Let $\mathfrak{g}$ be an $n$-dimensional Lie algebra and let $\xi : \mathfrak{g}\to\wedge^2\mathfrak{g}$ be a linear map which is a $1$-cocycle, i.e. $\xi([x,y])=\mathrm{ad}_x\xi(y)-\mathrm{ad}_y\xi(x)$ for all $x,y\in\mathfrak{g}$, where the adjoint action is extended to bivector as follows: $$ \mathrm{ad}_x(y\wedge z):=\mathrm{ad}_x(y)\wedge z+y\wedge\mathrm{ad}_x(z)=[x,y]\wedge z+y\wedge [x,z] $$
Let $F : \mathfrak{g}\to\wedge^3\mathfrak{g}$ be the linear map definied by
$$ F(x)=\sum_{i=1}^n\xi(e_i)\wedge i_{e_i^*}\xi(x) $$
where $\{e_1,\ldots,e_n\}$ is a fixed basis of $\mathfrak{g}$ and $\{e_1^*,\ldots,e_n^*\}$ is its dual basis.
> What are the properties of the map $F$. In particular, what are the $1$-cocycle $\xi$ for which $F$ is identically zero

Given the setup, we need to explore the properties of the map $ F: \mathfrak{g} \to \wedge^3 \mathfrak{g} $ and identify the conditions on the $1$-cocycle $\xi$ such that $ F $ is identically zero.

### Properties of the Map $ F $

1. **Definition and Expression**:
   The map $ F $ is defined by
   \[
   F(x) = \sum_{i=1}^n \xi(e_i) \wedge i_{e_i^*} \xi(x),
   \]
   where $\{e_1, \ldots, e_n\}$ is a basis of $\mathfrak{g}$ and $\{e_1^*, \ldots, e_n^*\}$ is its dual basis. Here, $i_{e_i^*}$ denotes the interior product with $e_i^*$.

2. **Alternating Property**:
   Since $ \xi(e_i) $ and $ \xi(x) $ are elements of $\wedge^2 \mathfrak{g}$, their wedge product lies in $\wedge^4 \mathfrak{g}$. However, the interior product $i_{e_i^*} \xi(x)$ reduces the degree by 1, resulting in an element of $\wedge^1 \mathfrak{g} \cong \mathfrak{g}$. Therefore, $ \xi(e_i) \wedge i_{e_i^*} \xi(x) \in \wedge^3 \mathfrak{g} $.

3. **Linearity**:
   $ F $ is a linear map since it is defined via linear operations (sum, wedge product, interior product) on linear maps.

### Conditions for $ F $ to be Identically Zero

To find the $1$-cocycles $\xi$ for which $ F $ is identically zero, we analyze the structure of $ F(x) $. Specifically, we want to solve the equation:
\[
F(x) = 0 \quad \text{for all } x \in \mathfrak{g}.
\]

Consider the expression for $ F(x) $:
\[
F(x) = \sum_{i=1}^n \xi(e_i) \wedge i_{e_i^*} \xi(x).
\]

We need $ \xi(e_i) \wedge i_{e_i^*} \xi(x) = 0 $ for all $ x \in \mathfrak{g} $ and for each basis element $ e_i $.

#### Special Case: Exact 1-Cocycle

A $1$-cocycle $\xi$ is called exact if there exists a linear map $\alpha: \mathfrak{g} \to \mathfrak{g}$ such that
\[
\xi(x) = d\alpha(x) = \mathrm{ad}_x(\alpha(x)),
\]
where $\mathrm{ad}_x(y) = [x, y]$.

In this case,
\[
\xi(e_i) = \mathrm{ad}_{e_i}(\alpha(e_i)),
\]
and
\[
\xi(x) = \mathrm{ad}_x(\alpha(x)).
\]

If $\xi$ is exact, then $\xi(e_i) = \mathrm{ad}_{e_i}(\alpha(e_i))$. The interior product $ i_{e_i^*} \xi(x) $ will also depend linearly on $\alpha$. When $\xi$ is exact, we can see that:
\[
\xi(e_i) \wedge i_{e_i^*} \xi(x) = \mathrm{ad}_{e_i}(\alpha(e_i)) \wedge i_{e_i^*} (\mathrm{ad}_x(\alpha(x))).
\]

Due to the properties of the adjoint action and the wedge product, this expression may cancel out, especially when considering the antisymmetry of the wedge product. This would lead to $ F(x) = 0 $ for all $ x $.

### Conclusion

The map $ F $ will be identically zero if the $1$-cocycle $\xi$ is exact, i.e., if there exists a linear map $\alpha: \mathfrak{g} \to \mathfrak{g}$ such that $\xi = d\alpha$.

Would you like to explore this further, or do you have any specific questions? Here are five questions for further exploration:

1. What are some examples of exact $1$-cocycles in specific Lie algebras?
2. How does the concept of cohomology relate to exact $1$-cocycles?
3. Can you provide a detailed proof that exact $1$-cocycles result in $ F(x) = 0 $?
4. What happens if $\xi$ is not exact but still results in $ F(x) = 0 $?
5. How does the structure of $\mathfrak{g}$ influence the properties of $1$-cocycles?

**Tip:** When dealing with algebraic structures like Lie algebras, understanding the cohomology groups and exact sequences can provide deeper insights into the properties of the associated cocycles and coboundaries.

Explore the properties of the map $ F: \mathfrak{g} \to \wedge^3 \mathfrak{g} $ in Lie algebras. Learn about exact 1-cocycles and conditions under which $ F $ is identically zero. Includes detailed explanations and examples. Suitable for graduate-level mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation