Alphabeta MathAffine Lie Algebras and Loop Central Extensions
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Commutative Algebra
Lie Theory
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Loop algebra of a simple Lie algebra
Definition
Residue two cocycle on a loop algebra
Lemma
The loop residue form is alternating
Lemma
The loop residue form satisfies the Lie two cocycle identity
Definition
Untwisted affine central extension
Definition
Degree derivation and full untwisted affine algebra
Proposition
The derived affine algebra omits only the degree derivation
Definition
Null root, central coroot, and affine level
Lemma
The affine simple root alpha zero is delta minus the highest root
Theorem
Loop and affine GCM presentations are isomorphic
Proposition
Affine Weyl group is a coroot lattice semidirect product
Proposition
Roots of an untwisted affine Lie algebra
Definition
Evaluation module at a nonzero loop parameter
Proposition
Evaluation modules have level zero and do not extend canonically over d
Definition
Twisted loop algebra from a diagram automorphism
Sources
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Section 7.1
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Section 12.2.1
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Sections 7.1-7.2
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Lemma 12.2.5
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, equations (7.1)-(7.2)
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Lemma 12.2.5 and Corollary 12.2.6
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Definition 12.2.3 and Corollary 12.2.6
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Definition 12.2.3 and Fact 12.2.7
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Section 7.2
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Section 12.2.2
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Sections 6.3 and 7.2
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Sections 12.1-12.2.2
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Theorem 7.2.1
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Theorem 12.2.15
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Section 6.4
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Theorem 12.2.19 and Lemma 12.2.20
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Section 7.2 and Corollary 7.2.2
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Proposition 12.2.14 and Corollary 12.2.16
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Section 7.1 loop-algebra conventions
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Section 12.2.1 loop-algebra conventions
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Definition 12.2.3
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Section 8.2
- Perrin, Introduction to Kac-Moody Groups and Lie Algebras, Definition 14.1.2 and Proposition 14.1.3