4.4 The Levi decomposition theorem

The Levi decomposition theorem is a powerful tool in Lie algebra theory. It breaks down any finite-dimensional Lie algebra into a solvable part and a semisimple part, allowing for deeper analysis of their structure.

This decomposition connects solvable and semisimple Lie algebras, two key types studied in the course. It helps simplify complex Lie algebras by separating their components, making them easier to understand and work with.

Levi Decomposition Theorem

Statement and Key Components

Top images from around the web for Statement and Key Components

• 

  File:Lie algebra extension figure 1.svg - Wikipedia View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Lie Groups [The Physics Travel Guide] View original

  Is this image relevant?

• 

  File:Lie algebra extension figure 1.svg - Wikipedia View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

1 of 3

Top images from around the web for Statement and Key Components

• 

  File:Lie algebra extension figure 1.svg - Wikipedia View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

• 

  Lie Groups [The Physics Travel Guide] View original

  Is this image relevant?

• 

  File:Lie algebra extension figure 1.svg - Wikipedia View original

  Is this image relevant?

• 

  Representation Theory [The Physics Travel Guide] View original

  Is this image relevant?

1 of 3

• The Levi decomposition theorem states that any finite-dimensional Lie algebra $\mathfrak{g}$ over a field of characteristic zero can be written as a semidirect product $\mathfrak{g} = \mathfrak{r} \oplus \mathfrak{s}$
  • $\mathfrak{r}$ is the solvable radical of $\mathfrak{g}$, the maximal solvable ideal of the Lie algebra $\mathfrak{g}$
  • $\mathfrak{s}$ is a semisimple subalgebra of $\mathfrak{g}$, called a Levi subalgebra or Levi factor of $\mathfrak{g}$
• The Levi decomposition is unique up to conjugation by inner automorphisms of $\mathfrak{g}$
• The Levi decomposition generalizes the Jordan decomposition of a linear operator into semisimple and nilpotent parts ($\mathfrak{gl}(n,\mathbb{C})$)

Significance and Applications

• The Levi decomposition allows for the structural analysis of Lie algebras by separating the solvable and semisimple components
• It provides a way to study Lie algebras by separately analyzing the solvable radical and semisimple part using techniques specific to each type
• The Levi decomposition is used to prove various results about Lie algebras
  • Weyl's complete reducibility theorem
  • Cartan criterion for semisimplicity
• In applications, the Levi decomposition can be computed using algorithms from computational Lie theory to analyze concrete Lie algebras

Solvable vs Semisimple Lie Algebras

Definitions and Properties

• A Lie algebra is solvable if its derived series terminates at the zero subalgebra after a finite number of steps
  • Derived series: $\mathfrak{g}^{(0)} = \mathfrak{g}, \mathfrak{g}^{(i+1)} = [\mathfrak{g}^{(i)}, \mathfrak{g}^{(i)}]$
  • Solvable if $\mathfrak{g}^{(n)} = 0$ for some $n$
• A Lie algebra is semisimple if it has no non-zero solvable ideals
  • Equivalently, if its radical (maximal solvable ideal) is zero
• Examples of solvable Lie algebras: $\mathfrak{t}(n)$ (upper triangular matrices), $\mathfrak{n}$ (nilpotent matrices)
• Examples of semisimple Lie algebras: $\mathfrak{sl}(n,\mathbb{C})$, $\mathfrak{so}(n,\mathbb{C})$, $\mathfrak{sp}(n,\mathbb{C})$

Relationship via Levi Decomposition

• The Levi decomposition shows that any Lie algebra can be decomposed into a solvable part (the radical $\mathfrak{r}$) and a semisimple part (the Levi subalgebra $\mathfrak{s}$)
• The solvable radical $\mathfrak{r}$ is the largest solvable ideal of the Lie algebra, while the Levi subalgebra $\mathfrak{s}$ is a maximal semisimple subalgebra
• The radical and Levi subalgebra are complementary in the sense that their intersection is zero and they generate the entire Lie algebra: $\mathfrak{g} = \mathfrak{r} \oplus \mathfrak{s}$

Existence and Uniqueness of Levi Decomposition

Proving Existence

• The proof of existence relies on the fact that the sum of solvable ideals is again a solvable ideal, implying that the solvable radical $\mathfrak{r}$ exists as the maximal solvable ideal
• The quotient algebra $\mathfrak{g}/\mathfrak{r}$ is semisimple, and a theorem states that any semisimple Lie algebra is a direct sum of simple Lie algebras
• Using the semisimplicity of $\mathfrak{g}/\mathfrak{r}$, one can construct a semisimple subalgebra $\mathfrak{s}$ of $\mathfrak{g}$ such that $\mathfrak{g} = \mathfrak{r} \oplus \mathfrak{s}$, proving the existence of the Levi decomposition

Proving Uniqueness

• To prove uniqueness, suppose $\mathfrak{g} = \mathfrak{r} \oplus \mathfrak{s}_1 = \mathfrak{r} \oplus \mathfrak{s}_2$ are two Levi decompositions
• One can show that there exists an inner automorphism $\phi$ of $\mathfrak{g}$ such that $\phi(\mathfrak{s}_1) = \mathfrak{s}_2$
• This demonstrates that the Levi decomposition is unique up to conjugation by inner automorphisms of $\mathfrak{g}$

Applying Levi Decomposition to Lie Algebras

Structural Analysis

• The Levi decomposition allows one to study a Lie algebra by separately analyzing its solvable radical and semisimple part
• The structure of the solvable radical can be understood using techniques for solvable Lie algebras
  • Lie's theorem on nilpotent Lie algebras
  • Jordan decomposition
• The semisimple part can be studied using the classification of simple Lie algebras and their representations
  • Classification of simple Lie algebras ($A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2$)
  • Representation theory of semisimple Lie algebras

Computational Methods

• In practice, the Levi decomposition can be computed using algorithms from computational Lie theory
• These algorithms allow for the structural analysis of concrete Lie algebras arising in applications
• Examples of computational methods:
  • Gröbner basis techniques for nilpotent Lie algebras
  • Algorithms for computing the radical and Levi subalgebra
  • Methods for classifying and constructing representations of semisimple Lie algebras

Theoretical Applications

• The Levi decomposition can be used to prove various results about Lie algebras
• Examples of theoretical applications:
  • Weyl's complete reducibility theorem: Every finite-dimensional representation of a semisimple Lie algebra is completely reducible
  • Cartan criterion for semisimplicity: A Lie algebra is semisimple if and only if its Killing form is non-degenerate
  • Mostow decomposition: A connected Lie group can be decomposed into a solvable normal subgroup and a semisimple subgroup