7.4 Covering groups and the fundamental group

Covering groups and fundamental groups are crucial concepts in Lie theory. They help us understand the global structure of Lie groups by connecting local properties to overall topology. This relationship allows us to simplify complex problems and classify Lie groups more effectively.

Universal covering groups are unique, simply connected versions of Lie groups. Their relationship to the original group's fundamental group gives us powerful tools for studying Lie group structure and representations, and even for constructing new Lie groups with specific properties.

Covering Groups and Fundamental Groups

Definition and Properties of Covering Groups

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• A covering group is a Lie group that locally resembles the original Lie group but may have a different global structure
• It is defined as a surjective homomorphism with a discrete kernel
• The covering group is a topological space that locally looks like the base space (original Lie group) but may have a different global structure
• Covering groups are important in the study of Lie groups because they allow for the investigation of the global structure of a Lie group by studying its local properties

Definition and Properties of the Fundamental Group

• The fundamental group of a Lie group G, denoted π₁(G), is the group of homotopy classes of loops based at the identity element of G
• It encodes the global topological structure of the Lie group
• The fundamental group is a topological invariant, meaning that homeomorphic Lie groups have isomorphic fundamental groups
• For example, the fundamental group of the circle group S¹ is isomorphic to the integers ℤ, while the fundamental group of the real line ℝ is trivial
• The fundamental group of a simply connected Lie group is trivial, i.e., π₁(G) = {e} if G is simply connected (e.g., the special unitary group SU(n))
• The fundamental group of a connected Lie group is always abelian

Constructing Universal Covering Groups

Definition and Uniqueness 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 covering homomorphism φ: G̃ → G
• The universal covering group is unique up to isomorphism
• The covering homomorphism φ: G̃ → G is a surjective Lie group homomorphism with discrete kernel
• For example, the universal covering group of the special orthogonal group SO(3) is the spin group Spin(3), which is isomorphic to the unit quaternions

Relationship between the Kernel of the Covering Homomorphism and the Fundamental Group

• The kernel of the covering homomorphism is isomorphic to the fundamental group of G, i.e., ker(φ) ≅ π₁(G)
• This means that the fundamental group of G can be realized as a subgroup of the universal covering group G̃
• The universal covering group of a simply connected Lie group is itself (e.g., the universal cover of SU(n) is SU(n) itself)
• The covering homomorphism provides a way to relate the structure of the original Lie group to its universal cover

Fundamental Group vs Center of Covering Group

Definition and Properties of the Center of a Lie Group

• The center of a Lie group G, denoted Z(G), is the subgroup of elements that commute with every element in G
• The center is a normal subgroup of G and is always abelian
• For example, the center of the general linear group GL(n, ℂ) consists of scalar matrices of the form λI, where λ is a non-zero complex number and I is the identity matrix

Isomorphism between the Fundamental Group and the Center of the Universal Covering Group

• For a connected Lie group G with universal covering group G̃, there is a natural isomorphism between the fundamental group of G and the center of G̃, i.e., π₁(G) ≅ Z(G̃)
• This isomorphism is given by mapping a loop in G to the unique element in G̃ that covers it and stays in the center of G̃
• As a consequence, the center of the universal covering group is always discrete
• This relationship allows for the study of the fundamental group of a Lie group through the center of its universal cover

Covering Groups for Lie Group Structure

Classification of Lie Groups using Covering Groups

• Covering groups can be used to classify Lie groups up to local isomorphism
• Lie groups with isomorphic Lie algebras have the same universal covering group
• This allows for the classification of Lie groups based on their universal covers and the discrete subgroups that can be factored out to obtain the original groups
• For example, the special unitary group SU(2) and the spin group Spin(3) have isomorphic Lie algebras and share the same universal cover, which is SU(2) itself

Simplification of Lie Group Problems using Covering Groups

• The study of covering groups allows for the reduction of many questions about Lie groups to the case of simply connected Lie groups, which are often easier to understand
• Properties of the original Lie group can be studied by investigating the corresponding properties of its universal cover and then descending back to the original group
• For example, representation theory of a Lie group is closely related to the representation theory of its universal covering group
• Representations of the universal cover that are trivial on the kernel descend to representations of the original group

Construction of New Lie Groups using Covering Groups

• Covering groups provide a way to construct new Lie groups from known ones
• By taking quotients of the universal covering group by discrete subgroups, one can obtain new Lie groups with desired properties
• For example, the spin groups Spin(n) are double covers of the special orthogonal groups SO(n)
• The metaplectic group Mp(2n) is a double cover of the symplectic group Sp(2n) and has applications in quantum mechanics