Representation theory of Lie groups and Lie algebras - Lec 17 - Frederic Schuller

Frederic Schuller

21 Sept 201592:38

EducationalLearning

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TLDRThe video script delves into the concept of representations of Lie algebras and Lie groups, a topic of significant interest in physics. It begins by defining a Lie algebra and its representation through a linear map into the endomorphisms of a vector space, ensuring compatibility with the algebra's bracket. The lecture introduces the representation space and explains how it acts on vectors to produce new vectors. It provides examples, including the SL(2,C) algebra and the rotation algebra SO(3), demonstrating how these algebras can be represented on different spaces like R^3 and C^2. The script also discusses the adjoint representation and its importance in classifying semi-simple and simple algebras. Furthermore, it explores reducibility and irreducibility of representations, the concept of a Casimir operator, and its significance in quantum mechanics and particle physics. The lecture concludes with a look at Lie group representations, the adjoint representation of a Lie group, and a teaser for the next topic: the exponential map, a vital tool for transitioning from the algebra level to the group level in the study of Lie groups.

Takeaways

📚 The concept of representations is central to understanding Lie algebras and Lie groups, often defining them in terms of their representations in physics.
🔍 A representation of a Lie algebra is a linear map from the algebra into the endomorphisms of a vector space, compatible with the algebra's bracket operation.
📏 The representation space is the vector space on which the Lie algebra acts via the representation, and it is crucial for understanding how elements of the algebra can act on vectors.
🌐 The importance of the Killing form in defining the structure constants of the Lie algebra and its role in constructing the Casimir operator for a given representation.
🧩 The adjoint representation is a special case where the Lie algebra is represented on its own vector space, providing a way to study the algebra's structure.
⚙️ The definition of a reducible representation, which can be decomposed into simpler representations, and an irreducible representation, which cannot be broken down further.
🤔 The concept of a faithful representation, which is injective and ensures that different algebra elements map to distinct endomorphisms.
🔑 The Casimir operator, unique to each representation, commutes with all elements of the algebra and is a constant multiple of the identity on the representation space if the representation is irreducible.
🧮 The calculation of the Casimir operator involves the trace of the product of representation matrices and is essential in quantum mechanics for determining particle properties.
🏞️ The use of the exponential map to transition from the Lie algebra to the Lie group, a topic to be explored further in subsequent lectures.
❓ The open question of whether every Lie algebra has a non-trivial representation, and the significance of this inquiry in the study of algebraic structures.

Q & A

What is a representation of a Lie algebra?
-A representation of a Lie algebra is a linear map that takes elements from the algebra and maps them into the endomorphisms of a vector space V, such that the map is compatible with the Lie bracket of the algebra.
Why are representations of Lie groups and algebras important in physics?
-Representations are important in physics because they allow us to understand the action of Lie groups and algebras on various spaces. They are often used to describe symmetries in physical systems and are fundamental in quantum mechanics and particle physics.
What is the difference between a reducible and an irreducible representation?
-A representation is called reducible if there exists a non-trivial subspace of the representation space that is invariant under the action of the Lie algebra elements. If no such subspace exists, the representation is called irreducible.
What is the adjoint representation of a Lie algebra?
-The adjoint representation is a map that takes each element of the Lie algebra and maps it to an endomorphism of the algebra itself, using the Lie bracket operation. It is a way to represent the Lie algebra on its own vector space.
What is a Casimir operator?
-A Casimir operator is an element of the endomorphism algebra associated with a given representation of a Lie algebra. It has the property of commuting with all elements of the representation, and for irreducible representations, it is a scalar multiple of the identity.
How is the dimension of a Lie algebra related to the Casimir operator in an irreducible representation?
-In an irreducible representation, the eigenvalue of the Casimir operator is given by the dimension of the Lie algebra divided by the dimension of the representation space.
What is the significance of the Killing form in the context of Lie algebras?
-The Killing form is a bilinear form on the Lie algebra that can be used to define the structure constants of the algebra. It is also used in the construction of the Casimir operator and helps in determining whether a Lie algebra is semisimple.
What is the role of the exponential map in the study of Lie groups?
-The exponential map is a function that maps elements of the Lie algebra to the Lie group, allowing us to move from the algebraic structure to the group structure. It is particularly useful in the study of one-parameter subgroups and in the integration of differential equations on Lie groups.
What is the adjoint representation of a Lie group?
-The adjoint representation of a Lie group is a map that takes a group element and maps it to an invertible linear transformation (an element of the general linear group) on the Lie algebra of the group, preserving the group operation.
How does the script differentiate between the adjoint representation of a Lie algebra and a Lie group?
-The script differentiates by using the lowercase 'ad' for the adjoint representation on the Lie algebra, which is a map from the algebra to its endomorphisms, and the capital 'Ad' for the adjoint representation on the Lie group, which is a map from the group to the general linear group over the algebra.
What is the significance of the trivial representation in the context of Lie algebras?
-The trivial representation is a representation where every element of the Lie algebra is mapped to the zero endomorphism on the representation space. It is significant because it always exists for any Lie algebra and serves as a baseline or reference point for more complex representations.

