{ "course": "RT-LIE", "course_complete": false, "scope": "Current explicit proof coverage; stated supplementary theorems do not become proved by citation.", "lessons": [ { "lesson": "RT-LIE-01", "proved_core": "Lie brackets, ideals/quotients, representations/tensors, classical forms and dimensions, small-dimension classification, enveloping universal property and module correspondence.", "supplementary_results": [ { "name": "PBW", "statement": "For any Lie algebra over any field and any totally ordered basis, the ordered enveloping monomials, including 1, form a basis; the symmetric algebra identifies with the associated graded algebra.", "proof_status": "closed_internal", "internal_proof": "RT-LIE-01.html#theorem-6-2; full ordered-word normal-form proof for arbitrary dimension and characteristic. Further applications: RT-LIE-13.html#theorem-2-1.", "proved_components": [], "remaining_component": null } ], "foundation_providers": [ { "result": "rank–nullity", "proof": "RT-LIE-01.html#lemma-0-1" }, { "result": "quotient-algebra factorization", "proof": "RT-LIE-01.html#lemma-0-2" }, { "result": "complex square roots", "proof": "RT-LIE-01.html#lemma-0-3" }, { "result": "finite bases and complements", "proof": "../../human/linear-algebra-bridges/from-bases-to-projections.html#extending-a-basis-and-choosing-a-complement" }, { "result": "determinants over commutative rings", "proof": "../../human/determinants/index.html" } ], "axiomatic_boundary": "Field and vector-space axioms; a specified totally ordered basis for arbitrary-dimensional PBW; the complete ordered real field for complex scalars. Not a claim of all foundational course closure." }, { "lesson": "RT-LIE-02", "proved_core": "Lower central/derived series, radical, Engel over every field and Lie over algebraically closed characteristic-zero fields, with descent of the solvable-derived nilpotence equivalence.", "supplementary_results": [], "foundation_providers": [ { "result": "Every field has an algebraic closure", "programme": "AI Integrated Stacks", "proof": "https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/fields.html#fields-theorem-existence-algebraic-closure", "pinned_native_proof": "https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/565b10e987aba5969b21145a0833f42d69f96790/fields.tex#L999", "revision": "565b10e987aba5969b21145a0833f42d69f96790", "native_sha256": "cfdee16ef341d3ecc079f774e9a4290813b36761b615221de8f74831bf4c4161", "native_label": "theorem-existence-algebraic-closure", "statement_and_proof_lines": [ 999, 1055 ], "conditions": "Arbitrary field; usual set theory with choice (Zorn). No characteristic or cardinality restriction.", "use": "Corollary 3.2; base extension followed by Proposition 1.2 descent", "scope": "Exact existing programme proof route; no new independent whole-chapter review or course-closure assertion." } ], "axiomatic_boundary": "The algebraic-closure construction uses Zorn’s lemma. No canonical choice of closure is claimed." }, { "lesson": "RT-LIE-03", "proved_core": "Polynomial Jordan decomposition (including Bézout residue construction), Cartan criteria, simple-ideal splitting, inner derivations and intrinsic Jordan decomposition.", "supplementary_results": [ { "name": "representation Jordan compatibility", "statement": "Finite-dimensional representations of a semisimple algebra preserve the intrinsic Jordan parts over characteristic-zero fields.", "proof_status": "closed_internal", "internal_proof": "RT-LIE-04 §4 Lemma 4.2 L163 and §5 Theorem 5.1 L194.", "proved_components": [], "remaining_component": null } ] }, { "lesson": "RT-LIE-04", "proved_core": "Casimir centrality, Weyl complete reducibility over every characteristic-zero field, Jordan compatibility, adjoint Killing Casimir scalar and boundary examples.", "supplementary_results": [] }, { "lesson": "RT-LIE-05", "proved_core": "Finite simple sl2 modules, multiplicities, Clebsch–Gordan, Casimir scalars and the direct sl2 Verma basis/submodule classification.", "supplementary_results": [ { "name": "sl2 PBW", "statement": "The ordered monomials f^a h^b e^c for a,b,c >= 0 form a basis of U(sl2).", "proof_status": "closed_internal", "internal_proof": "RT-LIE-13 §2 Theorem 2.1 L100, proof L42–104; direct Verma independence already proved in RT-LIE-05 L257–277.", "proved_components": [], "remaining_component": null } ] }, { "lesson": "RT-LIE-06", "proved_core": "Chevalley–Eilenberg d^2=0, H0/H1/H2 interpretations, full cochain contraction, both Whitehead lemmas including trivial coefficients/field descent, Levi existence. Current §7 also proves single-exponential Levi conjugacy and Ado over every characteristic-zero