Dynkin diagrams from Lie algebras, and vice versa - Lec 16 - Frederic Schuller

Frederic Schuller

21 Sept 2015100:32

EducationalLearning

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TLDRThe video script delves into the intricate details of Lie algebra and Lie groups, specifically focusing on the special linear group SL(2, C) and its associated Lie algebra. The presenter guides the audience through the construction of the tangent space at the identity element of the group, the creation of left-invariant vector fields, and the derivation of the Lie bracket structure. The exploration continues with the calculation of structure constants and the Killing form, leading to the determination of the algebra's semi-simplicity. The lecture further investigates the simplicity of the algebra by examining the absence of non-trivial ideals. The concept of roots and fundamental roots is introduced, with an example demonstrating how to reconstruct a Lie algebra from its Dynkin diagram. The script is a rich resource for those interested in advanced mathematical physics, providing a deep understanding of Lie theory and its applications.

Takeaways

📚 The study focuses on the special linear group SL(2,C) and its Lie algebra, specifically constructing the tangent space at the identity element and left invariant vector fields.
🔍 The tangent space at the identity is equipped with a Lie bracket, which is isomorphic to the set of left invariant vector fields on the Lie group.
🧮 The Lie bracket is defined by its action on functions, which requires calculating differential geometric derivatives and using the product rule for differentiation.
🎯 By evaluating the Lie bracket at the identity, a vector at the identity is obtained, which is crucial for defining the Lie bracket on the tangent space.
📐 The calculation of the Lie bracket for the SL(2,C) algebra involves evaluating specific brackets and using the anti-symmetry and Jacobi identity properties.
🔑 The structure constants of the Lie algebra are determined by the commutation relations of the basis vectors and are essential for defining the algebra's properties.
🔍 The Killing form is used to check for semi-simplicity of the Lie algebra, with non-degeneracy indicating semi-simplicity.
🔬 The Lie algebra SL(2,C) is confirmed to be simple, not just semi-simple, as it contains no non-trivial ideals.
🌱 The concept of roots and fundamental roots is introduced, with the roots being elements of the dual space of the Cartan subalgebra.
🌟 The Weyl group transformations act on the roots and are used to generate the entire root space from the fundamental roots.
📈 The Cartan matrix and the Dynkin diagram are derived from the root space and used to classify the Lie algebra, in this case, identifying it as of type A1.

Q & A

What is the special linear group SL(2, C) and why is it significant?
-The special linear group SL(2, C) is a group consisting of 2x2 complex matrices with determinant 1. It is significant because it is a fundamental example in the study of Lie groups and Lie algebras, which are key concepts in modern geometry and theoretical physics.
What is a left invariant vector field?
-A left invariant vector field is a vector field on a Lie group that remains unchanged under left translation by any element of the group. In other words, if you translate the vector field by a group element, the new vector field is the same as the original one.
How does one construct a Lie algebra from the tangent space at the identity element of a Lie group?
-One constructs a Lie algebra from the tangent space at the identity element of a Lie group by equipping the tangent space with a Lie bracket operation. This operation is derived from the differential geometric bracket and satisfies properties such as bilinearity, antisymmetry, and the Jacobi identity.
What are the structure constants in the context of a Lie algebra?
-Structure constants are coefficients that appear when expanding the Lie bracket of two basis elements of a Lie algebra in terms of the basis. They are important because they encode the algebraic structure of the Lie algebra and are dependent on the choice of basis.
What is the Killing form and how is it used in the study of Lie algebras?
-The Killing form is a bilinear form on a Lie algebra that is constructed using the Killing form on the dual of the Cartan subalgebra. It is used to determine the semi-simplicity of a Lie algebra. If the Killing form is non-degenerate, the Lie algebra is semisimple.
What is the significance of the semisimplicity of a Lie algebra?
-Semisimplicity of a Lie algebra is significant because it implies that the algebra can be expressed as a direct sum of simple Lie algebras. Simple Lie algebras are those that do not have nontrivial ideals, making them the building blocks for more complex algebraic structures.
How does the Killing form help in determining the semisimplicity of a Lie algebra?
-The Killing form helps in determining the semisimplicity of a Lie algebra by providing a criterion known as Cartan's criterion. If the Killing form is non-degenerate (i.e., its determinant is nonzero), it indicates that the Lie algebra is semisimple.
What is the relationship between the roots of a Lie algebra and the structure of its root space?
-The roots of a Lie algebra are elements of the dual space of the Cartan subalgebra that act as eigenvalues for the adjoint action on the root space. The structure of the root space is determined by the roots, and each root corresponds to an eigenvector in the root space.
What are the fundamental roots and how do they relate to the root space of a Lie algebra?
-Fundamental roots are a subset of the roots of a Lie algebra that are linearly independent and generate the entire root space when acted upon by the Weyl group. They provide a basis for the root space and are crucial for understanding the structure of the Lie algebra.
How can one reconstruct a Lie algebra from its Dynkin diagram?
-One can reconstruct a Lie algebra from its Dynkin diagram by identifying the fundamental roots, which correspond to the nodes of the diagram. The edges between nodes indicate the relationships between the roots, and the structure of the diagram guides the reconstruction of the Lie bracket relations in the algebra.
What is the significance of the Cartan matrix in the study of Lie algebras?
-The Cartan matrix is a square matrix that encodes the structure constants of a Lie algebra with respect to a basis of its Cartan subalgebra. It is used to determine the root system and the structure of the Lie algebra, particularly in the classification of simple Lie algebras.

