Learn the Structure and Classification of Semi-Simple Lie Algebras with Bourbaki
Bourbaki Lie Groups and Lie Algebras Chapters 13 PDF 26
Lie groups and Lie algebras are fundamental objects in mathematics and physics that capture the essence of symmetry, transformation, and linearization. They have applications in many areas such as differential geometry, topology, number theory, representation theory, quantum mechanics, particle physics, and more. In this article, we will explore some of the main topics covered in Chapters 13 and 26 of Bourbaki's book on Lie groups and Lie algebras, which deal with the structure and classification of semi-simple Lie algebras. We will assume some basic familiarity with abstract algebra, linear algebra, and differential calculus.
Bourbaki Lie Groups And Lie Algebras Chapters 13 Pdf 26
Introduction
What are Lie groups and Lie algebras? A Lie group is a smooth manifold that also has a group structure, such that the group operations (multiplication and inversion) are smooth maps. For example, the circle group S^1, the real line group R, the general linear group GL(n,R), and the special orthogonal group SO(n) are all examples of Lie groups. A Lie algebra is a vector space that also has a bilinear operation called the Lie bracket, which satisfies some properties such as anti-symmetry ([x,y] = -[y,x]) and Jacobi identity ([x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0). For example, the space of n x n matrices with the commutator bracket ([A,B] = AB - BA) is a Lie algebra. There is a natural correspondence between Lie groups and Lie algebras, given by the exponential map, which maps a Lie algebra element to a Lie group element by taking the limit of repeated multiplication. For example, the exponential map for the matrix Lie algebra is given by exp(A) = I + A + A^2/2! + A^3/3! + ..., where I is the identity matrix.
Who are Bourbaki and what is their contribution to mathematics? Bourbaki is the collective pseudonym of a group of French mathematicians who started in the 1930s with the aim of writing a comprehensive and rigorous treatise on modern mathematics. Their work, titled Elements of Mathematics, consists of several volumes covering topics such as set theory, algebra, topology, analysis, integration, commutative algebra, algebraic geometry, differential geometry, spectral theory, and Lie groups and Lie algebras. Their style is abstract, axiomatic, and formal, and their influence is immense in shaping the language and foundations of mathematics. One of their most famous contributions is the concept of structures, which are sets equipped with some operations or relations that satisfy certain axioms. For example, a group is a structure with one operation (multiplication) that satisfies associativity, identity, and inverse axioms. A vector space is a structure with two operations (addition and scalar multiplication) that satisfy commutativity, associativity, identity, inverse, distributivity, and compatibility axioms.
What are the main topics covered in Chapters 13 of Bourbaki's book on Lie groups and Lie algebras? Chapter 13 is titled The Structure of Semi-Simple Lie Algebras, and it covers some of the most important results about the structure and representation theory of semi-simple Lie algebras. A semi-simple Lie algebra is a Lie algebra that has no non-trivial abelian ideals (an ideal is a subalgebra that is closed under the Lie bracket with any element of the algebra, and abelian means that the Lie bracket is zero). Equivalently, a semi-simple Lie algebra is a direct sum of simple Lie algebras, where a simple Lie algebra is one that has no non-trivial ideals at all. For example, the matrix Lie algebra gl(n,R) is semi-simple, but not simple, because it has a one-dimensional ideal consisting of scalar multiples of the identity matrix. The simple Lie algebras are classified into four infinite families (A-D) and five exceptional cases (E-G). The main topics covered in Chapter 13 are:
The Killing form and the Cartan criterion: The Killing form is a bilinear form on a Lie algebra that measures its non-degeneracy. The Cartan criterion is a theorem that characterizes semi-simple Lie algebras by the non-degeneracy of their Killing form.
The root system of a semi-simple Lie algebra: A root system is a set of vectors in a Euclidean space that encodes the structure of a semi-simple Lie algebra. It has some properties such as being finite, spanning the space, being invariant under reflections, and satisfying some integrality conditions.
The Weyl group and the fundamental system of roots: The Weyl group is a finite group that acts on the root system by reflections. It preserves some properties such as the length and angle of the roots. A fundamental system of roots is a subset of the root system that generates it under the action of the Weyl group. It determines a Cartan subalgebra, which is a maximal abelian subalgebra that consists of diagonalizable elements.
