4.1 Solvable and nilpotent Lie algebras

Solvable and nilpotent Lie algebras are key concepts in understanding Lie algebra structure. They represent special types of algebras where repeated commutators eventually vanish, offering insights into the algebra's complexity and behavior.

These concepts play a crucial role in classifying Lie algebras. By examining how quickly commutators reach zero, we can determine if an algebra is solvable, nilpotent, or neither, helping us understand its properties and relationships to other algebras.

Solvable and Nilpotent Lie Algebras

Defining Solvable Lie Algebras

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• A Lie algebra $\mathfrak{g}$ is solvable if its derived series $\mathfrak{g}^{(i)}$, defined by $\mathfrak{g}^{(0)} = \mathfrak{g}$ and $\mathfrak{g}^{(i+1)} = [\mathfrak{g}^{(i)}, \mathfrak{g}^{(i)}]$, terminates at zero after a finite number of steps
• The derived series of a Lie algebra $\mathfrak{g}$ forms a decreasing sequence of ideals $\mathfrak{g} \supseteq \mathfrak{g}^{(1)} \supseteq \mathfrak{g}^{(2)} \supseteq ... \supseteq \mathfrak{g}^{(n)} = \{0\}$, where each $\mathfrak{g}^{(i)}$ is an ideal of $\mathfrak{g}$
• The quotient algebra $\mathfrak{g}/\mathfrak{g}^{(1)}$ of a solvable Lie algebra $\mathfrak{g}$ by its derived subalgebra $\mathfrak{g}^{(1)}$ is abelian, meaning $[\mathfrak{g}/\mathfrak{g}^{(1)}, \mathfrak{g}/\mathfrak{g}^{(1)}] = \{0\}$
• A Lie algebra $\mathfrak{g}$ is solvable if and only if $[\mathfrak{g}, \mathfrak{g}]$ is nilpotent

Defining Nilpotent Lie Algebras

• A Lie algebra $\mathfrak{g}$ is nilpotent if its lower central series $\mathfrak{g}^i$, defined by $\mathfrak{g}^1 = \mathfrak{g}$ and $\mathfrak{g}^{i+1} = [\mathfrak{g}, \mathfrak{g}^i]$, terminates at zero after a finite number of steps
• The lower central series of a Lie algebra $\mathfrak{g}$ forms a decreasing sequence of ideals $\mathfrak{g} \supseteq \mathfrak{g}^2 \supseteq \mathfrak{g}^3 \supseteq ... \supseteq \mathfrak{g}^n = \{0\}$, where each $\mathfrak{g}^i$ is an ideal of $\mathfrak{g}$
• The nilpotency class of a nilpotent Lie algebra $\mathfrak{g}$ is the smallest positive integer $c$ such that $\mathfrak{g}^{c+1} = \{0\}$
• The quotient algebra $\mathfrak{g}/\mathfrak{g}^2$ of a nilpotent Lie algebra $\mathfrak{g}$ by its second term in the lower central series $\mathfrak{g}^2$ is abelian, meaning $[\mathfrak{g}/\mathfrak{g}^2, \mathfrak{g}/\mathfrak{g}^2] = \{0\}$

Solvable vs Nilpotent Lie Algebras

Relationship between Solvable and Nilpotent Lie Algebras

• Every nilpotent Lie algebra is solvable, but the converse is not true in general
• If a Lie algebra $\mathfrak{g}$ is nilpotent, its derived series terminates at zero faster than its lower central series, meaning $\mathfrak{g}^{(i)} \subseteq \mathfrak{g}^i$ for all $i \geq 0$
• The Heisenberg algebra is a fundamental example of a nilpotent Lie algebra that is the unique (up to isomorphism) non-abelian nilpotent Lie algebra of dimension 3

Comparing Properties of Solvable and Nilpotent Lie Algebras

• Solvable Lie algebras can be represented by upper triangular matrices under a suitable basis (Lie's Theorem), while nilpotent Lie algebras can be represented by strictly upper triangular matrices (Engel's Theorem)
• The center of a Lie algebra $\mathfrak{g}$, denoted by $Z(\mathfrak{g}) = \{x \in \mathfrak{g} | [x, y] = 0 \text{ for all } y \in \mathfrak{g}\}$, is a nilpotent ideal of $\mathfrak{g}$
• The normalizer of a nilpotent ideal in a Lie algebra is a solvable subalgebra containing the ideal

Solvable and Nilpotent Ideals

Existence of Solvable and Nilpotent Ideals

• Lie's Theorem states that every finite-dimensional Lie algebra $\mathfrak{g}$ over a field of characteristic zero contains a unique maximal solvable ideal called the radical of $\mathfrak{g}$ and denoted by $\operatorname{rad}(\mathfrak{g})$
  • The quotient algebra $\mathfrak{g}/\operatorname{rad}(\mathfrak{g})$ is semisimple, meaning it has no non-zero solvable ideals
• Engel's Theorem states that a finite-dimensional Lie algebra $\mathfrak{g}$ is nilpotent if and only if for every $x \in \mathfrak{g}$, the linear map $\operatorname{ad}_x: \mathfrak{g} \to \mathfrak{g}$ defined by $\operatorname{ad}_x(y) = [x, y]$ is nilpotent

Levi Decomposition

• Every finite-dimensional Lie algebra $\mathfrak{g}$ can be written as a semidirect sum $\mathfrak{g} = \mathfrak{s} \oplus \operatorname{rad}(\mathfrak{g})$, where $\mathfrak{s}$ is a semisimple subalgebra (called a Levi subalgebra) and $\operatorname{rad}(\mathfrak{g})$ is the solvable radical of $\mathfrak{g}$
  • This decomposition shows the relationship between solvable and semisimple Lie algebras within a given Lie algebra

Classifying Lie Algebras

Low-Dimensional Lie Algebras

• In dimensions 1 and 2, all Lie algebras are solvable
• In dimension 3, there are 9 classes of Lie algebras, of which only $\mathfrak{sl}(2, \mathbb{C})$ is semisimple (non-solvable)
  • The Heisenberg algebra is a nilpotent Lie algebra of dimension 3

Classifying Solvable and Nilpotent Lie Algebras

• Lie's Theorem provides a way to identify solvable Lie algebras by representing them using upper triangular matrices under a suitable basis
• Engel's Theorem allows for identifying nilpotent Lie algebras by representing them using strictly upper triangular matrices under a suitable basis