Advanced Topics in Mathematics: Representation
Theory, Lie Groups, and Applications in Physics
and Geometry
Mathematical Researcher
November 21, 2024
1 Representation Theory: An Introduction
Representation theory is a branch of mathematics that studies abstract algebraic
structures by representing their elements as linear transformations of vector
spaces. The key objects of study are groups, algebras, and other structures,
with the goal of understanding their symmetries by analyzing how they act on
vector spaces.
In its simplest form, a representation of a group G is a homomorphism from
G to the general linear group GL(V ), where V is a vector space. This means
that each element of G is associated with an invertible linear transformation of
V , and group operations correspond to composition of these transformations.
A representation of a group G on a vector space V is a homomorphism
ρ : G → GL(V ), where GL(V ) is the group of invertible linear transformations
on V .
Representations provide a powerful tool to study groups by turning abstract
algebraic objects into concrete linear operations. Understanding the structure
of representations helps uncover the symmetry properties of the group, which
can be crucial in both pure mathematics and theoretical physics.
1.1 Irreducible Representations
An important concept in representation theory is that of irreducible represen-
tations. A representation ρ is said to be irreducible if there is no nontrivial
subspace W ⊆ V such that ρ(g)W ⊆ W for all g ∈ G. In other words, V cannot
be decomposed into smaller invariant subspaces under the action of the group.
A representation ρ : G → GL(V ) is irreducible if there are no proper non-zero
subspaces W ⊆ V such that ρ(g)W ⊆ W for all g ∈ G.
The classification of irreducible representations is a central problem in rep-
resentation theory, as they form the ”building blocks” of more general represen-
tations.
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1.2 The Peter-Weyl Theorem
In the context of compact Lie groups, the Peter-Weyl theorem plays a crucial
role in the understanding of representations. It states that the regular repre-
sentation of a compact Lie group decomposes into a direct sum of irreducible
representations, each of which is finite-dimensional.
[Peter-Weyl] Let G be a compact Lie group. The regular representation of G
on the space L2 (G) decomposes into a direct sum of irreducible representations,
with each irreducible representation appearing finitely many times.
This theorem is fundamental in understanding the harmonic analysis on
compact Lie groups and provides a powerful tool for decomposing functions on
groups into more manageable components.
2 Lie Groups and Lie Algebras
Lie groups are smooth manifolds that also have a group structure, and they play
a central role in geometry and theoretical physics, especially in the study of sym-
metries. The corresponding Lie algebra of a Lie group captures the infinitesimal
structure of the group and provides a way to study its local properties.
2.1 Basic Definitions of Lie Groups and Lie Algebras
A Lie group is a group that is also a smooth manifold, such that the group
operations (multiplication and inversion) are smooth maps. The Lie algebra
associated with a Lie group G is the tangent space at the identity element,
equipped with a Lie bracket.
A Lie group is a group G that is also a smooth manifold, with the group
operations µ : G × G → G and ι : G → G being smooth maps.
The Lie algebra g of a Lie group G is the tangent space at the identity
element e ∈ G:
g = Te G.
The Lie algebra is equipped with a binary operation called the Lie bracket,
which satisfies the Jacobi identity:
[x, y] = −[y, x] and [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z ∈ g.
2.2 Lie Groups in Physics
Lie groups and their associated Lie algebras are central in the study of symme-
tries in physics. For example, the group SO(3), the special orthogonal group
in three dimensions, describes rotational symmetries, and its Lie algebra so(3)
describes infinitesimal rotations.
Lie groups also appear in quantum mechanics and particle physics, where
they describe the symmetry properties of physical systems. For instance, the
group SU(2), the special unitary group in two dimensions, is important in the
study of quantum spin and gauge theories.
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2.3 Representations of Lie Algebras
The representation theory of Lie algebras is the study of how the Lie algebra g
can act on vector spaces. A representation of a Lie algebra g is a homomorphism
from g to the space of endomorphisms of a vector space V , i.e., a map ρ : g →
End(V ).
A representation of a Lie algebra g on a vector space V is a linear map
ρ : g → End(V ) such that:
ρ([x, y]) = ρ(x)ρ(y) − ρ(y)ρ(x) for all x, y ∈ g.
Lie algebra representations play an important role in understanding the
structure of physical systems and quantum fields, as they allow one to study how
symmetries (represented by Lie groups) manifest in the behavior of particles and
fields.
3 Applications in Geometry and Physics
Lie groups and their representations have widespread applications in both geom-
etry and theoretical physics. In geometry, Lie groups describe the symmetries
of geometric objects and spaces. In physics, they provide the framework for
understanding symmetries in quantum mechanics, particle physics, and general
relativity.
3.1 Symmetry Groups in Geometry
In differential geometry, the symmetry group of a smooth manifold M is the Lie
group of diffeomorphisms of M , and its Lie algebra captures the infinitesimal
transformations of the manifold. The study of Lie groups and their actions
on manifolds is essential for understanding geometric structures such as fiber
bundles, homogeneous spaces, and differential equations.
3.2 Gauge Theory and Quantum Mechanics
In theoretical physics, Lie groups are used to describe the symmetry properties
of gauge theories, which are the foundation of the Standard Model of particle
physics. The gauge group SU(3) × SU(2) × U(1) describes the symmetries of
the strong, weak, and electromagnetic forces, respectively.
In quantum mechanics, the representation theory of Lie groups allows one
to classify the states of quantum systems, especially when the system possesses
symmetry. For example, the angular momentum states of particles are classified
according to the representations of the rotation group SO(3), and the spin states
are classified according to the representations of SU(2).
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4 Conclusion
Representation theory and the study of Lie groups are essential areas of mod-
ern mathematics that connect algebra, geometry, and physics. They provide
the language for understanding symmetries and structure in both mathematical
and physical systems. From the classification of finite-dimensional representa-
tions to the use of Lie groups in gauge theories, the concepts explored in this
field have far-reaching implications in both pure mathematics and the natural
sciences. The study of these structures continues to drive progress in areas such
as quantum mechanics, string theory, and the geometry of high-dimensional
spaces.
