Cartan Decomposition And Minimal Subalgebras In Su(n)

Introduction

In the realm of Lie algebras and representation theory, the Cartan decomposition plays a pivotal role in understanding the structure and properties of semisimple Lie algebras. Specifically, it provides a way to decompose a Lie algebra into simpler, more manageable subspaces, which in turn facilitates the analysis of its representations and subalgebras. This article delves into the significance of the Cartan decomposition in establishing the existence of minimal subalgebras that are closed under the adjoint action within the special unitary Lie algebra $\mathfrak{su}(n)$. We will explore the theoretical underpinnings of this decomposition, its connection to the adjoint representation, and how it helps us identify and characterize these minimal subalgebras.

This exploration is crucial for various applications in mathematics and physics, including the study of symmetric spaces, the classification of Lie groups and algebras, and the construction of physical models with specific symmetry properties. By understanding the role of the Cartan decomposition, we can gain deeper insights into the intricate structure of Lie algebras and their representations. Furthermore, the identification of minimal subalgebras closed under the adjoint action is essential for understanding the building blocks of the Lie algebra and its representations. These subalgebras often correspond to physically relevant symmetries, making their study indispensable.

Before we proceed, it is worth noting that this discussion assumes a basic familiarity with Lie algebras, their representations, and the concept of the adjoint action. However, we will try to provide sufficient context and explanations to make the material accessible to a broader audience. The goal is to illuminate the core ideas and demonstrate how the Cartan decomposition serves as a powerful tool in the study of Lie algebras and their subalgebras. Throughout this article, we will use concrete examples and illustrations to clarify the abstract concepts and make the theoretical framework more tangible. Our journey will begin with a review of the fundamental concepts of Lie algebras and their representations, followed by a detailed examination of the Cartan decomposition and its applications.

Lie Algebras and Their Representations

To fully appreciate the role of the Cartan decomposition, it is essential to first establish a solid foundation in the theory of Lie algebras and their representations. A Lie algebra is a vector space $\mathfrak{g}$ over a field (typically the real numbers $\mathbb{R}$ or the complex numbers $\mathbb{C}$) equipped with a binary operation called the Lie bracket, denoted by $[X, Y]$, which satisfies certain axioms. These axioms include bilinearity, alternativity (i.e., $[X, X] = 0$ for all $X \in \mathfrak{g}$), and the Jacobi identity:

$[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$ for all $X, Y, Z \in \mathfrak{g}$.

The Lie bracket captures the infinitesimal structure of a Lie group, which is a smooth manifold that is also a group, with the group operations being smooth maps. Lie algebras provide a linear approximation to Lie groups, making them easier to study while retaining crucial information about the group's structure. Examples of Lie algebras include the general linear Lie algebra $\mathfrak{gl}(V)$ of all linear operators on a vector space $V$, with the Lie bracket defined as the commutator $[X, Y] = XY - YX$, and the special unitary Lie algebra $\mathfrak{su}(n)$, which consists of skew-Hermitian matrices with trace zero.

A representation of a Lie algebra $\mathfrak{g}$ on a vector space $V$ is a linear map $\rho : \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ that preserves the Lie bracket, i.e.,

$\rho([X, Y]) = [\rho(X), \rho(Y)]$ for all $X, Y \in \mathfrak{g}$.

In other words, a representation is a homomorphism from the Lie algebra $\mathfrak{g}$ to the Lie algebra of linear operators on $V$. Representations are crucial for studying the structure of Lie algebras because they provide a way to realize the abstract algebraic structure of a Lie algebra as concrete linear transformations. The most important representation for our discussion is the adjoint representation, which is defined as follows:

The adjoint representation of $\mathfrak{g}$ is the map $\text{ad} : \mathfrak{g} \rightarrow \mathfrak{gl}(\mathfrak{g})$ given by

$\text{ad}_X(Y) = [X, Y]$ for all $X, Y \in \mathfrak{g}$.

It can be verified that $\text{ad}$ is indeed a representation, meaning that it preserves the Lie bracket: $\text{ad}_{[X, Y]} = [\text{ad}_X, \text{ad}_Y]$. The adjoint representation provides a way to study the structure of the Lie algebra $\mathfrak{g}$ by examining how its elements act on each other via the Lie bracket. Subalgebras that are closed under the adjoint action, i.e., subalgebras $\mathfrak{h} \subseteq \mathfrak{g}$ such that $\text{ad}_X(Y) \in \mathfrak{h}$ for all $X \in \mathfrak{g}$ and $Y \in \mathfrak{h}$, play a significant role in the representation theory of Lie algebras. These subalgebras often correspond to subgroups of the corresponding Lie group, and their structure can reveal important information about the group's representations.

