# Tagged Questions

**4**

votes

**3**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**4**answers

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

**1**answer

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, ...