# Groups, Lie algebras, & its Representations

This wiki book is based on Wikipedia articles. No originality is claimed. The purpose of this book is to provide a pedagogical arrangement of the topics in Group, Lie algebra, and its representation theory. This means it starts from the very basics to the advanced (and contemporary) topics in these subjects. *The primary readers would be mathematical (or theoretical) physicists and mathematicians. But, any interested readers may benefit from this book.*

**Book Recommendations:**

▛ 2003 - Fuchs, Schweigert - Symmetries, Lie Algebras and Representations $\mid$ For mathematically minded people

▟ 2018 - Georgi - Lie Algebras In Particle Physics

▛ 1990 - Kac - Infinite-Dimensional Lie Algebras

▛ 1998 - Kac - Vertex algebras for beginners**Lecture Note Recommendations:**

▛ 2010 - Lüdeling - Group Theory (for Physicists)**Note:**

▛ Use capital letters to denote Lie groups and lowercase fraktur to denote corresponding Lie algebras.

■ See this animation here, taken from here

## Basics

- Set
- Element
- Binary Relation
- Reflexive Relation
- Symmetric Relation
- Transitive Relation
- Equivalence Relation
- Partial Order $\mid$ Partially Ordered Set (poset)
- Total Order

- Map
- Domain
- Co-domain
- Range
- Image
- Injective $\mid$ one-to-one $\mid$ 1-1
- Surjective $\mid$ onto
- Bijective $\mid$ one-to-one and onto $\mid$ one-to-one correspondence $\mid$ 1-1 correspondence $\mid$ invertible
- Homomorphism

- Operand
- Binary Operation
- Algebraic structure
- Point Set Topology

- Set
- Finite Field $\mid$ Galois field
- Field extension
- Algebra
- Unital algebra
- Zero algebra
- Associative algebra
- Distributive algebra $\mid$ Non-associative algebra

- Integral Domain
- Quotient ring $\mid$ Factor Ring
- Fermat’s Little Theorem
- Euler’s Theorem
- Ideal
- Prime Ideal
- Maximal Ideal
- Principal ideal
- Unique factorization domain $\mid$ Factorial Ring
- Euclidean domain $\mid$ Euclidean ring

- Basis
- Dual basis
- Dimension
- Linear combination
- Bilinear map
- Sesquilinear form $\mid$ Hermitian Product
- Inner product $\mid$ Scalar product
- Tensor product
- Subspace
- Lie Algebra

## Groups

Group Ho

**mo**morphism $\mid$ Not to be confused with ho**meo**morphism- Group Monomorphism $\mid$ Injective Homomorphism
- Group Epimorphism $\mid$ Surjective Homomorphism
- Group Isomorphism $\mid$ Bijective Homomorphism
- Cayley’s Theorem
- Isomorphism Theorem
- First Isomorphism Theorem $\mid$ Fundamental Theorem of Homomorphism $\mid$ See its proof at page 18 of Lüdeling’s note.

- Group Endomorphism $\mid$ Homomorphism from a group to itself
- Group Automorphism $\mid$ Bijective Endomorphism

Product of Groups

- Direct Product
- Kronecker product $\mid$ matrix direct product

- Direct Sum
- Free Product
- Semidirect Product

- Direct Product

## Representations

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Published onMay 25, 2023

Last revised onFeb 16, 2024