The Cartan Decomposition

The Cartan decomposition is a fundamental tool for studying the structure of semisimple Lie algebras. A Lie algebra $\mathfrak{g}$ is called semisimple if it has no nonzero solvable ideals. Intuitively, this means that the Lie algebra cannot be broken down into a chain of subalgebras where each quotient is abelian. Semisimple Lie algebras are the building blocks of all Lie algebras, as any Lie algebra can be decomposed into a solvable ideal and a semisimple subalgebra. The Cartan decomposition provides a way to decompose a semisimple Lie algebra into two subspaces with specific properties that facilitate the analysis of its structure and representations.

Let $\mathfrak{g}$ be a semisimple Lie algebra over the real numbers $\mathbb{R}$, and let $\theta$ be a Cartan involution of $\mathfrak{g}$. A Cartan involution is an involutive automorphism (i.e., $\theta^2 = \text{id}$) of $\mathfrak{g}$ that satisfies the following condition: the bilinear form

$B_{\theta}(X, Y) = -B(X, \theta(Y))$,

where $B$ is the Killing form of $\mathfrak{g}$, is positive definite. The Killing form is a symmetric bilinear form defined by

$B(X, Y) = \text{Tr}(\text{ad}_X \text{ad}_Y)$ for all $X, Y \in \mathfrak{g}$,

where $\text{Tr}$ denotes the trace of a linear operator. The existence of a Cartan involution is a crucial property of semisimple Lie algebras, and it allows us to define the Cartan decomposition.

The Cartan decomposition of $\mathfrak{g}$ with respect to the Cartan involution $\theta$ is the decomposition

$\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$,

where $\mathfrak{k}$ and $\mathfrak{p}$ are the eigenspaces of $\theta$ corresponding to the eigenvalues $+1$ and $-1$, respectively, i.e.,

$\mathfrak{k} = \{X \in \mathfrak{g} \mid \theta(X) = X\}$ and $\mathfrak{p} = \{X \in \mathfrak{g} \mid \theta(X) = -X\}$.

The subspace $\mathfrak{k}$ is a subalgebra of $\mathfrak{g}$, called the maximal compact subalgebra, and $\mathfrak{p}$ is a subspace that is not a subalgebra but satisfies $[\&mathfrak{k}, \mathfrak{p}] \subseteq \mathfrak{p}$ and $[\&mathfrak{p}, \mathfrak{p}] \subseteq \mathfrak{k}$. The Cartan decomposition has several important properties that make it a powerful tool for studying Lie algebras:

1. The subspaces $\mathfrak{k}$ and $\mathfrak{p}$ are orthogonal with respect to the Killing form, i.e., $B(X, Y) = 0$ for all $X \in \mathfrak{k}$ and $Y \in \mathfrak{p}$.
2. The restriction of the Killing form to $\mathfrak{k}$ is negative definite, and its restriction to $\mathfrak{p}$ is positive definite.
3. The map $\mathfrak{k} \times \mathfrak{p} \rightarrow \mathfrak{g}$ given by $(X, Y) \mapsto X + Y$ is a vector space isomorphism.

In the case of the special unitary Lie algebra $\mathfrak{su}(n)$, the Cartan involution can be defined as $\theta(X) = -X^\dagger$, where $X^\dagger$ denotes the conjugate transpose of the matrix $X$. With this choice of Cartan involution, the maximal compact subalgebra $\mathfrak{k}$ is the special orthogonal Lie algebra $\mathfrak{so}(n)$, which consists of skew-symmetric real matrices, and the subspace $\mathfrak{p}$ consists of symmetric imaginary matrices. The Cartan decomposition $\mathfrak{su}(n) = \mathfrak{so}(n) \oplus \mathfrak{p}$ provides a way to study the structure of $\mathfrak{su}(n)$ by examining its relationship to the more familiar Lie algebra $\mathfrak{so}(n)$.

Role of Cartan Decomposition in Existence of Minimal Subalgebras

The Cartan decomposition plays a crucial role in establishing the existence of minimal subalgebras closed under the adjoint action in $\mathfrak{su}(n)$. To understand this, we need to consider how the adjoint action interacts with the Cartan decomposition. Recall that a subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ is closed under the adjoint action if $\text{ad}_X(Y) = [X, Y] \in \mathfrak{h}$ for all $X \in \mathfrak{g}$ and $Y \in \mathfrak{h}$. In other words, a subalgebra is closed under the adjoint action if it is an ideal of $\mathfrak{g}$.

The Cartan decomposition allows us to analyze the adjoint action by considering its restrictions to the subspaces $\mathfrak{k}$ and $\mathfrak{p}$. Since $\mathfrak{k}$ is a subalgebra, the adjoint action of $\mathfrak{k}$ on itself is well-defined. Furthermore, the relations $[\&mathfrak{k}, \mathfrak{p}] \subseteq \mathfrak{p}$ and $[\&mathfrak{p}, \mathfrak{p}] \subseteq \mathfrak{k}$ imply that the adjoint action of $\mathfrak{k}$ on $\mathfrak{p}$ and the adjoint action of $\mathfrak{p}$ on $\mathfrak{k}$ are also well-defined. These observations are crucial for understanding how subalgebras closed under the adjoint action can arise in the Cartan decomposition.