Outlines

00:00

📚 Introduction to SL(2,C) and its Lie Algebra

The video begins with an introduction to the special linear group SL(2,C) and its associated Lie algebra. The focus is on constructing the thinking diagram for the group and analyzing the tangent space at the identity element. The lecture builds upon the theory from a previous lecture, aiming to apply it to the example of SL(2,C). The concept of left invariant vector fields and their construction from coordinate-induced tangent vectors is discussed, leading to the understanding that the tangent space at the identity can be equipped with a Lie bracket, making it a Lie algebra.

05:02

🔍 Calculation of Lie Brackets

The paragraph delves into the calculation of Lie brackets for the tangent space at the identity of the manifold. The process involves pushing forward the tangent vectors to a point in the manifold and then applying the differential geometric bracket. The calculation is executed by evaluating the action on an arbitrary function, with a particular focus on the derivatives at the identity. The goal is to define the Lie bracket on the tangent space and to understand its structure in relation to the left-invariant vector fields.

10:05

🧮 Evaluation of Brackets and Vector Fields

This section focuses on the evaluation of specific brackets and the resulting vector fields. The calculation is meticulous, involving the action of vector fields on functions and the application of the product rule. The expectation is that the Lie bracket of two vector fields will result in another vector field, which is confirmed through the calculations. The process also involves dropping second derivative terms to simplify the calculation, with the assurance that they will cancel out.

15:06

🔗 Defining the Lie Bracket and Structure Constants

The paragraph discusses the definition of the Lie bracket in the tangent space at the identity of the SL(2,C) group. The results of the calculations are used to define the Lie bracket, and the structure constants are identified. The importance of the basis in determining the structure constants is highlighted. The Lie algebra of SL(2,C) is characterized by these structure constants, which are complex numbers in this context.

20:06

📏 Calculation of the Killing Form

The Killing form is introduced as an important object for studying the Lie algebra. The calculation of its components is shown, with the understanding that the Killing form is symmetric and can be used to determine the semi-simplicity of the Lie algebra. The calculation involves summing over structure coefficients and taking into account the anti-symmetry of the indices. The Killing form is found to be non-degenerate, confirming the semi-simplicity of the SL(2,C) algebra.

25:08

🔍 Analysis of the Lie Algebra's Decomposability

The paragraph explores whether the Lie algebra SL(2,C) is simple or can be decomposed into simpler algebras. A simple Lie algebra is defined as one that contains no non-trivial ideals. By considering possible restrictions on linear combinations of basis vectors, it is shown that the only possible ideals are the trivial ones, indicating that SL(2,C) is indeed a simple Lie algebra and not just semi-simple.

30:11

🌱 Roots and Fundamental Roots of the Lie Algebra

The concept of roots and fundamental roots of the Lie algebra is introduced. The roots are elements of the dual space of the Cartan subalgebra, and the fundamental roots form a linearly independent subset of the root space. The paragraph explains how the entire root space can be generated from the fundamental roots using the Weyl group. The choice of fundamental roots is not unique, and the example of SL(2,C) is used to illustrate this concept.

35:13

📈 Weyl Group Transformations and Root Space

This section discusses the Weyl group transformations and their action on the root space. The Weyl group is generated by reflections through the hyperplanes orthogonal to the roots. The transformations are shown to be linear in the roots, and the action of the Weyl group on the fundamental roots is used to generate the entire root space. The paragraph also touches on the concept of the Cartan matrix and its relation to the Killing form.

40:14

🏗️ Reconstruction of Lie Algebra from the Dynkin Diagram

The final part of the script outlines the process of reconstructing a Lie algebra from its Dynkin diagram. Starting with a given Dynkin diagram, the fundamental roots are identified, and the Cartan matrix is constructed. The angle between the roots is calculated, and the root space is generated using the Weyl group transformations. The commutation relations of the Lie algebra are then derived from the root diagram, and the entire Lie bracket structure is recovered. The process demonstrates that the Dynkin diagram contains all the information needed to define a simple Lie algebra.

Mindmap

Keywords

💡Special Linear Group (SL(2,C))

The Special Linear Group (SL(2,C)) is a group consisting of 2x2 complex matrices with determinant one. It is a fundamental object in the study of Lie groups and Lie algebras. In the video, the presenter delves into the structure of this group and its associated Lie algebra, which is central to understanding the concepts discussed.

