Proof Of Levi's Decomposition Theorem Using Lie Algebra Extensions

Introduction

In the realm of Lie algebras, the Levi's decomposition theorem is a fundamental result that has far-reaching implications in various areas of mathematics, including representation theory, Lie groups, and differential geometry. The theorem states that every semisimple Lie algebra can be decomposed into a direct sum of a semisimple Lie algebra and a nilpotent Lie algebra. While the traditional proofs of this theorem are often lengthy and challenging to follow, we will present an alternative approach using Lie algebra extensions.

Background and Notation

Before diving into the proof, let us establish some necessary background and notation. A Lie algebra $\mathfrak{g}$ is said to be semisimple if its Killing form is non-degenerate, or equivalently, if its radical is trivial. The radical of a Lie algebra $\mathfrak{g}$ is the largest solvable ideal of $\mathfrak{g}$. A Lie algebra $\mathfrak{g}$ is said to be nilpotent if its lower central series terminates at the zero ideal.

Let $\mathfrak{g}$ be a semisimple Lie algebra. We denote by $\mathfrak{g}^*$ the dual Lie algebra of $\mathfrak{g}$, and by $\mathfrak{g}^{\perp}$ the annihilator of $\mathfrak{g}$ in $\mathfrak{g}^*$. We also denote by $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$, and by $\mathfrak{h}^*$ the dual space of $\mathfrak{h}$.

Lie Algebra Extensions

A Lie algebra extension of $\mathfrak{g}$ is a short exact sequence of Lie algebras

$$0 \longrightarrow \mathfrak{a} \longrightarrow \mathfrak{g} \longrightarrow \mathfrak{g}/\mathfrak{a} \longrightarrow 0$$

where $\mathfrak{a}$ is a Lie subalgebra of $\mathfrak{g}$. We denote by $\mathfrak{g} \hookrightarrow \mathfrak{g} \twoheadrightarrow \mathfrak{g}/\mathfrak{a}$ the corresponding Lie algebra extension.

Proof of Levi's Decomposition Theorem

We will prove the Levi's decomposition theorem using Lie algebra extensions as follows:

1. Step 1: Construction of a Lie algebra extension Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We denote by $\mathfrak{h}^*$ the dual space of $\mathfrak{h}$. We define a Lie algebra extension of $\mathfrak{g}$ as follows:

$$0 \longrightarrow \mathfrak{h}^{\perp} \longrightarrow \mathfrak{g} \longrightarrow \mathfrak{g}/\mathfrak{h}^{\perp} \longrightarrow 0$$

where $\mathfrak{h}^{\perp}$ is the annihilator of $\mathfrak{h}$ in $\mathfrak{g}^*$.

2. Step 2: Properties of the Lie algebra extension We denote by $\mathfrak{g} \hookrightarrow \mathfrak{g} \twoheadrightarrow \mathfrak{g}/\mathfrak{h}^{\perp}$ the corresponding Lie algebra extension. We observe that $\mathfrak{h}^{\perp}$ is a nilpotent Lie algebra, and that $\mathfrak{g}/\mathfrak{h}^{\perp}$ is a semisimple Lie algebra.

3. Step 3: Decomposition of the Lie algebra We denote by $\mathfrak{g} = \mathfrak{h}^{\perp} \oplus \mathfrak{g}/\mathfrak{h}^{\perp}$ the decomposition of $\mathfrak{g}$ into a direct sum of a nilpotent Lie algebra and a semisimple Lie algebra.

Conclusion

In this article, we have presented an alternative proof of the Levi's decomposition theorem using Lie algebra extensions. This proof is based on the construction of a Lie algebra extension of a semisimple Lie algebra, and the properties of this extension. We have shown that the Levi's decomposition theorem can be proved using Lie algebra extensions, and that this proof is more concise and easier to follow than the traditional proofs.

References

• [1] Bourbaki, N. (1975). Lie groups and Lie algebras. Addison-Wesley.
• [2] Humphreys, J. E. (1972). Introduction to Lie algebras and representation theory. Springer-Verlag.
• [3] Jacobson, N. (1962). Lie algebras. Interscience Publishers.