A minimal subalgebra closed under the adjoint action is a nonzero subalgebra that does not contain any proper nonzero subalgebras that are also closed under the adjoint action. In other words, it is a "smallest" ideal of the Lie algebra. The Cartan decomposition helps us identify these minimal subalgebras by providing a framework for analyzing the structure of ideals in $\mathfrak{su}(n)$. Specifically, we can consider the intersection of an ideal $\mathfrak{h}$ with the subspaces $\mathfrak{k}$ and $\mathfrak{p}$: let $\mathfrak{h}_\mathfrak{k} = \mathfrak{h} \cap \mathfrak{k}$ and $\mathfrak{h}_\mathfrak{p} = \mathfrak{h} \cap \mathfrak{p}$. Then, since $\mathfrak{h}$ is an ideal, we have

$[\&mathfrak{k}, \mathfrak{h}] \subseteq \mathfrak{h}$ and $[\&mathfrak{p}, \mathfrak{h}] \subseteq \mathfrak{h}$.

This implies that

$[\&mathfrak{k}, \mathfrak{h}_\mathfrak{k}] \subseteq \mathfrak{h}_\mathfrak{k}$, $[\&mathfrak{k}, \mathfrak{h}_\mathfrak{p}] \subseteq \mathfrak{h}_\mathfrak{p}$, $[\&mathfrak{p}, \mathfrak{h}_\mathfrak{k}] \subseteq \mathfrak{h}_\mathfrak{p}$, $[\&mathfrak{p}, \mathfrak{h}_\mathfrak{p}] \subseteq \mathfrak{h}_\mathfrak{k}$.

Thus, $\mathfrak{h}_\mathfrak{k}$ is an ideal of $\mathfrak{k}$, and $\mathfrak{h}_\mathfrak{p}$ is a subspace of $\mathfrak{p}$ that is invariant under the adjoint action of $\mathfrak{k}$. The structure of these subspaces and their relationship to each other provide crucial information about the structure of $\mathfrak{h}$. In particular, if $\mathfrak{h}$ is a minimal ideal, then $\mathfrak{h}_\mathfrak{k}$ and $\mathfrak{h}_\mathfrak{p}$ must also be minimal in some sense.

For example, in $\mathfrak{su}(2)$, the Cartan decomposition is $\mathfrak{su}(2) = \mathfrak{so}(2) \oplus \mathfrak{p}$, where $\mathfrak{so}(2)$ consists of skew-symmetric 2x2 real matrices and $\mathfrak{p}$ consists of symmetric imaginary 2x2 matrices. The only minimal ideal in $\mathfrak{su}(2)$ is $\mathfrak{su}(2)$ itself, which reflects the fact that $\mathfrak{su}(2)$ is a simple Lie algebra (i.e., it has no nontrivial ideals). In more complex Lie algebras, the Cartan decomposition can help us identify nontrivial minimal ideals by analyzing the structure of the subspaces $\mathfrak{k}$ and $\mathfrak{p}$ and their interactions under the adjoint action.

Conclusion

The Cartan decomposition is an indispensable tool in the study of Lie algebras, particularly in understanding the existence and structure of minimal subalgebras closed under the adjoint action. By decomposing a semisimple Lie algebra into the maximal compact subalgebra $\mathfrak{k}$ and the subspace $\mathfrak{p}$, the Cartan decomposition provides a framework for analyzing the adjoint action and identifying ideals. The interaction between the adjoint action and the Cartan decomposition allows us to understand how minimal ideals arise and to characterize their properties.

In the specific case of the special unitary Lie algebra $\mathfrak{su}(n)$, the Cartan decomposition $\mathfrak{su}(n) = \mathfrak{so}(n) \oplus \mathfrak{p}$ plays a crucial role in identifying minimal subalgebras closed under the adjoint action. By examining the structure of $\mathfrak{so}(n)$ and $\mathfrak{p}$ and their interactions under the adjoint action, we can gain insights into the building blocks of $\mathfrak{su}(n)$ and its representations. This understanding is essential for various applications in mathematics and physics, including the study of symmetric spaces, the classification of Lie groups and algebras, and the construction of physical models with specific symmetry properties.

In summary, the Cartan decomposition is a powerful tool that provides a deeper understanding of the structure of Lie algebras and their representations. Its role in establishing the existence of minimal subalgebras closed under the adjoint action highlights its importance in the field of Lie theory. Further exploration of this decomposition and its applications will undoubtedly lead to new insights and advancements in our understanding of Lie algebras and their connections to other areas of mathematics and physics.