💡Lie Algebra

A Lie algebra is a mathematical structure that generalizes the idea of Lie groups by focusing on the algebraic properties of the group's elements. It is characterized by a binary operation called the Lie bracket. In the script, the Lie algebra of SL(2,C) is explored through its tangent space and structure constants.

💡Tangent Space

The tangent space at the identity element of a Lie group is a vector space that can be equipped with a Lie algebra structure. It is the space of all tangent vectors at that point. The script discusses the construction of the tangent space for SL(2,C) and how it relates to the Lie algebra.

💡

💡Left Invariant Vector Fields

Left invariant vector fields are vector fields on a Lie group that are invariant under left multiplication by group elements. They are used to define the structure of the Lie algebra. In the video, the construction of left invariant vector fields on SL(2,C) is explained.

💡Differential Geometric Bracket

The differential geometric bracket, or Lie bracket, is an operation that combines two vector fields to produce a third. It is fundamental in defining the algebraic structure of a Lie algebra. The script describes how this bracket is used to define the Lie bracket on the tangent space of SL(2,C).

💡Structure Constants

Structure constants are numerical coefficients that appear in the commutation relations of a Lie algebra. They encode the algebraic properties of the algebra. The video includes a discussion on how to calculate these constants for the Lie algebra of SL(2,C).

💡Killing Form

The Killing form is a bilinear form on a Lie algebra that is used to study its structure, particularly its semi-simplicity. The video explains how the Killing form is calculated for the Lie algebra of SL(2,C) and its implications for the algebra's semi-simplicity.

💡Semisimplicity

A Lie algebra is called semisimple if it has no non-trivial abelian ideals. This property is important as it implies the algebra can be broken down into a direct sum of simple Lie algebras. The script discusses the semisimplicity of the Lie algebra of SL(2,C) using the Killing form.

💡Simple Lie Algebra

A simple Lie algebra is a Lie algebra that does not have any non-trivial ideals other than itself and the zero ideal. It is a building block for more complex Lie algebras. The video shows that the Lie algebra of SL(2,C) is not only semisimple but also simple.

💡Roots and Fundamental Roots

In the context of Lie algebras, roots are certain elements of the dual space of a Cartan subalgebra. Fundamental roots are a subset of roots that form a basis for the root space. The video explains how to identify the roots and fundamental roots of the Lie algebra of SL(2,C).

💡Dixmier Conjecture

The Dixmier conjecture is a statement in the theory of Lie algebras that concerns the eigenvalues of the adjoint map of an element in a simple Lie algebra. While not explicitly mentioned in the script, the concept is related to the discussion on the structure of the Lie algebra of SL(2,C).

💡Differential Geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus to study problems in geometry. In the video, differential geometry is used in the context of defining and working with tangent vectors and Lie brackets in the Lie algebra of SL(2,C).

Highlights

Continued study of the special linear group SL(2,C) and its Lie algebra, focusing on constructing the tangent diagram.

Construction of left invariant vector fields on SL(2,C) by taking left translations and push forwards.

Derivation of the vector field left invariant by the action of the group on the underlying manifold.

The tangent space at the identity element of the group can be equipped with a Lie bracket, isomorphic to the set of left invariant vector fields.

Introduction of the double bracket symbol to distinguish the Lie bracket from the differential geometric bracket.

Calculation of the brackets by evaluating the action on functions and then at the identity to obtain a vector at the identity.

The Lie bracket of two vector fields at the identity results in a vector at the identity, indicating the structure of the Lie algebra.

Calculation of the differential geometric derivatives and use of the product rule in the context of the Lie algebra of SL(2,C).

Evaluation of the brackets at the identity and simplification by dropping second derivative terms.

The Lie algebra of SL(2,C) is shown to be simple, not just semi-simple, through the absence of non-trivial ideals.

Introduction of the Killing form and its significance in determining the semi-simplicity of the Lie algebra.

Calculation of the Killing form components using the structure constants and the dual Killing form.

Determination that the Killing form is not negative definite, indicating that the group SL(2,C) is not compact.

Identification of the roots and fundamental roots of the Lie algebra, crucial for understanding its structure.

Use of the Weyl group transformations to generate the entire root space from fundamental roots.

Construction of the Cartan matrix and the Dynkin diagram from the root space and fundamental roots.

Reconstruction of a Lie algebra from its Dynkin diagram, demonstrated through an example with two fundamental roots.

Derivation of the commutation relations (Lie bracket relations) from the root diagram and Dynkin diagram.

Application of the Jacobi identity to determine the Lie bracket relations between elements of the root space.

Complete recovery of the Lie algebra structure from the root diagram, showcasing the comprehensiveness of the Dynkin diagram.

Transcripts

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Lie Algebra SL(2,C)Thinking Diagram Root Diagram Differential Geometry Mathematical Analysis Complex Numbers Vector Fields Algebraic Structures Theoretical Physics