4
votes
3answers
239 views

Poincare duality for (co)homology of Lie algebras?

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. In Algebra, Geometry, and Software Systems by Joswig & Takayama on p.200 it ...
13
votes
2answers
657 views

Is homology finitely generated as an algebra?

If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra? Is it easier if we impose any of the three conditions: characteristic zero; ...
1
vote
0answers
100 views

A Isomorphism between the extension group and cohomology group of Lie algebras [closed]

Within the book An introduction to homological algebra by Weibel, I am trying to prove the following isomorphism, but I am not sure this is true. But I really want to know how to prove or disprove ...
6
votes
0answers
161 views

The meaning of a “subcomplex” of the Cartan-Eilenberg of a Lie algebra

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra, and $\mathfrak{g}^* $ be the dual vector space. We have the standard Cartan-Eilenberg complex $(\wedge^{\cdot} \mathfrak{g}^* ...
4
votes
2answers
576 views

Whitehead lemmas in Lie algebra cohomology for non-algebraically closed fields

I read in Weibel's homological algebra that Whitehead's first and second lemmas are true for any characteristic 0 field. I mean the following: Whitehead Lemma(s): Let g be a semisimple Lie algebra ...
0
votes
0answers
248 views

cokernel for $L_\infty$-algebra morphisms

As I have asked a wrong question previously, I edited a bit. It is obvious that the cokernel construction does not work well for the category of Lie algebras. The cokernel exists only for a normal ...
3
votes
2answers
550 views

How is this observation related to Koszul duality?

Let $X$ be a smooth variety, $\mathcal D$ the sheaf of algebraic differentail operators, $\Omega$ the algebraic deRham complex and $\mathcal M$ a quasi coherent $\mathcal O_X$-module. Now there is a ...
3
votes
2answers
508 views

Describing the kernel of the exponential map as a homology group

I am reading Deligne: Hodge III, and am puzzled by a certain statement in section 10. If anyone could give a reference or a hint for how to prove this, I would be grateful. Maybe it is obvious and I ...
1
vote
0answers
195 views

exactness of n homology functor

Let $G$ be a real reductive group, $P=MN$ a parabolic subgroup with its Levi decomposition, let $\mathfrak{n}$ be the nilpotent Lie algebra of $N$. Now given a smooth representation $(\pi,V)$ of $G$, ...
7
votes
4answers
1k 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 ...
3
votes
1answer
272 views

Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$. The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$ has the bracket $$[xt^r, ...