# Knapp — Lie Groups Beyond an Introduction — Candidate Nodes (Extractor output)

## Scout summary of EXISTING coverage (well-covered areas — likely already present):
Lie algebra/group basics, exponential map, BCH, adjoint rep, Engel/Lie theorems,
solvable/nilpotent/semisimple/reductive, radical, Levi decomposition, Killing form,
Cartan criteria, Casimir, Cartan subalgebra, root space decomposition, roots,
Weyl group/chambers, Cartan matrix, Dynkin diagrams, classification A-G,
universal enveloping algebra, PBW, highest weight theorem, Verma modules,
Harish-Chandra isomorphism, Weyl character/dimension/integration formula,
Kostant multiplicity/partition function, Freudenthal formula, weight/root lattices,
fundamental weights, Weyl vector rho, maximal tori, Peter-Weyl, Iwasawa/Bruhat
(named, may be shallow), compact/noncompact symmetric spaces (Cartan classification),
exponential, complexification (general).

SPARSE/ABSENT per Scout: restricted/relative roots (NOT systematic), nilpotent orbits
(only Jacobson-Morozov), Borel-Weil-Bott (Peter-Weyl present, Borel-Weil sparse),
covering groups (1 node), detailed harmonic analysis, non-compact group structure,
classical domains.

## CANDIDATE LIST (from 3 Extractors — dedupe against graph, KEEP only genuinely new)

### A. Lie algebra structure (Extractor 1) — most likely already present, verify:
- Lie's three theorems / Lie correspondence (theorem)
- Closed Subgroup Theorem (Cartan) (theorem)
- Subgroup-Subalgebra Correspondence (theorem)
- Abstract Jordan Decomposition (theorem)
- Preservation of Jordan Decomposition (theorem)
- Malcev-Harish-Chandra Theorem (conjugacy of Levi factors) (theorem)
- Whitehead's Lemmas (cohomology vanishing) (theorem)
- Regular Element (state)
- Existence of Cartan Subalgebras (theorem)
- Conjugacy of Cartan Subalgebras (theorem)
- sl(2)-triple / coroot H_alpha (state)
- Integrality / root string property (theorem)
- Reduced Root System (axiom)
- Triangular Decomposition of U(g) (theorem)
- Harish-Chandra Projection / rho-shift (technique)
- Chevalley Restriction theorem S(h)^W polynomial (theorem)
- Serre Relations (axiom)
- Chevalley Generators / Chevalley Basis (state)
- Serre's Existence Theorem (theorem)
- Isomorphism Theorem for Semisimple Lie Algebras (theorem)
- Existence Theorem (root system => Lie algebra) (theorem)
- Exceptional isomorphisms low rank (theorem)
- Weyl Denominator Formula (theorem)
- Brauer-Klimyk / Steinberg tensor product formula (theorem)
- Dot Action of W (state)
- Inner Automorphism Group Int g (state)
- Longest Element w_0 (state)
- Length Function / reduced words (state)

### B. Compact groups, real forms, Cartan decomposition (Extractor 2) — KEY GAP AREA:
- Compact Lie algebra (Ad-invariant pos-def form) (axiom)
- Negative-definiteness of Killing form on compact semisimple (theorem)
- Maximal Torus Theorem (every element in a torus; conjugacy) (theorem)
- Surjectivity of exp for compact connected (theorem)
- Weyl group of compact group N(T)/T (state)
- Identification W(G,T) = W(Delta) (theorem)
- Weyl's theorem: finiteness of fundamental group of compact semisimple (theorem)
- Compactness of universal cover of compact semisimple (theorem)
- Real Form of complex Lie algebra (axiom)
- Conjugation w.r.t. a real form (state)
- Compact Real Form (state)
- Existence of Compact Real Form (theorem)
- Uniqueness/conjugacy of Compact Real Form (theorem)
- Weyl's Unitary Trick (technique)
- Complexification of a compact Lie group (state)
- Cartan Involution (axiom)
- Existence of Cartan Involution (theorem)
- Conjugacy of Cartan Involutions (theorem)
- Cartan Decomposition g = k + p (theorem/state)
- B_theta inner product (state)
- Global Cartan Decomposition G = K exp(p) (theorem)
- Maximal Compact Subgroup (state)
- Cartan-Iwasawa-Malcev (conjugacy of maximal compacts) (theorem)
- Polar Decomposition (group/matrix) (theorem)
- Reductive Lie Group (Knapp's class, 4-tuple) (axiom)
- Structure theorem for reductive groups (theorem)
- Cartan subalgebra of real semisimple (axiom)
- theta-stable Cartan subalgebra (state)
- Existence of theta-stable Cartan subalgebra (theorem)
- Maximally compact / maximally noncompact Cartan subalgebras (state)
- Finiteness of conjugacy classes of Cartan subalgebras (theorem)
- Cayley Transform (technique)
- Real / Imaginary / Complex roots (rel. theta-stable CSA) (state)
- Compact vs noncompact imaginary roots (state)
- Borel-Weil Theorem (theorem)
- Borel-Weil-Bott Theorem (theorem)
- Flag Manifold G^C/B (state)
- Classification of Real Forms (theorem)
- Correspondence real forms <-> involutions of u_0 (theorem)
- Vogan Diagram (state)
- Vogan Diagram Theorem (theorem)
- Borel-de Siebenthal Theorem (theorem)
- Satake Diagram (state)
- Automorphism group Aut g / Int g (state)
- Out(g) = Aut(Dynkin diagram) (theorem)
- M = Z_K(a) centralizer (state)

