Alphabeta MathIntegrable Highest Weight Kac Moody Modules
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1 · Prerequisites
Abstract Algebra
Lie Theory
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
Set Theory
2 · Logical flowchart
3 · Items
Theorem
Universal property and pbw character of kac moody verma modules
Theorem
Kac moody verma module has a unique simple quotient
Definition
Kac moody integral and dominant integral weights
Definition
Integrable kac moody module
Lemma
Integrability can be checked on simple root sl2 subalgebras
Lemma
Dominance is necessary for an integrable highest weight module
Lemma
Simple root power relations generate the integrable quotient
Theorem
Integrability criterion for simple highest weight kac moody modules
Proposition
Integrable weight sets and multiplicities are weyl invariant
Definition
Real coroot signs, word length and inversion sets
Lemma
Reduced words, root signs and finite coroot inversions
Lemma
Dominant representatives, wall stabilizers and terminating reflection descent
Proposition
Every integrable weight is weyl conjugate toward the dominant chamber
Lemma
Maximal and primitive weights in integrable category O modules
Lemma
Casimir separates comparable dominant primitive weights
Theorem
Complete reducibility of integrable kac moody o modules
Corollary
Tensor products of integrable highest weight modules decompose
Lemma
Affine central coroot from the transpose null ray
Proposition
Integrable affine highest weights have nonnegative integral level
Proposition