Topological groups combine group theory with topology, allowing us to study continuity and convergence in algebraic structures. They're key in noncommutative geometry, providing a framework for spaces with algebraic properties.
These groups have a group structure and a compatible topology. Examples include Lie groups, real numbers under addition, and matrix groups. Properties like continuity of operations and completeness make them powerful tools in mathematics.
Definition of topological groups
A is a mathematical structure that combines the algebraic properties of a group with the topological properties of a topological space
This combination allows for a rich interplay between algebra and topology, enabling the study of continuity, convergence, and other analytical properties in the context of group theory
Topological groups play a fundamental role in various branches of mathematics, including noncommutative geometry, where they provide a framework for studying spaces with additional algebraic structure
Algebraic structure
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The algebraic structure of a topological group is that of a group, which consists of a set G equipped with a binary operation (usually denoted as multiplication) that satisfies the group axioms:
Closure: For any a,b∈G, the product ab is also in G
Associativity: For any a,b,c∈G, (ab)c=a(bc)
Identity: There exists an element e∈G such that ae=ea=a for all a∈G
Inverses: For each a∈G, there exists an element a−1∈G such that aa−1=a−1a=e
The group operation in a topological group is often referred to as multiplication, even if it is not the usual multiplication of numbers (e.g., matrix multiplication in the general linear group)
Topology on group
In addition to the algebraic structure, a topological group is equipped with a topology on the underlying set G
The topology on G is a collection of subsets, called open sets, that satisfy certain axioms:
The empty set ∅ and the entire set G are open
The union of any collection of open sets is open
The intersection of any finite collection of open sets is open
The topology allows for the notion of continuity, convergence, and other analytical properties to be defined on the group
The choice of topology can vary depending on the specific group and the desired properties (e.g., discrete topology, Euclidean topology, Zariski topology)
Compatibility of group operations
For a group G with a topology to be a topological group, the group operations must be compatible with the topology in the following sense:
The multiplication map m:G×G→G, defined by m(a,b)=ab, is continuous with respect to the product topology on G×G and the topology on G
The inversion map i:G→G, defined by i(a)=a−1, is continuous with respect to the topology on G
Compatibility ensures that the algebraic structure of the group is preserved under continuous deformations and limits, allowing for the study of the group's properties using topological tools
Examples of topological groups
Topological groups appear in various areas of mathematics and have numerous applications in noncommutative geometry
Some notable examples of topological groups include Lie groups, the additive group of real numbers, and the general linear group
These examples demonstrate the diversity of topological groups and their relevance in different mathematical contexts
Lie groups
Lie groups are a class of topological groups that are also smooth manifolds, meaning they have a differentiable structure compatible with the group operations
Examples of Lie groups include:
The real numbers R under addition
The circle group S1={z∈C:∣z∣=1} under multiplication
The special orthogonal group SO(n) of n×n orthogonal matrices with determinant 1
The special unitary group SU(n) of n×n unitary matrices with determinant 1
Lie groups are of particular importance in noncommutative geometry, as they give rise to interesting examples of noncommutative spaces through their actions on manifolds and their associated C*-algebras
Additive group of real numbers
The additive group of real numbers, denoted by (R,+), is a topological group under addition with the standard Euclidean topology on R
The group operation is addition, and the inverse of an element x∈R is its additive inverse −x
The Euclidean topology on R is generated by open intervals (a,b), making it a Hausdorff space
(R,+) is a locally compact abelian group, which is a key example in and Fourier theory
General linear group
The general linear group, denoted by GL(n,R) or GL(n,C), is the group of all invertible n×n matrices over the real numbers R or complex numbers C, respectively
The group operation is matrix multiplication, and the identity element is the n×n identity matrix
GL(n,R) and GL(n,C) are topological groups with the topology induced by the Euclidean topology on the space of matrices
The general linear group is a non-compact and plays a crucial role in the theory of representations and the study of vector bundles in noncommutative geometry
Properties of topological groups
Topological groups possess several important properties that distinguish them from general topological spaces
These properties, such as continuity of group operations, the Hausdorff property, uniform space structure, and completeness, provide a rich framework for studying the interplay between algebra and topology
Understanding these properties is essential for working with topological groups in the context of noncommutative geometry and other areas of mathematics
Continuity of group operations
In a topological group G, the group operations of multiplication and inversion are continuous functions with respect to the topology on G