The Chevalley basis and the universal enveloping algebra: A Chevalley basis is a basis of a semi-simple Lie algebra that consists of elements corresponding to roots and Cartan subalgebra elements. It gives a concrete realization of a semi-simple Lie algebra in terms of matrices or polynomials. The universal enveloping algebra is an associative algebra that contains a given Lie algebra as a subalgebra and satisfies a universal property with respect to representations.
Chapter 13: The Structure of Semi-Simple Lie Algebras
The Killing Form and the Cartan Criterion
The Killing form is a bilinear form on a Lie algebra that measures its non-degeneracy. It is defined by B(x,y) = tr(ad(x)ad(y)), where tr denotes the trace of an endomorphism and ad(x) denotes the adjoint map defined by ad(x)(y) = [x,y]. The Killing form has some properties such as being symmetric (B(x,y) = B(y,x)), invariant (B([x,y],z) = B(x,[y,z])), and associative (B(x,[y,z]) = B([x,y],z)). The Killing form can be used to measure the non-degeneracy of a Lie algebra, i.e., whether there exists a non-zero element that commutes with every element of the algebra. The Cartan criterion is a theorem that states that a Lie algebra is semi-simple if and only if its Killing form is non-degenerate, i.e., B(x,y) = 0 for all y implies x = 0.
The Root System of a Semi-Simple Lie Algebra
A root system is a set of vectors in a Euclidean space that encodes the structure of a semi-simple Lie algebra. It is obtained by choosing a Cartan subalgebra of the Lie algebra and decomposing the rest of the algebra into eigenspaces of the adjoint action of the Cartan subalgebra. A Cartan subalgebra is a maximal abelian subalgebra that consists of diagonalizable elements. For example, for the matrix Lie algebra gl(n,R), a Cartan subalgebra is the set of diagonal matrices. The eigenspaces of the adjoint action are called root spaces, and the corresponding eigenvalues are called roots. For example, for the matrix Lie algebra gl(n,R), a root space is the set of matrices with only one non-zero entry outside the diagonal, and a root is the difference of the diagonal entries corresponding to that entry. A root system is then defined as the set of all roots, considered as vectors in a Euclidean space spanned by a basis of the Cartan subalgebra. A root system has some properties such as being finite, spanning the space, being invariant under reflections, and satisfying some integrality conditions.
The Weyl Group and the Fundamental System of Roots
The Weyl group is a finite group that acts on the root system by reflections. It preserves some properties such as the length and angle of the roots. A reflection is a linear transformation that fixes a hyperplane and reverses the direction of vectors orthogonal to it. For example, for the matrix Lie algebra gl(n,R), a reflection is given by swapping two rows and columns of a matrix and changing their signs. The Weyl group is generated by the reflections corresponding to the simple roots, which are roots that cannot be written as a positive linear combination of other roots. A fundamental system of roots is a subset of the root system that generates it under the action of the Weyl group. It determines a Cartan subalgebra, which is spanned by the elements corresponding to the simple roots. For example, for the matrix Lie algebra gl(n,R), a fundamental system of roots is given by e1 - e2, e2 - e3, ..., en-1 - en, where ei denotes the standard basis vector.
The Chevalley Basis and the Universal Enveloping Algebra
A Chevalley basis is a basis of a semi-simple Lie algebra that consists of elements corresponding to roots and Cartan subalgebra elements. It gives a concrete realization of a semi-simple Lie algebra in terms of matrices or polynomials. For example, for the matrix Lie algebra gl(n,R), a Chevalley basis is given by the matrices Eij with one non-zero entry equal to one at the (i,j)-th position, and Hk with one non-zero entry equal to one at the (k,k)-th position and another non-zero entry equal to minus one at the (k+1,k+1)-th position. The universal enveloping algebra is an associative algebra that contains a given Lie algebra as a subalgebra and satisfies a universal property with respect to representations. It is constructed by taking the tensor algebra of the Lie algebra and quotienting by the ideal generated by the elements of the form x \otimes y - y \otimes x - [x,y], where x and y are elements of the Lie algebra. For example, for the matrix Lie algebra gl(n,R), the universal enveloping algebra is the algebra of polynomials in n^2 variables with coefficients in R.
Chapter 26: The Classification of Semi-Simple Lie Algebras
The Dynkin Diagrams and the Cartan Matrix
The Dynkin diagrams and the Cartan matrix are two ways of encoding the structure of root systems. A Dynkin diagram is a graph whose vertices are labeled by simple roots and whose edges are determined by the angles between them. For example, for the matrix Lie algebra gl(n,R), the Dynkin diagram is a chain of n-1 vertices connected by single edges. A Cartan matrix is a matrix whose entries are given by the inner products of simple roots and their duals. For example, for the matrix Lie algebra gl(n,R), the Cartan matrix is a tridiagonal matrix with -1s on the subdiagonal and superdiagonal, and 2s on the diagonal. The Dynkin diagram and the Cartan matrix are related by some rules, such as:
If two simple roots are orthogonal, there is no edge between them.
