7.3 Maximal tori and the Weyl group of a Lie group

Maximal tori and the Weyl group are key to understanding Lie groups. Maximal tori are the largest abelian subgroups, while the Weyl group captures symmetries of the root system.

These concepts help classify compact Lie groups and reveal their structure. The interplay between maximal tori, the Weyl group, and root systems is crucial for grasping Lie group theory.

Maximal tori in Lie groups

Definition and properties of maximal tori

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• A torus is a compact, connected, abelian Lie subgroup of a Lie group
  • Isomorphic to a direct product of circles (S^1 × S^1 × ... × S^1)
  • Examples of tori include the circle group (S^1) and the 2-torus (S^1 × S^1)
• A maximal torus is a torus that is not contained in any other torus
  • It is a maximal abelian subgroup of the Lie group
  • Every element of a compact, connected Lie group is contained in a maximal torus
• All maximal tori in a Lie group are conjugate to each other
  • If T_1 and T_2 are maximal tori, then there exists g in G such that gT_1g^(-1) = T_2
• The dimension of a maximal torus is called the rank of the Lie group
  • For example, the rank of SU(n) is n-1, and the rank of SO(2n) is n

Centralizer and normalizer of maximal tori

• The centralizer of a maximal torus is the maximal torus itself
  • C_G(T) = {g in G : gtg^(-1) = t for all t in T} = T
  • This property distinguishes maximal tori from other abelian subgroups
• The normalizer of a maximal torus is a larger subgroup containing the torus
  • N_G(T) = {g in G : gTg^(-1) = T}
  • The normalizer plays a crucial role in defining the Weyl group of the Lie group

Weyl group action on tori

Definition and properties of the Weyl group

• The Weyl group of a Lie group G with respect to a maximal torus T is the quotient group N_G(T)/C_G(T)
  • It is a finite group that encodes the symmetries of the root system
  • The Weyl group acts on the maximal torus by conjugation
• The Weyl group is generated by reflections in the roots of the Lie algebra corresponding to the Lie group
  • These reflections are orthogonal transformations with respect to the Killing form
  • The length of a Weyl group element is the minimum number of reflections needed to express it

Action of the Weyl group on the character lattice

• The action of the Weyl group on the maximal torus induces an action on the character group of the torus
  • The character group is a lattice (a discrete subgroup of the dual of the Lie algebra of the torus)
  • The Weyl group acts on the character lattice by reflections in the roots
• The orbits of the Weyl group action on the character lattice are called root orbits
  • Each root orbit contains a unique dominant weight (a weight in the positive Weyl chamber)
  • The number of dominant weights is equal to the order of the Weyl group divided by the number of elements in the root orbit

Lie group structure and Weyl group

Relationship between the Weyl group and the root system

• The root system of a Lie algebra is determined by the action of the Weyl group on the character lattice of a maximal torus
  • The roots are the weights of the adjoint representation of the Lie algebra
  • The simple roots are a basis for the root system, and their number is equal to the rank of the Lie algebra
• The Weyl chambers are fundamental domains for the action of the Weyl group on the Lie algebra
  • They are convex cones bounded by the hyperplanes orthogonal to the roots
  • The positive Weyl chamber is the one containing the dominant weights
• The Weyl group acts transitively on the Weyl chambers, and the stabilizer of a Weyl chamber is trivial
  • This means that the Weyl chambers are in one-to-one correspondence with the elements of the Weyl group

Bruhat decomposition and the Weyl group

• The Bruhat decomposition expresses a Lie group as a disjoint union of double cosets of a Borel subgroup, indexed by elements of the Weyl group
  • A Borel subgroup is a maximal connected solvable subgroup of the Lie group
  • The double cosets are of the form BwB, where B is a Borel subgroup and w is an element of the Weyl group
• The Bruhat order on the Weyl group is a partial order defined by the inclusion of Bruhat cells
  • The Bruhat cell corresponding to w is the double coset BwB
  • The Bruhat order encodes important combinatorial and geometric properties of the Lie group

Classifying compact Lie groups

Classification of compact, simple Lie groups

• Compact, connected Lie groups can be classified up to isomorphism by their root systems
  • The root system is determined by the Weyl group and the character lattice of a maximal torus
  • Two compact, connected Lie groups are isomorphic if and only if their root systems are isomorphic
• The classification of compact, simple Lie groups corresponds to the classification of irreducible root systems
  • There are four infinite families of irreducible root systems: A_n (n ≥ 1), B_n (n ≥ 2), C_n (n ≥ 3), and D_n (n ≥ 4)
  • There are five exceptional irreducible root systems: G_2, F_4, E_6, E_7, and E_8
• The Weyl group of a compact, simple Lie group is the reflection group associated with its root system
  • For example, the Weyl group of SU(n) is the symmetric group S_n, and the Weyl group of SO(2n) is the signed permutation group

Dynkin diagrams and the fundamental group

• The Dynkin diagram of a compact, simple Lie group encodes the structure of its root system and Weyl group
  • The nodes of the Dynkin diagram represent the simple roots, and the edges represent the angles between them
  • The Dynkin diagram determines the Cartan matrix, which encodes the inner products between the simple roots
• The fundamental group of a compact, simple Lie group can be determined from the character lattice of a maximal torus and the action of the Weyl group
  • The fundamental group is isomorphic to the center of the universal cover of the Lie group
  • For simply connected Lie groups (such as SU(n) and Sp(n)), the character lattice is the weight lattice
  • For non-simply connected Lie groups (such as SO(n) and PU(n)), the character lattice is a sublattice of the weight lattice, determined by the action of the Weyl group