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Results tagged with ra.rings-and-algebras
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user 4231
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
9
votes
Can one show the equivalence of the abstract and classical Jordan decompositions for simple ...
Here are a few extended comments. First, it's always desirable to re-examine basic material as more of it accumulates and makes the research frontier look impossibly remote to students. The handy b …
3
votes
About enveloping algebras of direct sums
There are probably not many textbook references for these generalities, but I'd suggest looking at the first chapter of the Bourbaki treatise Groupes et algebres de Lie (whose chapters I-IX have been …
- 52.9k
2
votes
Lie algebra embeddings and the center of their enveloping algrabras
I'm doubtful about getting nice relationships between centers of universal enveloping algebras, if you look at embeddings for arbitrary pairs of semisimple Lie algebras; maybe there are subtle connect …
- 52.9k
6
votes
Irreducible representations of $\text{SL}(2, \mathbb{F}_q)$ which don't exist in decomposition?
I'll add an overlong comment to provide more perspective, in community-wiki format. The question is probably a bit misguided, even taken as a purely pedagogical one (the older mathematics involved h …
6
votes
What is the correct formulation of the CDE triangle?
To revisit Bruce's earlier question, it might be useful to suggest a more sceptical alternative to Ben's answer and the related comments. I doubt that there will be a "correct" version of the CDE-tr …
- 52.9k
4
votes
Is the Cartan matrix of a finite-dimensional (Hopf) algebra invertible over the rationals?
Perhaps it's worthwhile to mention one very large and natural class of finite dimensional Hopf algebras (over fields of prime characteristic $p$) for which the Cartan matrix is certainly not invertibl …
- 52.9k
2
votes
Accepted
Extension of an involutive automorphism
[REVISED]
The answer to the first question is NO. It's difficult as a rule to show that an involutive automorphism of $\mathfrak{g'}$ fails to extend to $\mathfrak{g}$,
so it may be better to consi …
- 52.9k
9
votes
German term for "restricted Lie algebra" ?
The correct answer to the narrow question asked about terminology has been given by quid, but I'm tempted to add some broader semi-historical remarks as well. As usual, the development of mathematic …
- 52.9k
1
vote
translation functors in parabolic category $\mathcal{O}$
I hope the question can be reformulated more precisely, with all notation specified, since there may well be more to say about the parabolic subcategories of the BGG category $\mathcal{O}$. (Also, a …
- 52.9k
1
vote
BGG-like resolutions and translations
Ben's somewhat informal answer to Q1 seems to address the main issue well (apart from evading the uncertain nature of "blocks"). If I read Jantzen's Satz 2.25 correctly, it includes this kind of inf …
- 52.9k
5
votes
Is there a 'nice' interpretation of virtual representations?
Probably the general answer to your question is "no", especially if you are working in a classical situation where all representations are completely reducible. Your example involving Adams operati …
- 52.9k
30
votes
0
answers
3k
views
Greatly expanded new edition of a Bourbaki chapter on algebra?
Recently I discovered by accident that Bourbaki issued in 2012 a radically expanded version of their 1958 Chapter 8 Modules et anneaux semi-simples (like other chapters, initially in French) within th …
- 52.9k
1
vote
Accepted
Coinduced modules in the BGG category $\mathcal O$ over complex semisimple Lie algebras
This line of questioning has been pursued in greater generality. starting in prime characteristic by Ron Irving (and myself) and then in the analogous setting of category $\mathcal{O}$ for a semisimpl …
- 52.9k
3
votes
Is there an almost-direct product decomposition for disconnected reductive algebraic groups?
Maybe it's useful to add a few further clarifications to the original question, which Will has answered for solvable or nilpotent algebraic groups.
1) The basic definitions are the same over any alge …
- 52.9k
13
votes
4
answers
3k
views
What is a "block" in an abelian category?
In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which …
- 52.9k