If two simple roots have an acute angle, there is a single edge between them.
If two simple roots have an obtuse angle, there are multiple edges between them.
The number of edges between two simple roots is equal to twice the absolute value of their Cartan matrix entry.
The direction of an edge (if any) between two simple roots is determined by their relative lengths.
The Classification Theorem and its Consequences
The classification theorem for semi-simple Lie algebras is a theorem that states that every semi-simple Lie algebra over a field of characteristic zero has a root system that belongs to one of four infinite families (A-D) or five exceptional cases (E-G). The theorem also gives a correspondence between semi-simple Lie algebras and Dynkin diagrams or Cartan matrices up to isomorphism. The theorem can be proved by using some criteria such as:
The determinant of a Cartan matrix must be non-zero.
The entries of a Cartan matrix must satisfy some inequalities.
The eigenvalues of a Cartan matrix must be real and positive.
The rank of a semi-simple Lie algebra must be less than or equal to its dimension.
Some consequences of the classification theorem for representations, automorphisms, and extensions of semi-simple Lie algebras are:
The irreducible representations of a semi-simple Lie algebra are determined by its highest weights, which are elements of the dual space of its Cartan subalgebra that satisfy some conditions.
The automorphism group of a semi-simple Lie algebra is generated by inner automorphisms (conjugation by elements of the Lie algebra) and diagram automorphisms (permutations or reflections of the Dynkin diagram).
The extensions of a semi-simple Lie algebra by another Lie algebra are classified by its cohomology groups, which measure the obstructions to splitting or triviality.
Conclusion
In this article, we have given an overview of some of the main topics covered in Chapters 13 and 26 of Bourbaki's book on Lie groups and Lie algebras, which deal with the structure and classification of semi-simple Lie algebras. We have seen how the theory of Lie algebras is based on the notion of root systems, which are sets of vectors that encode the structure of a semi-simple Lie algebra. We have also seen how the root systems are classified by Dynkin diagrams or Cartan matrices, which are graphical or algebraic representations of their properties. We have also mentioned some applications of the theory of Lie algebras to representation theory, automorphism theory, and extension theory. We hope that this article has given the reader some insight into the beauty and richness of Lie algebras and their connections to other areas of mathematics and physics.
FAQs
Here are some frequently asked questions about Bourbaki Lie Groups and Lie Algebras Chapters 13 PDF 26:
Q: Where can I find the PDF file of Bourbaki Lie Groups and Lie Algebras Chapters 13? A: You can find it online at https://www.maths.ed.ac.uk/v1ranick/papers/bourbakilie.pdf.
Q: What are the prerequisites for reading Bourbaki Lie Groups and Lie Algebras Chapters 13? A: You should have some background in abstract algebra, linear algebra, and differential calculus. You should also be familiar with some basic concepts of category theory, such as functors, natural transformations, and adjoints.
Q: What are some other references for learning about Lie groups and Lie algebras? A: Some other references are:
Lie Groups, Lie Algebras, and Representations: An Elementary Introduction by Brian C. Hall.
An Introduction to Lie Groups and Lie Algebras by Alexander Kirillov Jr.
Lie Algebras in Particle Physics by Howard Georgi.
Q: What are some applications of Lie groups and Lie algebras in physics? A: Some applications are:
Lie groups are used to describe the symmetries of physical systems, such as rotations, translations, gauge transformations, etc.
Lie algebras are used to describe the generators of infinitesimal symmetries, such as angular momentum, momentum, charge, etc.
Representations of Lie groups and Lie algebras are used to describe the states and observables of physical systems, such as spin, energy, mass, etc.
Q: What are some open problems or challenges in the theory of Lie groups and Lie algebras? A: Some open problems or challenges are:
The classification of finite simple groups is complete, but the classification of finite simple Lie algebras is not.
The classification of simple complex Lie algebras is complete, but the classification of simple real or p-adic Lie algebras is not.
The representation theory of semi-simple Lie algebras is well-understood, but the representation theory of non-semi-simple or infinite-dimensional Lie algebras is not.
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