4.3 Semisimple Lie algebras and their structure

Semisimple Lie algebras are a crucial class in Lie theory, characterized by having no non-zero solvable ideals. They're the building blocks of complex Lie algebras, decomposing into simple ideals. This structure reveals deep connections between algebra and geometry.

The Killing form plays a key role in understanding semisimple Lie algebras. It's non-degenerate for semisimple algebras, allowing for root space decomposition and classification. This leads to the powerful tools of root systems and Dynkin diagrams.

Semisimple Lie algebras

Definition and properties

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• A Lie algebra is semisimple when it contains no non-zero solvable ideals
• Semisimple Lie algebras are expressed as a direct sum of simple Lie algebras, non-abelian Lie algebras whose only ideals are the trivial ideal {0} and the algebra itself
• The Killing form of a semisimple Lie algebra is non-degenerate, implying that if $B(x,y) = 0$ for all $y$ in the Lie algebra, then $x = 0$
• Common examples of semisimple Lie algebras include the special linear algebra $sl(n,C)$, the orthogonal algebra $so(n,C)$, and the symplectic algebra $sp(n,C)$

Killing form and Cartan's criterion

• The Killing form is a symmetric bilinear form defined on a Lie algebra $L$ as $B(x,y) = tr(ad_x \circ ad_y)$, where $ad_x$ is the adjoint representation of $x$
• Cartan's criterion states that a Lie algebra is semisimple if and only if its Killing form is non-degenerate
• The non-degeneracy of the Killing form allows for the construction of a root space decomposition and the study of the structure of semisimple Lie algebras
• The Killing form is invariant under automorphisms of the Lie algebra, which leads to important symmetries in the theory of semisimple Lie algebras

Structure of semisimple Lie algebras

Root space decomposition

• Semisimple Lie algebras admit a root space decomposition with respect to a Cartan subalgebra, a maximal abelian subalgebra that acts diagonalizably on the Lie algebra via the adjoint representation
• The root space decomposition of a semisimple Lie algebra $L$ is given by $L = H \oplus (\oplus_{\alpha \in \Phi} L_{\alpha})$, where $H$ is the Cartan subalgebra, $\Phi$ is the set of roots (eigenvalues of the adjoint representation), and $L_{\alpha}$ is the root space corresponding to the root $\alpha$
• Each root space $L_{\alpha}$ is one-dimensional, and the elements of $L_{\alpha}$ are eigenvectors of the adjoint representation of elements in the Cartan subalgebra
• The root space decomposition provides a way to study the structure of semisimple Lie algebras in terms of their root systems

Root systems and Weyl groups

• The set of roots $\Phi$ forms a root system, a finite subset of a Euclidean space satisfying certain axioms related to symmetry and closure under reflections
• Root systems have a basis of simple roots, which are roots that cannot be written as a sum of other positive roots
• The Weyl group of a semisimple Lie algebra is the group generated by reflections in the hyperplanes orthogonal to the roots, acting on the root system and preserving the Killing form
• The Weyl group is a finite reflection group and plays a crucial role in the classification of semisimple Lie algebras
• The length of elements in the Weyl group corresponds to the length of reduced expressions in terms of simple reflections, which is related to the geometry of the root system

Decomposition of semisimple Lie algebras

Ideals and direct sums

• The proof of the decomposition of semisimple Lie algebras into simple ideals relies on the non-degeneracy of the Killing form
• If $L$ is a semisimple Lie algebra and $I$ is a non-zero ideal of $L$, then both $I$ and its orthogonal complement $I^{\perp}$ (with respect to the Killing form) are ideals of $L$
• This leads to the decomposition $L = I \oplus I^{\perp}$, where both $I$ and $I^{\perp}$ are semisimple Lie algebras
• By repeatedly applying this decomposition, a semisimple Lie algebra can be expressed as a direct sum of simple ideals, since a simple Lie algebra cannot be further decomposed

Proof of the decomposition theorem

• The proof proceeds by induction on the dimension of the semisimple Lie algebra $L$
• If $L$ is not simple, then it contains a non-zero proper ideal $I$, and the Killing form restricted to $I$ is non-degenerate
• The orthogonal complement $I^{\perp}$ is also an ideal of $L$, and $L = I \oplus I^{\perp}$ as a direct sum of ideals
• By the induction hypothesis, both $I$ and $I^{\perp}$ can be decomposed into a direct sum of simple ideals, yielding the desired decomposition of $L$
• The base case of the induction is when $L$ is simple, in which case no further decomposition is possible

Classification of semisimple Lie algebras

Root systems and Dynkin diagrams

• Semisimple Lie algebras are classified by their root systems, which can be represented using Dynkin diagrams
• A Dynkin diagram encodes the angular relationships between the simple roots of a root system
• The nodes of a Dynkin diagram represent the simple roots, and the edges represent the angles between them, with different types of edges corresponding to different angle values (e.g., single edge for 120°, double edge for 135°, triple edge for 150°)
• The classification of semisimple Lie algebras over the complex numbers leads to four infinite families: $A_n$ ($sl(n+1,C)$), $B_n$ ($so(2n+1,C)$), $C_n$ ($sp(n,C)$), and $D_n$ ($so(2n,C)$), as well as five exceptional Lie algebras: $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$

Real forms and Satake diagrams

• The classification of semisimple Lie algebras over the real numbers involves additional subtleties related to the real forms of the complex Lie algebras
• A real form of a complex Lie algebra is a real Lie algebra whose complexification is isomorphic to the original complex Lie algebra
• Real forms can be studied using Satake diagrams, a variant of Dynkin diagrams that encode the involution on the root system corresponding to the real form
• Compact and split real forms play a special role in the theory of semisimple Lie algebras, with the compact form having a negative definite Killing form and the split form having a root space decomposition over the real numbers
• The classification of real semisimple Lie algebras is more intricate than the complex case, but it builds upon the same principles of root systems and Dynkin diagrams, with additional data to capture the real structure