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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 …
Martin Sleziak's user avatar
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 …
Leonid Positselski's user avatar
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 …
Leonid Positselski's user avatar
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 …
Leonid Positselski's user avatar
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 …
Leonid Positselski's user avatar
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 …
Leonid Positselski's user avatar
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 …
Leonid Positselski's user avatar
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 …
Leonid Positselski's user avatar
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 …
Leonid Positselski's user avatar
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 …
Leonid Positselski's user avatar
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 ( …
Leonid Positselski's user avatar
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 …
Leonid Positselski's user avatar
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 …
Leonid Positselski's user avatar

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