Outlines

00:00

😀 Introduction to Representations of Lie Algebras and Groups

The lecture begins with an introduction to the concept of representations of Lie algebras and Lie groups, emphasizing their importance in physics. The speaker explains that Lie groups are defined as smooth manifolds with a compatible group law, while Lie algebras are defined as left invariant vector fields. The section 4.6 is introduced, focusing on linear representations of these structures. The key concept of a representation as a linear map from the Lie algebra to the endomorphisms of a vector space is presented, along with the requirement of compatibility with the Lie bracket.

05:04

📚 Detailed Explanation of Lie Algebra Representations

The paragraph delves into the specifics of Lie algebra representations, using the example of the SL(2,C) algebra. It discusses how representations act on a vector space and the importance of the compatibility condition with the algebra's bracket. The representation space is defined, and the idea of acting on this space using endomorphisms is explored. The paragraph also provides an example of a representation for SL(2,C) using 2x2 matrices.

10:05

🔍 Compatibility Checks and the Concept of Representations

This section discusses the need to check the compatibility of the commutator with the representation map. It uses the basis representations of the Lie algebra to demonstrate this process. The paragraph also touches on the concept of the adjoint representation and the idea that every Lie algebra has a trivial representation. It concludes with the question of whether every Lie algebra has a non-trivial representation and introduces the concept of a faithful representation.

15:07

🧩 Exploring Reducible and Irreducible Representations

The lecture continues by defining reducible and irreducible representations. It explains that a representation is reducible if there exists a non-trivial subspace that is invariant under the action of the Lie algebra elements. If no such subspace exists, the representation is deemed irreducible. The paragraph provides examples to illustrate these concepts and discusses the significance of irreducible representations in particle physics.

20:09

🤔 The Role of the Adjoint Representation

The adjoint representation is introduced as a specific type of representation where the Lie algebra is mapped linearly into its own endomorphism space. The paragraph discusses the construction of the adjoint representation and its significance in the classification of semi-simple and simple Lie algebras. It also explores the concept of the Casimir operator in the context of representations.

25:10

📏 Properties of the Casimir Operator

The paragraph focuses on the properties of the Casimir operator, explaining that it commutes with all elements of the Lie algebra when acting on the representation space. It discusses the Schur's lemma in the context of irreducible representations and how the Casimir operator is related to the dimension of the Lie algebra and the representation space.

30:11

🤓 Calculation of the Casimir Operator for Specific Representations

This section provides a detailed calculation of the Casimir operator for specific representations, including the vector representation and the spin representation. It highlights the discrepancy between the mathematical definition and the物理 (physical) definition used in quantum mechanics, noting the factor of two difference and its implications.

35:12

🌐 Representations of Lie Groups

The lecture concludes with a discussion on representations of Lie groups, distinguishing them from Lie algebra representations. It introduces the concept of the adjoint representation for Lie groups and explains how it can be constructed. The paragraph also touches on the importance of the determinant condition for the general linear group and the compatibility of the group law with the representation.

40:13

🔬 Examples of Lie Group Representations

The paragraph provides examples of Lie group representations, specifically for the SO(2) group. It explains how to represent elements of the group using matrices and how the group operation corresponds to the addition of angles in the representation space. The concept of the adjoint representation for Lie groups is also introduced, highlighting its properties and construction.

45:18

🚀 Conclusion and Preview of Upcoming Topics

The lecture concludes with a summary of the topics covered, including the representations of finite-dimensional Lie algebras and Lie groups. It previews the next topic, which will be the exponential map and its role in transitioning from the Lie algebra level back to the Lie group level, a concept with significant implications in differential geometry.

Mindmap

Keywords

💡Representation

In the context of the video, a representation refers to a mathematical structure that encapsulates the properties of Lie algebras and Lie groups. Specifically, a representation of a Lie algebra is a linear map that preserves the algebraic structure, including the bracket operation. It is a fundamental concept in the study of Lie theory and is used to understand the symmetries in physics. The video discusses linear representations of Lie algebras and Lie groups, emphasizing their importance in physics and mathematics.

