Alphabeta MathKac Moody Algebras from Generalized Cartan Matrices
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1 · Prerequisites
Abstract Algebra
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
Definition
Generalized cartan matrix
Definition
Symmetrizable generalized cartan matrix
Definition
Realization of a generalized cartan matrix
Proposition
Minimal realizations exist and are unique up to isomorphism
Definition
Kac Moody root lattice height and positive cone
Lemma
PBW for countably presented Kac Moody Lie algebras
Lemma
Free Lie construction for finite Kac Moody generators
Definition
Contragredient lie algebra before the maximal ideal quotient
Proposition
Contragredient algebra has a triangular decomposition
Lemma
The sum of triangularly disjoint graded ideals is disjoint from h
Definition
Kac moody algebra associated to a gcm
Proposition
Kac moody root spaces are finite dimensional
Lemma
The opposite simple centralizer in a Kac Moody half vanishes
Lemma
Serre elements vanish before Serre generation
Definition
Simple reflections and the kac moody weyl group
Proposition
The weyl group preserves roots and root multiplicities
Theorem
Invariant bilinear form for a symmetrizable kac moody algebra
Definition
Kac moody category o
Definition
Kac moody verma module
Definition
Generalized casimir on restricted kac moody modules
Theorem
Generalized kac moody casimir is central and scalar on highest weight modules
Lemma
Bounded above kac moody weight modules are generated by primitive vectors
Lemma
Enveloping quotient kernels and augmentation intersections
Lemma
Kac moody relation module embeds in verma modules and obeys the casimir constraint
Lemma
The serre quotient has weyl symmetry and no residual kac moody kernel
Theorem
Serre presentation of a kac moody algebra
Definition
Real and imaginary kac moody roots
Proposition
Real root spaces are one dimensional sl2 roots
Lemma
Strict linear alternative for GCM trichotomy
Theorem
Finite affine indefinite trichotomy for indecomposable gcms
Lemma
Finite-type Kac–Moody roots descend to simple roots
Lemma
Nonsingular indecomposable Kac–Moody algebras are simple
Proposition
Finite type kac moody algebras recover the dg semisimple algebras
Sources
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §1.2, p.10
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §2.1, pp.26–27
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Definition 1.2.2 and Proposition 1.2.4, pp.10–12
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Proposition 1.2.4, pp.11–12; independent row-image construction replaces a principal-minor assumption
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — end of §1.2, pp.12–13
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — local PBW reduction supporting §1.3 and §9.3 (the finite-dimensional PBW theorem is not imported)
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §1.3, Theorem 1.3.3(ii), pp.14–16; explicit universal-property construction
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Definition 1.3.1, p.13
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Theorem 1.3.3, pp.13–16; full tensor-module argument
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Lemma 1.3.2 and Theorem 1.3.3(v), pp.13–16
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Definition 1.4.1, p.16
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Theorem 1.3.3(iv), §1.4, pp.14–19
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Perrin Lemma 4.2.8, p.36; maximal-ideal argument as in Kleshchev §1.4
- Nicolas Perrin, Introduction to Kac-Moody groups and Lie algebras, section 4.2
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §1.4 Serre vanishing and Lemma 3.1.1; Perrin Propositions 4.2.6–4.2.7, pp.35–37
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §3.2, pp.39–42
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Lemma 3.1.2 and §3.2, pp.37–42
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Lemma 2.2.1 and Theorem 2.2.3, pp.28–32; complete height induction and zero-height invariance
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §9.1, pp.116–118
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §9.1, pp.116–117; local countable PBW verification
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Definition 2.3.3 and equations (2.18)–(2.20), pp.33–34
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Lemma 2.3.1, Theorem 2.3.5 and Corollary 2.3.6, pp.32–36
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Lemma 9.1.3 and preceding primitive-vector definition, pp.117–118
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Lemmas 9.3.1–9.3.3, pp.122–124; corrected left U(R)-module proof for Lemma 9.3.3
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Proposition 9.3.4, pp.124–125; corrected associative last-letter coefficients and augmentation proof
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Theorem 9.3.5, pp.125–126; direct Serre-quotient exponential construction from §3.2
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Theorem 9.3.5, pp.125–126; half-ideal and PBW consequences
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §5.1, pp.68–69, and §5.3, pp.73–74
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — §5.1, pp.68–69, with §3.2 transport
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Lemma 4.1.4 and Proposition 4.1.5, pp.51–52; minimum taken directly on the coefficient simplex
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Definition 4.1.1, Lemmas 4.1.6–4.1.7, Theorem 4.1.12, Lemma 4.1.13, Lemma 4.2.2 and Theorem 4.2.3, pp.50–60; direct quadratic expansion replaces spectral theory
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Proposition 4.3.2, pp.63–64; local height descent and explicit dual-basis lattice bound
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Proposition 1.4.8(i), pp.19–20; direct ideal propagation
- Kleshchev, Lectures on Infinite Dimensional Lie Algebras — Propositions 1.4.3, 1.4.8(i), 4.3.2, pp.17–20 and 63–64; local inverse generator maps