2.4 Direct sums and semidirect products of Lie algebras

Direct sums and semidirect products are key ways to build bigger Lie algebras from smaller ones. They help us understand how complex Lie algebras are put together and how they work.

These constructions are super useful for breaking down tricky Lie algebras into simpler parts. By looking at how Lie algebras combine, we can figure out a lot about their structure and properties.

Direct sum and semidirect product of Lie algebras

Definition and properties

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• Define the direct sum of two Lie algebras $\mathfrak{g}_1$ and $\mathfrak{g}_2$ as a Lie algebra, denoted $\mathfrak{g}_1 \oplus \mathfrak{g}_2$, which is the direct sum of the underlying vector spaces with the Lie bracket defined componentwise
• Introduce the semidirect product of two Lie algebras $\mathfrak{h}$ and $\mathfrak{k}$, denoted $\mathfrak{h} \ltimes \mathfrak{k}$, as a Lie algebra constructed using a Lie algebra homomorphism $\phi: \mathfrak{h} \to \operatorname{Der}(\mathfrak{k})$, where $\operatorname{Der}(\mathfrak{k})$ is the Lie algebra of derivations of $\mathfrak{k}$
• Specify the underlying vector space of the semidirect product as the direct sum $\mathfrak{h} \oplus \mathfrak{k}$, and define the Lie bracket by $[(h_1, k_1), (h_2, k_2)] = ([h_1, h_2], [k_1, k_2] + \phi(h_1)(k_2) - \phi(h_2)(k_1))$
• Recognize the direct sum as a special case of the semidirect product where the homomorphism $\phi$ is the zero map

Examples and applications

• Provide examples of direct sums and semidirect products of Lie algebras, such as the direct sum of two abelian Lie algebras or the semidirect product of a semisimple Lie algebra and a solvable Lie algebra
• Discuss the role of direct sums and semidirect products in the classification of Lie algebras, particularly in the context of the Levi decomposition
• Illustrate the use of direct sums and semidirect products in physics, such as the Poincaré algebra as a semidirect product of the Lorentz algebra and the spacetime translation algebra

Constructing Lie algebras

Direct sum construction

• Explain how to construct the direct sum $\mathfrak{g}_1 \oplus \mathfrak{g}_2$ of two Lie algebras $\mathfrak{g}_1$ and $\mathfrak{g}_2$ by taking the direct sum of the underlying vector spaces and defining the Lie bracket componentwise
• Provide a step-by-step example of constructing the direct sum of two specific Lie algebras, such as $\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{so}(3)$
• Discuss the properties of the direct sum, such as the fact that $\mathfrak{g}_1$ and $\mathfrak{g}_2$ are ideals of $\mathfrak{g}_1 \oplus \mathfrak{g}_2$, and the Lie bracket between elements of $\mathfrak{g}_1$ and $\mathfrak{g}_2$ is always zero

Semidirect product construction

• Describe the process of constructing a semidirect product $\mathfrak{h} \ltimes \mathfrak{k}$ using a Lie algebra homomorphism $\phi: \mathfrak{h} \to \operatorname{Der}(\mathfrak{k})$
• Provide examples of semidirect products, such as the Euclidean Lie algebra $\mathfrak{e}(n) = \mathfrak{so}(n) \ltimes \mathbb{R}^n$ and the Poincaré Lie algebra $\mathfrak{p}(n) = \mathfrak{so}(n) \ltimes \mathbb{R}^{n+1}$, where $\mathfrak{so}(n)$ is the Lie algebra of the special orthogonal group $\operatorname{SO}(n)$
• Explain how the semidirect product construction can be used to build larger Lie algebras from smaller ones, providing a way to study the structure of Lie algebras

Lie algebra structure from decomposition

Direct sum decomposition

• State that if a Lie algebra $\mathfrak{g}$ can be written as a direct sum $\mathfrak{g}_1 \oplus \mathfrak{g}_2$, then $\mathfrak{g}_1$ and $\mathfrak{g}_2$ are ideals of $\mathfrak{g}$, and the Lie bracket between elements of $\mathfrak{g}_1$ and $\mathfrak{g}_2$ is always zero
• Provide an example of a Lie algebra that can be decomposed as a direct sum, such as the complex general linear algebra $\mathfrak{gl}(n,\mathbb{C}) = \mathfrak{sl}(n,\mathbb{C}) \oplus \mathbb{C}$
• Discuss the implications of a direct sum decomposition on the structure and properties of a Lie algebra, such as the fact that the direct summands are invariant under the adjoint representation

Semidirect product decomposition

• Explain that in a semidirect product $\mathfrak{h} \ltimes \mathfrak{k}$, the subalgebra $\mathfrak{k}$ is an ideal, while $\mathfrak{h}$ is a subalgebra but not necessarily an ideal
• Provide an example of a Lie algebra that can be decomposed as a semidirect product, such as the Poincaré algebra $\mathfrak{p}(n) = \mathfrak{so}(n) \ltimes \mathbb{R}^{n+1}$
• Discuss the Levi decomposition, which states that any finite-dimensional Lie algebra over a field of characteristic zero can be written as a semidirect product of a solvable ideal (the radical) and a semisimple subalgebra (the Levi subalgebra)
• Explain how the structure of a Lie algebra can be determined by studying its decomposition into direct sums or semidirect products of simpler Lie algebras

Lie bracket computation in direct sums and semidirect products

Direct sum Lie bracket

• Define the Lie bracket in a direct sum $\mathfrak{g}_1 \oplus \mathfrak{g}_2$ componentwise: $[(x_1, y_1), (x_2, y_2)] = ([x_1, x_2], [y_1, y_2])$, where $x_1, x_2 \in \mathfrak{g}_1$ and $y_1, y_2 \in \mathfrak{g}_2$
• Provide an example of computing the Lie bracket in a direct sum of two specific Lie algebras, such as $\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{so}(3)$
• Explain why the Jacobi identity for the Lie bracket in a direct sum follows from the Jacobi identity in the constituent Lie algebras

Semidirect product Lie bracket

• Give the formula for the Lie bracket in a semidirect product $\mathfrak{h} \ltimes \mathfrak{k}$ with homomorphism $\phi: \mathfrak{h} \to \operatorname{Der}(\mathfrak{k})$: $[(h_1, k_1), (h_2, k_2)] = ([h_1, h_2], [k_1, k_2] + \phi(h_1)(k_2) - \phi(h_2)(k_1))$, where $h_1, h_2 \in \mathfrak{h}$ and $k_1, k_2 \in \mathfrak{k}$
• Provide an example of computing the Lie bracket in a semidirect product, such as the Euclidean Lie algebra $\mathfrak{e}(n) = \mathfrak{so}(n) \ltimes \mathbb{R}^n$
• Explain that to compute the Lie bracket in a semidirect product, one needs to know the Lie brackets in the constituent Lie algebras $\mathfrak{h}$ and $\mathfrak{k}$, as well as the homomorphism $\phi$
• Discuss how the Jacobi identity for the Lie bracket in a semidirect product follows from the Jacobi identity in the constituent Lie algebras and the properties of the homomorphism $\phi$