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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 …

The book "Enveloping algebras" by Dixmier appears to contain some material on Harish-Chandra modules in the classical sense (i.e., with respect to the maximal compact subgroup) and also on Verma modul …

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 …

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 …

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 …

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 …

The right definition is: take the free associative (tensor) algebra generated by $V$; divide out the ideal generated by the elements $xy-(-1)^{|x||y|}yx$ for all homogeneous $x$, $y\in V$ and $z^2=0$ …

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 …

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 …

The assertion that k[G] is isomorphic to the direct sum of End(V) as a k-algebra is not true for an arbitrary field k of the characteristic not dividing |G|. Some additional condition must be impose …

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 …

This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary c …

The answer to the question in the title is "no". Semisimplicity is an open condition; however, it is not a dense open condition. Indeed, the variety of Lie algebras is reducible. There is one equat …

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 …

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 …