Specifically, the multiplication map m:G×G→G, defined by m(a,b)=ab, is continuous when G×G is equipped with the product topology
Similarly, the inversion map i:G→G, defined by i(a)=a−1, is continuous with respect to the topology on G
Continuity of group operations ensures that the algebraic structure of the group is preserved under continuous deformations and limits
Hausdorff property
A topological group G is Hausdorff (or separated) if for any two distinct elements a,b∈G, there exist disjoint open sets U,V⊂G such that a∈U and b∈V
The Hausdorff property is a separation axiom that ensures distinct elements can be separated by open sets
Many important topological groups, such as Lie groups and the additive group of real numbers, are Hausdorff
The Hausdorff property is crucial for studying the local structure of topological groups and their quotient spaces
Uniform space
A topological group G is naturally equipped with a uniform structure, which is a collection of subsets of G×G called entourages that satisfy certain axioms
The uniform structure on G is generated by the sets {(a,b)∈G×G:a−1b∈U}, where U is an open neighborhood of the identity element in G
The uniform structure allows for the study of uniform continuity, Cauchy sequences, and completeness in the context of topological groups
Uniform spaces provide a general framework for studying the notions of proximity and uniformity in topological spaces, and topological groups are a prime example of uniform spaces
Completeness
A topological group G is complete if every Cauchy sequence in G converges to an element of G
Completeness is a desirable property for topological groups, as it ensures that the group has no "gaps" or "missing limits"
Many important topological groups, such as the additive group of real numbers and the unitary group U(H) of a Hilbert space H, are complete
Completeness is particularly relevant in the study of C*-algebras associated to topological groups, as it relates to the existence of certain operator-valued integrals and the construction of crossed products
Subgroups and quotient groups
Subgroups and quotient groups are fundamental constructions in the theory of topological groups
These constructions allow for the study of smaller or simpler topological groups within a larger group and the formation of new topological groups by identifying elements based on a certain equivalence relation
Understanding subgroups and quotient groups is essential for analyzing the structure of topological groups and their applications in noncommutative geometry
Topological subgroups
A subgroup H of a topological group G is called a topological subgroup if H is a topological group with respect to the subspace topology inherited from G
For H to be a topological subgroup, it must satisfy two conditions:
H is a subgroup of G in the algebraic sense, i.e., closed under the group operations of G
H is a closed subset of G with respect to the topology on G
Examples of topological subgroups include:
The special orthogonal group SO(n) as a subgroup of the general linear group GL(n,R)
The integers Z as a subgroup of the additive group of real numbers (R,+)
Topological subgroups inherit many properties from the ambient group, such as the Hausdorff property and completeness
Quotient topological groups
Given a topological group G and a closed N, the quotient group G/N can be given a topology that makes it a topological group
The quotient topology on G/N is defined as the finest topology that makes the quotient map π:G→G/N, given by π(a)=aN, continuous
With this topology, G/N becomes a topological group, called the quotient topological group
Examples of quotient topological groups include:
The circle group S1 as the quotient of the additive group of real numbers R by the subgroup of integers Z
The projective special linear group PSL(n,C) as the quotient of the special linear group SL(n,C) by its center
Projection maps
The quotient map π:G→G/N, which sends an element a∈G to its coset aN in the quotient group G/N, is a continuous surjective homomorphism of topological groups
The quotient map π is open, meaning that it maps open sets in G to open sets in G/N
The kernel of the quotient map π is precisely the normal subgroup N, i.e., ker(π)=N
Projection maps play a crucial role in the study of topological groups, as they allow for the construction of new topological groups from existing ones and the analysis of their relationships
Isomorphism theorems
The isomorphism theorems for topological groups are analogous to the isomorphism theorems for algebraic groups, but they take into account the topological structure
The first isomorphism theorem states that if f:G→H is a continuous surjective homomorphism of topological groups, then the quotient group G/ker(f) is isomorphic to H as a topological group
The second isomorphism theorem states that if H and N are closed subgroups of a topological group G with N being normal, then the quotient group H/(H∩N) is isomorphic to the subgroup HN/N of the quotient group G/N
The third isomorphism theorem states that if N and M are closed normal subgroups of a topological group G with N⊂M, then the quotient group (G/N)/(M/N) is isomorphic to the quotient group G/M
These isomorphism theorems provide a powerful tool for understanding the relationships between topological groups and their subgroups and quotients
Compact topological groups
Compact topological groups are a special class of topological groups that exhibit nice properties due to their compactness