### C. Restricted roots, parabolics, symmetric spaces, integration (Extractor 3) — KEY GAP AREA:
- Maximal Abelian Subspace a of p (state)
- Real Rank of G (state)
- Restricted Roots (state)
- Restricted Root Space Decomposition (theorem)
- Restricted Root Multiplicity (state)
- Restricted Root System Sigma (possibly non-reduced) (theorem)
- m = centralizer Z_k(a) (state)
- Positivity/ordering of restricted roots (technique)
- Nilpotent subalgebra n (state)
- Iwasawa Decomposition g = k+a+n (theorem)
- Iwasawa Decomposition G = KAN (theorem)
- Uniqueness in Iwasawa Decomposition (theorem)
- A = exp a vector group (state)
- N = exp n nilpotent group (state)
- Weyl group of restricted root system W(Sigma) (state)
- W(Sigma) = N_K(a)/Z_K(a) (theorem)
- KAK Decomposition (Cartan/polar) (theorem)
- Closed positive Weyl chamber a+ / A+ (state)
- Minimal Parabolic Subgroup P = MAN (state)
- Parabolic Subalgebra (axiom)
- Standard parabolic subgroups from subsets of simple roots (theorem)
- Langlands Decomposition of a parabolic (theorem)
- Bruhat Decomposition (theorem)
- Integral formula for Iwasawa KAN decomposition (theorem)
- Half-sum of positive restricted roots rho (state)
- Integral formula for KAK decomposition (sinh Jacobian) (theorem)
- Integral formula over G/K symmetric space (theorem)
- Haar Measure on a Lie Group (axiom)
- Modular Function Delta_G (state)
- Unimodularity of semisimple/reductive groups (theorem)
- Non-unimodularity of AN (theorem)
- Riemannian Symmetric Space (axiom)
- Geodesic Symmetry (state)
- Symmetric Pair (G,K) (axiom)
- Involutive automorphism sigma (axiom)
- Globally Symmetric Space G/K (theorem)
- Locally Symmetric Space (parallel curvature) (axiom)
- Cartan's characterization symmetric <=> nabla R = 0 (theorem)
- Symmetric space of noncompact/compact/Euclidean type (state)
- Duality compact <-> noncompact symmetric spaces (theorem)
- Rank of a symmetric space (state)
- Cartan classification of irreducible symmetric spaces (theorem)
- Hermitian Symmetric Space (axiom)
- Criterion for Hermitian type (Z(k) != 0) (theorem)
- Bounded Symmetric Domain (state)
- Harish-Chandra Decomposition g^C = p- + k^C + p+ (theorem)
- Harish-Chandra Embedding G/K -> p+ (theorem)
- Strongly Orthogonal Roots (state)
- Kostant Convexity Theorem (theorem)
- Iwasawa Projection H(g) (state)
- Opposite nilpotent n-bar = theta n (state)
- Open Bruhat cell / big cell (theorem)
- G-invariant metric on G/K from B_theta (state)
- Curvature formula R(X,Y)Z = -[[X,Y],Z] (theorem)
- de Rham Decomposition (theorem)
- Siegel domain / tube domain realization (state)