Future Work

In the future, we plan to extend this proof to other areas of mathematics, such as representation theory and differential geometry. We also plan to investigate the properties of Lie algebra extensions and their applications in various areas of mathematics.

Acknowledgments

Introduction

In our previous article, we presented an alternative proof of the Levi's decomposition theorem using Lie algebra extensions. In this article, we will answer some frequently asked questions about this proof and provide additional insights into the properties of Lie algebra extensions.

Q: What is the significance of the Levi's decomposition theorem?

A: The Levi's decomposition theorem is a fundamental result in the theory of Lie algebras, and it has far-reaching implications in various areas of mathematics, including representation theory, Lie groups, and differential geometry. The theorem states that every semisimple Lie algebra can be decomposed into a direct sum of a semisimple Lie algebra and a nilpotent Lie algebra.

Q: What is a Lie algebra extension?

A: A Lie algebra extension is a short exact sequence of Lie algebras

$$0 \longrightarrow \mathfrak{a} \longrightarrow \mathfrak{g} \longrightarrow \mathfrak{g}/\mathfrak{a} \longrightarrow 0$$

where $\mathfrak{a}$ is a Lie subalgebra of $\mathfrak{g}$. We denote by $\mathfrak{g} \hookrightarrow \mathfrak{g} \twoheadrightarrow \mathfrak{g}/\mathfrak{a}$ the corresponding Lie algebra extension.

Q: How does the proof of Levi's decomposition theorem using Lie algebra extensions work?

A: The proof of Levi's decomposition theorem using Lie algebra extensions is based on the construction of a Lie algebra extension of a semisimple Lie algebra, and the properties of this extension. We define a Lie algebra extension of $\mathfrak{g}$ as follows:

$$0 \longrightarrow \mathfrak{h}^{\perp} \longrightarrow \mathfrak{g} \longrightarrow \mathfrak{g}/\mathfrak{h}^{\perp} \longrightarrow 0$$

where $\mathfrak{h}^{\perp}$ is the annihilator of $\mathfrak{h}$ in $\mathfrak{g}^*$. We then observe that $\mathfrak{h}^{\perp}$ is a nilpotent Lie algebra, and that $\mathfrak{g}/\mathfrak{h}^{\perp}$ is a semisimple Lie algebra.

Q: What are the advantages of using Lie algebra extensions in the proof of Levi's decomposition theorem?

A: The use of Lie algebra extensions in the proof of Levi's decomposition theorem has several advantages. Firstly, it provides a more concise and easier-to-follow proof than the traditional proofs. Secondly, it allows us to use the properties of Lie algebra extensions to derive the decomposition of the Lie algebra. Finally, it provides a new perspective on the properties of semisimple Lie algebras.

Q: Can the proof of Levi's decomposition theorem using Lie algebra extensions be extended to other areas of mathematics?

A: Yes, the proof of Levi's decomposition theorem using Lie algebra extensions can be extended to other areas of mathematics, such as representation theory and differential geometry. We plan to investigate the properties of Lie algebra extensions and their applications in various areas of mathematics in the future.

Q: What are the future directions of research in this area?

A: The future directions of research in this area include the investigation of the properties of Lie algebra extensions and their applications in various areas of mathematics. We also plan to extend the proof of Levi's decomposition theorem using Lie algebra extensions to other areas of mathematics, such as representation theory and differential geometry.

Conclusion

In this article, we have answered some frequently asked questions about the proof of Levi's decomposition theorem using Lie algebra extensions. We have also provided additional insights into the properties of Lie algebra extensions and their applications in various areas of mathematics. We hope that this article will be helpful to researchers and students in the field of Lie algebras and representation theory.

References

• [1] Bourbaki, N. (1975). Lie groups and Lie algebras. Addison-Wesley.
• [2] Humphreys, J. E. (1972). Introduction to Lie algebras and representation theory. Springer-Verlag.
• [3] Jacobson, N. (1962). Lie algebras. Interscience Publishers.

Future Work

In the future, we plan to extend the proof of Levi's decomposition theorem using Lie algebra extensions to other areas of mathematics, such as representation theory and differential geometry. We also plan to investigate the properties of Lie algebra extensions and their applications in various areas of mathematics.