The study of compact topological groups is a rich area of research, with connections to various branches of mathematics, including noncommutative geometry
Compact topological groups have a well-behaved structure and admit a unique invariant measure called the Haar measure, which plays a crucial role in harmonic analysis and
Definition and examples
A topological group G is called compact if it is compact as a topological space, i.e., every open cover of G has a finite subcover
Equivalently, a topological group G is compact if and only if it is Hausdorff and every sequence in G has a convergent subsequence
Examples of compact topological groups include:
Finite groups with the discrete topology
The circle group S1 with the topology induced by the Euclidean topology on the complex plane
The special orthogonal group SO(n) with the topology induced by the Euclidean topology on the space of matrices
The unitary group U(n) with the topology induced by the operator norm topology on the space of matrices
Compact topological groups are ubiquitous in mathematics and have applications in various areas, including topology, analysis, and geometry
Properties of compact groups
Compact topological groups possess several desirable properties that distinguish them from general topological groups:
Every compact topological group is Hausdorff and regular
Every compact subgroup of a Hausdorff topological group is closed
The product of compact topological groups is compact with respect to the product topology
Continuous homomorphisms between compact topological groups are automatically uniformly continuous
Compact topological groups also have a rich representation theory, as they admit a complete classification of their irreducible unitary representations
Haar measure on compact groups
Every compact topological group G admits a unique left-invariant regular Borel probability measure, called the Haar measure
The Haar measure μ on G satisfies the following properties:
μ(G)=1
μ(gE)=μ(E) for every Borel set E⊂G and every g∈G, where gE={ga:a∈E}
μ is outer regular: μ(E)=inf{μ(U):E⊂U,U open} for every Borel set E⊂G
μ is inner regular: μ(E)=sup{μ(K):K⊂E,K compact} for every Borel set E⊂G
The existence and uniqueness of the Haar measure on compact groups is a powerful tool in harmonic analysis and the study of representations of topological groups
Peter-Weyl theorem
The is a fundamental result in the representation theory of compact topological groups
It states that for a compact topological group G, the matrix coefficients of irreducible unitary representations of G form a complete orthonormal system in the Hilbert space L2(G) of square-integrable functions on G with respect to the Haar measure
Consequently, every function in L2(G) can be approximated by linear combinations of matrix coefficients of irreducible unitary representations
The Peter-Weyl theorem provides a powerful tool for studying the structure of compact topological groups and their representations, and it has applications in various areas of mathematics, including noncommutative geometry and quantum group theory
Representations of topological groups
Representations of topological groups are a fundamental tool in the study of these groups and their applications in various areas of mathematics
A representation of a topological group G is a continuous homomorphism from G to the group of bounded linear operators on a topological vector space
Representations allow for the study of topological groups through their actions on vector spaces, which can provide valuable insights into the structure and properties of the group
Continuous representations
A continuous representation of a topological group G on a topological vector space V is a continuous π:G→GL(V), where GL(V) is the group of bounded linear operators on V with the strong operator topology
Continuity of the representation means that for each $v \
Key Terms to Review (18)
Birkhoff's Theorem: Birkhoff's Theorem states that every continuous function defined on a compact convex set can be represented as the supremum of a family of affine functions. This theorem connects with various aspects of analysis and topology, especially in the context of topological groups, as it emphasizes the significance of convexity and continuity in understanding function spaces and their properties.
Closed Subgroup: A closed subgroup is a subgroup of a topological group that is also a closed set in the topology of that group. This means that if you take any sequence of elements from the subgroup that converges to a limit in the topological group, that limit must also be in the subgroup. Closed subgroups play an essential role in understanding the structure and properties of topological groups, often helping to establish important results related to continuity and compactness.
Compact group: A compact group is a topological group that is both compact as a topological space and a group under the operation of multiplication. This means that every open cover of the group has a finite subcover, ensuring that the group is closed and bounded. Compact groups have strong implications in various areas of mathematics, especially in analysis and geometry, where they often exhibit nice properties like the existence of invariant measures.
Connectedness: Connectedness refers to a property of a topological space where the space cannot be divided into two disjoint, nonempty open sets. Essentially, a space is connected if there are no gaps or separations within it. This concept plays a crucial role in understanding the structure of spaces and their continuous functions, especially when discussing homeomorphisms and the behavior of topological groups.
