7.1 Connected and simply connected Lie groups

Connected and simply connected Lie groups are crucial in understanding the structure of Lie groups. These groups can't be divided into separate pieces and have no holes, making them fundamental building blocks in the theory.

Studying these groups reveals deep connections between a group's topology and its algebraic properties. This knowledge is essential for classifying Lie groups and has wide-ranging applications in mathematics and physics.

Connected and Simply Connected Lie Groups

Defining Connected and Simply Connected Lie Groups

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• A connected Lie group is a Lie group where the underlying manifold is connected
  • Consists of a single piece
  • Cannot be divided into two or more disjoint open subsets
• A simply connected Lie group is a connected Lie group where every loop can be continuously shrunk to a point within the group
  • A loop is a continuous closed curve
  • The fundamental group of a simply connected Lie group is trivial
  • The fundamental group consists only of the identity element
• Examples of connected Lie groups
  • SO(n)
  • SL(n, R)
  • U(n)
• Examples of simply connected Lie groups
  • The universal cover of SL(2, R)
  • The Heisenberg group

Properties and Applications of Connected and Simply Connected Lie Groups

• The structure of connected and simply connected Lie groups is closely related to the structure of their Lie algebras
• Simply connected Lie groups are crucial in the study of representations and in the classification of Lie groups
• The universal covering group of a connected Lie group is always simply connected
• The study of connected and simply connected Lie groups has applications in various areas of mathematics and physics
  • Quantum mechanics
  • Gauge theories
  • Differential geometry

Exponential Map and Connectedness

Definition and Properties of the Exponential Map

• The exponential map exp: g → G is a smooth map from the Lie algebra g to the Lie group G
  • Defined by exp(X) = γ(1), where γ is the unique one-parameter subgroup of G with γ'(0) = X
  • The exponential map is a local diffeomorphism near the identity element of G
• For a connected Lie group G, the exponential map is surjective
  • Every element of G can be written as the exponential of some element in the Lie algebra g
• The exponential map may not be injective
  • Its kernel is a discrete subgroup of the Lie algebra g
  • The kernel is related to the fundamental group of the Lie group G

Proving the Surjectivity of the Exponential Map for Connected Lie Groups

• The surjectivity of the exponential map for connected Lie groups can be proved using the existence and uniqueness of solutions to ordinary differential equations on manifolds
• The proof involves constructing a smooth path from the identity element to any given element of the Lie group
  • This path is obtained by solving an appropriate differential equation
  • The solution to the differential equation is guaranteed by the existence and uniqueness theorems for ODEs on manifolds
• The exponential map is then used to show that the endpoint of the constructed path can be written as the exponential of some element in the Lie algebra

Classifying Low-Dimensional Lie Groups

One-Dimensional Connected and Simply Connected Lie Groups

• The only connected and simply connected one-dimensional Lie group (up to isomorphism) is the additive group of real numbers (R, +)
  • This group is abelian and has a trivial Lie algebra
• Other one-dimensional Lie groups, such as the circle group S^1, are connected but not simply connected
  • The circle group has a non-trivial fundamental group isomorphic to the integers Z

Two-Dimensional Connected and Simply Connected Lie Groups

• In two dimensions, there are two connected and simply connected Lie groups
  • The abelian group R^2
  • The non-abelian group known as the affine group of the line, Aff(R) = {(a, b) | a > 0, b ∈ R}
• The Lie algebra of R^2 is abelian, while the Lie algebra of Aff(R) is solvable but not nilpotent

Three-Dimensional Connected and Simply Connected Lie Groups

• In three dimensions, there are four connected and simply connected Lie groups
  • The abelian group R^3
  • The Heisenberg group
  • The universal cover of SL(2, R)
  • The universal cover of SO(3), which is isomorphic to SU(2)
• The Lie algebras of these groups have different structures
  • The Lie algebra of R^3 is abelian
  • The Lie algebra of the Heisenberg group is nilpotent
  • The Lie algebras of the universal covers of SL(2, R) and SO(3) are semisimple

Higher-Dimensional Connected and Simply Connected Lie Groups

• The classification of connected and simply connected Lie groups in higher dimensions becomes increasingly complex
• The classification relies on the structure theory of Lie algebras
  • Involves the study of semisimple, solvable, and nilpotent Lie algebras
  • Uses the root system and the Dynkin diagram associated with a semisimple Lie algebra
• The classification of compact, connected, and simply connected Lie groups is given by the Cartan-Killing classification

Universal Covering Group of a Lie Group

Definition and Properties of the Universal Covering Group

• The universal covering group of a connected Lie group G is a simply connected Lie group G̃ together with a smooth covering homomorphism φ: G̃ → G
• The covering homomorphism φ is a local diffeomorphism and a group homomorphism
  • Makes G̃ a covering space of G in the topological sense
• The kernel of the covering homomorphism φ is a discrete central subgroup of G̃
  • Isomorphic to the fundamental group of G
• The universal covering group is unique up to isomorphism

Constructing the Universal Covering Group

• The universal covering group can be constructed using the path lifting property of covering spaces
  • Involves lifting paths and loops from the base space G to the covering space G̃
  • The lifted paths and loops are used to define the group structure on G̃
• Alternatively, the universal covering group can be constructed by integrating the Lie algebra of G
  • The Lie algebra is simply connected as a vector space
  • The group structure on G̃ is obtained by exponentiating the Lie algebra and using the Baker-Campbell-Hausdorff formula

Examples of Universal Covering Groups

• The real line R is the universal cover of the circle group S^1
  • The covering homomorphism is given by the exponential map exp: R → S^1, exp(t) = e^(it)
  • The kernel of the covering homomorphism is the subgroup of integer multiples of 2π
• The group SU(2) is the universal cover of SO(3)
  • The covering homomorphism is a two-to-one group homomorphism
  • The kernel of the covering homomorphism is the subgroup {±I}, where I is the 2×2 identity matrix
• The universal cover of SL(2, R) is a simply connected Lie group that is not isomorphic to any classical matrix group