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Results tagged with rt.representation-theory
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user 2106
Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
3
votes
0
answers
107
views
Modules over the unipotent subalgebra as direct summands of modules over a semisimple Lie al...
Let $\mathfrak g$ be a semisimple finite-dimensional Lie algebra over the field of complex numbers $\mathbb C$. Let $\mathfrak n\subset\mathfrak g$ be the maximal unipotent subalgebra of $\mathfrak g …
7
votes
Accepted
Gluing perverse sheaves?
Beilinson's How to glue perverse sheaves explains how one can glue perverse sheaves on a variety from perverse sheaves on a closed subvariety and its open complement (assuming that the closed subvarie …
7
votes
Accepted
Semisimple Abelian categories with infinite sums
A) It depends on what you are interested in. If you do not impose the finiteness condition, then it means that you are describing a different class of abelian categories. Which class is that, depend …
7
votes
Accepted
Dual of a projective module
You are right. There is no such a map as the one you are trying to describe.
Here is a map that actually exists. Let $R$ and $S$ be two noncommutative rings with units, and let $P$ be an $R$-$S$-bi …
6
votes
Accepted
On definitions and explicit examples of pure-injective modules
This is not pure-injectivity. The relevant concept is that of an fp-injective ("finitely presented-injective") module, otherwise known as an "absolutely pure" module. A left $R$-module $J$ is said t …
6
votes
Accepted
Graded quivers vs "ordinary" quivers and derived categories
I have not heard the slogan and perhaps do not understand the context, but it seems to me that this has nothing to do with the derived categories. For any graded quiver (with or without relations) th …
8
votes
Accepted
Compatibility of two definitions of Koszul dual
Let $A$ be a quadratic graded algebra over a field $k$ with finite-dimensional components $A_n$ and $A_0=k$. Then the construction of the quadratic dual algebra $A^!$ involves setting $A^!_1$ to be t …
4
votes
Accepted
Question about an exact sequence
For any abelian category $\mathcal{A}$, an object $M\in\mathcal{A}$ is called a generator if, given an injective nonsurjective morphism $U\to V$ in $\mathcal{A}$, there always exists a morphism $M\to …
15
votes
6
answers
2k
views
References for Harish-Chandra pairs and modules, category "O"?
I am looking for references related to the terms "Harish-Chandra pair" and "Harish-Chandra modules", and also to the term "category O". I know what these are, or I think I do (a Harish-Chandra pair i …
7
votes
What is the (Koszul? derived?) interpretation of a pair of Lie algebras with the same cohomo...
I know, basically, two answers to this question. The first one starts from the observation that what you call "a relatively easy theorem" generalizes to infinite-dimensional Lie algebras $L$ by repla …
16
votes
What are examples of cogenerators in R-mod?
For any ring $R$, the $R$-module $Hom_{\mathbb Z}(R,\mathbb Q/\mathbb Z)$ is an injective cogenerator of the category of $R$-modules. Here $\mathbb Q/\mathbb Z$ can be replaced with $\mathbb R/\mathb …
12
votes
Accepted
Koszul duality and modules over the Chevalley complex
The derived category of finite-dimensional $g$-modules is not a full subcategory of the derived category of arbitrary $g$-modules for a finite-dimensional Lie algebra $g$, in general (e.g., for a semi …
2
votes
In which categories is every coalgebra a sum of its finite-dimensional subcoalgebras?
Let us replace "dualizable" with "Noetherian". Then, I believe, for any locally Noetherian Grothendieck (abelian) category with an exact, associative tensor product functor preserving direct limits ( …
5
votes
Cohomology of Flag Varieties
Concerning the side question, the map K/T → G/B is actually an isomorphism of real manifolds, not just a homotopy equivalence. Not sure about the references, but this is essentially textbook material …
6
votes
Accepted
Indexing the Line Bundles of a Flag Manifold
Attempting to answer your last question: given a (topological, Lie, algebraic, etc.) group G and a closed subgroup P in G, the category of G-equivariant vector bundles on X=G/P is equivalent to the ca …