6.2 Introduction to Lie groups and Lie algebras

Lie groups and algebras are fundamental structures in mathematics, combining smooth manifolds with group operations. They're crucial for understanding symmetries in physics and geometry, forming a bridge between continuous and discrete mathematics.

The relationship between Lie groups and their associated Lie algebras is deep and powerful. Through tools like the exponential map and one-parameter subgroups, we can study complex group structures by examining their simpler, linearized versions in the Lie algebra.

Lie Groups and Algebras

Fundamental Concepts of Lie Groups and Algebras

• Lie group defines a smooth manifold with group structure where multiplication and inversion operations are smooth
• Lie algebra consists of tangent space at the identity element of a Lie group
• Exponential map connects Lie algebra to Lie group by mapping elements of the algebra to the group
• One-parameter subgroup represents a continuous path in the Lie group, parameterized by a real number

Relationship Between Lie Groups and Algebras

• Lie algebra captures local structure of Lie group near identity element
• Exponential map exp: g → G maps Lie algebra g to Lie group G
• One-parameter subgroups form integral curves of left-invariant vector fields on the Lie group
• Lie bracket operation in algebra corresponds to group commutator in Lie group

Examples and Applications

• Matrix Lie groups (GL(n,R), SL(n,R), O(n), SO(n)) serve as concrete examples of Lie groups
• Corresponding matrix Lie algebras (gl(n,R), sl(n,R), o(n), so(n)) demonstrate Lie algebra structure
• Exponential map for matrix Lie groups calculated using matrix exponential series
• Physics applications include describing symmetries in particle physics and quantum mechanics

Structure and Representation

Structural Components of Lie Algebras

• Structure constants define Lie bracket operation in terms of basis elements of Lie algebra
• Adjoint representation maps Lie algebra elements to linear transformations on the algebra itself
• Left-invariant vector fields remain unchanged under left translations of the Lie group
• Maurer-Cartan form provides canonical left-invariant 1-form on Lie group, taking values in Lie algebra

Representations and Their Significance

• Adjoint representation Ad: G → GL(g) maps Lie group elements to automorphisms of Lie algebra
• Infinitesimal adjoint representation ad: g → gl(g) derived from adjoint representation
• Left-invariant vector fields form basis for Lie algebra of vector fields on Lie group
• Maurer-Cartan form ω satisfies structural equation dω + 1/2[ω,ω] = 0

Computational Aspects and Applications

• Structure constants calculated by expanding Lie bracket in terms of basis elements
• Adjoint representation computed using matrix exponential for matrix Lie groups
• Left-invariant vector fields constructed by pushing forward basis vectors from identity element
• Maurer-Cartan form used in differential geometry to study geometric properties of Lie groups