💡Lie Algebra

A Lie algebra, denoted as L, is a vector space equipped with a binary operation called the Lie bracket that satisfies certain properties, including bilinearity, antisymmetry, and the Jacobi identity. Lie algebras are closely related to Lie groups and are used to study the structure of symmetries. In the video, the Lie algebra is defined in terms of left-invariant vector fields and its representations are explored.

💡Endomorphism

An endomorphism is a function that maps a vector space to itself. In the context of the video, the representation of a Lie algebra is a linear map into the endomorphisms of some vector space V. This means that each element of the Lie algebra is associated with a linear transformation of V, allowing for the study of the algebra's action on the vector space.

💡Vector Space

A vector space is a collection of objects (called vectors) that, along with scalar multiplication and vector addition, satisfy certain axioms. In the video, it is mentioned that the Lie algebra L is a vector space and that the representation map goes into the endomorphisms of some other vector space V, which could be real or complex.

💡Bilinear Map

A bilinear map is a function that is linear in each of two variables while the other is held fixed. In the video, the Lie bracket is described as a bilinear map from the product of the vector space V with itself to the endomorphisms of V, which is essential for the compatibility with the Lie algebra structure.

💡Lie Group

A Lie group is a group that is also a differentiable manifold, with the property that the group operations are differentiable. Lie groups are used to study continuous symmetries. The video touches on the representation of Lie groups and how they are related to Lie algebras through the concept of the adjoint representation.

💡Adjoint Representation

The adjoint representation, denoted as ad, is a specific type of representation of a Lie algebra or Lie group where the algebra or group acts on itself via the Lie bracket operation. In the video, it is discussed that the adjoint representation of a Lie group is a map that takes the group into the general linear group over its Lie algebra.

💡Casimir Operator

The Casimir operator is a special element of a universal enveloping algebra that commutes with all elements of the algebra. In the context of the video, it is constructed for a given representation of a Lie algebra and has the property of commuting with all elements of the representation. It is used in physics to label particles by their spin.

💡Irreducible Representation

An irreducible representation is a representation that cannot be broken down into smaller, simpler representations. It is a fundamental building block in the study of representations. In the video, it is mentioned that elementary particles in quantum mechanics are often associated with irreducible representations of the symmetry algebra.

💡Reducible Representation

A reducible representation is one that can be decomposed into smaller, simpler representations. It is called reducible because it can be broken down into invariant subspaces. In the video, the concept is introduced to illustrate the difference between complex representations that can be simplified and those that are fundamental and cannot be simplified further.

💡Killing Form

The Killing form is a bilinear form on a Lie algebra that is used to study the structure of the algebra. It is defined using the Killing metric and is related to the algebra's Cartan matrix. In the video, the Killing form is discussed in relation to the construction of the Casimir operator and the classification of representations.

Highlights

Introduction to the importance of representations in understanding Lie algebras and Lie groups, particularly in physics.

Definition of a Lie algebra representation as a linear map into the endomorphisms of a vector space.

Explanation of the compatibility condition for the Lie bracket with respect to the representation map.

Discussion on the representation space and its role in acting on vectors to produce new vectors.

Introduction of the concept of a linear representation through endomorphisms on a vector space.

Example of the SL(2,C) algebra and its basis vectors, highlighting the structure derived from left invariant vector fields.

Construction of a representation for SL(2,C) using 2x2 matrices and checking the compatibility condition.

Introduction of the rotation algebra SO(3) and its basis, with a focus on the structure constants epsilonijk.

Finding a representation for the three-dimensional rotation algebra that acts on R^3.

Mention of the spin representation of SO(3) and its action on two-component complex objects.

Discussion of the trivial representation of any Lie algebra and its significance in physics.

Definition and importance of the adjoint representation in understanding the structure of Lie algebras.

Explanation of reducible and irreducible representations and their significance in particle physics.

Introduction of the concept of a faithful representation and its injectivity property.

Discussion of the Casimir operator, its construction for a given representation, and its significance in physics.

Proof that the Casimir operator of an irreducible representation is a constant multiple of the identity.

Calculation of the Casimir operator for the vector and spin representations of SO(3).

Highlighting the mismatch between the physics and mathematics definitions of the Casimir operator and its resolution.

Introduction to the representation of Lie groups, their definition, and the distinction from Lie algebra representations.

Construction of the adjoint representation for Lie groups and its significance in the study of Lie group structure.

Transcripts

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Lie Algebras Lie Groups Representations Physics Mathematics Endomorphisms Vector Fields Manifold Group Law Differential Geometry