Continuous Group Action: A continuous group action is a way for a topological group to act on a topological space such that the group elements correspond to homeomorphisms of that space. This means that when you apply a group element to a point in the space, the result varies continuously with the point, respecting the topology of the space. This concept is essential in understanding how symmetries and transformations can be described within the framework of topology, connecting algebraic structures to topological properties.
Group Action: A group action is a formal way in which a group interacts with a set, where each element of the group corresponds to a function that transforms the set while preserving its structure. Essentially, it defines how the group's elements can 'act' on the elements of a set, providing a powerful framework for understanding symmetries and transformations in various mathematical contexts. This concept is key in connecting groups with geometric and topological structures, allowing us to study how these actions can influence the properties of spaces.
Group homomorphism: A group homomorphism is a function between two groups that preserves the group operation, meaning that if you take two elements from the first group, their operation in that group corresponds to the operation of their images in the second group. This concept is crucial for understanding how different groups relate to each other and allows for the study of their structures and properties. Group homomorphisms help in connecting algebraic structures through mappings that maintain essential properties.
Harmonic Analysis: Harmonic analysis is a branch of mathematics that studies functions or signals through the representation of their constituent frequencies. This approach allows for a deep understanding of various mathematical objects by analyzing their structure in terms of harmonic components, which are often represented using Fourier series and transforms. In the context of topological groups, harmonic analysis examines the interplay between group structure and functional properties, enabling the exploration of symmetry and periodicity in mathematical models.
Henri Cartan: Henri Cartan was a prominent French mathematician known for his significant contributions to algebraic topology, sheaf theory, and homological algebra. His work laid important foundations that intersect with various areas of mathematics, particularly in understanding the structure of topological groups and the development of cyclic cohomology, which has applications in noncommutative geometry.
Jean-Pierre Serre: Jean-Pierre Serre is a prominent French mathematician known for his contributions to various fields, including algebraic geometry, topology, and number theory. His work has significantly influenced the development of modern mathematics, particularly in the context of topological groups, where his insights into homotopy and cohomology theories have laid foundational groundwork.
Lie Group: A Lie group is a group that is also a differentiable manifold, which means that its elements can be smoothly transformed and have a smooth structure. This combination allows for the use of calculus in studying the group's properties, making it essential in areas like geometry and physics, especially when dealing with continuous symmetries and transformations.
Local Compactness: Local compactness refers to a property of a topological space where every point has a neighborhood that is compact. This concept is essential because it helps in understanding how spaces can behave similarly to compact spaces without being fully compact themselves. Local compactness is particularly significant when discussing properties like continuity and convergence in various mathematical contexts, as it allows for the extension of compactness properties to more general settings.
Normal subgroup: A normal subgroup is a subgroup that remains invariant under conjugation by elements of the group, meaning if 'N' is a normal subgroup of 'G', then for every element 'g' in 'G' and every element 'n' in 'N', the element 'gng^{-1}' is also in 'N'. This property is essential because it allows for the construction of quotient groups, which are fundamental in understanding the structure of groups and their relationships with each other.
Orbit space: An orbit space is the set of equivalence classes formed by a group action on a topological space, where each point in the space is identified with its orbit under the action of the group. This concept helps in understanding how symmetries and transformations affect the structure of a space, allowing for a more simplified view by collapsing orbits into single points. By studying orbit spaces, one can analyze properties like quotient topology and the behavior of continuous functions on these spaces.
Peter-Weyl Theorem: The Peter-Weyl Theorem is a fundamental result in the representation theory of compact topological groups, which states that the continuous functions on a compact group can be decomposed into a direct sum of matrix representations. This theorem connects the algebraic structure of groups to harmonic analysis and highlights how the representations can be used to analyze functions on these groups.
Representation Theory: Representation theory is the study of how algebraic structures, such as groups and algebras, can be realized as linear transformations of vector spaces. This branch of mathematics connects abstract algebra with linear algebra and has significant applications in various areas, including physics and geometry.
Topological Group: A topological group is a mathematical structure that combines the concepts of group theory and topology, where a group is equipped with a topology that makes the group operations continuous. This means that both the multiplication operation and the inverse operation are continuous functions with respect to the topology, allowing for the exploration of algebraic structures within a topological context.
Topological isomorphism: Topological isomorphism is a mapping between two topological spaces that preserves the structures of both spaces, meaning it maintains the properties of open sets. This concept is crucial in understanding the relationships between different topological groups, as it allows one to identify spaces that are fundamentally the same in terms of their topological properties despite potentially differing in their underlying sets or algebraic structures.