirely historical person, as Stone himself is now to most mathematicians).
Jan
25
answered Reference request: expository text on the structure of reductive groups over non-archimedean local fields
Jan
25
awarded  Good Question
Jan
22
revised Fundamental invariants for root subsystems
deleted 1360 characters in body
Jan
21
revised Greatly expanded new edition of a Bourbaki chapter on algebra?
added 334 characters in body
Jan
16
comment Is the Steinberg representation always irreducible?
@Tobias: You need to be more cautious about the characteristic of the field involved, which isn't always the defining characteristic for groups of Lie type.
Jan
15
answered Is the Steinberg representation always irreducible?
Jan
15
comment Is the Steinberg representation always irreducible?
Note that the Borel-Serre paper is now accessible online here: sciencedirect.com/science/article/pii/0040938376900379
Jan
13
comment Greatly expanded new edition of a Bourbaki chapter on algebra?
@quid: By "small" I meant that the actual text is about 150 pages, divided into various topics which were probably treated separately by various people. I wasn't aware of the announced new book, but it's always tricky to maintain a consistent standard with a committee of authors. I do understand the policy of anonymity, but their pace of new production has been erratic in recent decades. So I have no idea how viable the project continues to be, especially in an era when many books and many voices are out there.