field.", "supplementary_results": [ { "name": "Levi conjugacy", "statement": "Over a characteristic-zero field, any two Levi subalgebras of a finite-dimensional Lie algebra are conjugate by exp(ad a) for a in its nilradical.", "proof_status": "proved_in_owned_current_lesson", "internal_proof": "RT-LIE-06 §7.3, Theorem7.1", "proved_components": [], "remaining_component": null }, { "name": "Ado", "statement": "Every finite-dimensional Lie algebra over a characteristic-zero field has a faithful finite-dimensional representation.", "proof_status": "proved_in_owned_current_lesson", "internal_proof": "RT-LIE-06 §§7.4–7.9, Theorem7.2", "proved_components": [], "remaining_component": null } ] }, { "lesson": "RT-LIE-07", "proved_core": "Maximal toral existence/centralizer, root decomposition, rank-one triples, one-dimensional/reduced roots, nonproportional strings, bracket equality, Cartan integrality/reflections, rationality/positivity and classical matrix decompositions.", "supplementary_results": [ { "name": "Cartan equivalence and conjugacy", "statement": "Maximal toral subalgebras are Cartan subalgebras, and all Cartan subalgebras of a complex semisimple algebra are inner-conjugate.", "proof_status": "closed_internal", "internal_proof": "RT-LIE-10 §2 Theorem 2.2 L79–100 and §4 Theorem 4.2 L236–245; geometry inputs remain separate prerequisites below.", "proved_components": [], "remaining_component": null }, { "name": "sp4 so5", "statement": "The complex Lie algebras sp4 and so5 are isomorphic (stronger than the proved B2/C2 root comparison).", "proof_status": "closed_internal", "internal_proof": "RT-LIE-12 §2 Proposition 2.1 L91, explicit symplectic exterior-square construction L128–137; quotient form and basis Exercise 7.2.", "proved_components": [], "remaining_component": null } ] }, { "lesson": "RT-LIE-08", "proved_core": "Finite reduced crystallographic root systems: bases/chambers, strong exchange/length, simple transitivity, longest element, all rank-two cases, braid word property and Coxeter presentation.", "supplementary_results": [] }, { "lesson": "RT-LIE-09", "proved_core": "Root reconstruction/classification, every classical/exceptional coordinate model, root counts, highest roots, vector stabilizers, Weyl orders and Coxeter-number tables.", "supplementary_results": [ { "name": "Lie realization", "statement": "The corresponding finite root system is realized by a complex semisimple Lie algebra and determines its isomorphism class.", "proof_status": "closed_internal", "internal_proof": "RT-LIE-11 Theorem 3.1 L152–168 and Theorem 6.1 L268–302; explicit classical/exceptional models in RT-LIE-12 §§1–4.", "proved_components": [], "remaining_component": null } ] }, { "lesson": "RT-LIE-10", "proved_core": "General regular-element Cartans, semisimple Cartan/maximal-toral equivalence, Int(g)=Aut(g)^0(C), Cartan conjugacy and choice independence of roots/rank.", "supplementary_results": [] }, { "lesson": "RT-LIE-11", "proved_core": "Normalized simple generators, auxiliary free Lie embeddings/Cartan survival, isomorphism theorem, Serre-ideal stability, local nilpotence/reflections, finite-type Serre theorem and complex-simple classification.", "supplementary_results": [ { "name": "Aut split", "statement": "After choosing a base and normalized simple generators, Aut(g) is Int(g) semidirect Aut(Dyn(g)); diagram automorphisms preserve directed Cartan entries and may permute isomorphic components; every automorphism factors uniquely as an inner automorphism followed by the chosen diagram lift.", "proof_status": "proved_in_owned_current_lesson", "internal_proof": "RT-LIE-11 §9, Theorem9.1 and (9.8)", "proved_components": [], "remaining_component": null } ] }, { "lesson": "RT-LIE-12", "proved_core": "Classical matrix realization/simplicity/root types, actual low-rank maps, all dimensions, finite-type exceptional presentations, full complex octonion derivation space and simplicity, normalized exceptional-series identity and eight value checks.", "supplementary_results": [ { "name": "Chevalley integral basis", "statement": "For a complex semisimple algebra with chosen base, choose all root vectors simultaneously with [x_alpha,x_-alpha]=h_alpha, integral Cartan data, and [x_alpha,x_beta]=N_alpha,beta x_alpha+beta where N_alpha,beta=±(p+1), p=max{j>=0: beta-j alpha is a root}; absent nonopposite root sums have bracket zero.", "proof_status": "proved_in_owned_current_lesson", "internal_proof": "RT-LIE-12 §5, Theorem5.1", "proved_components": [], "remaining_component": null } ] }, { "lesson": "RT-LIE-13", "proved_core": "Full PBW over arbitrary fields and arbitrary dimension, injectivity/domain, finite-dimensional left/right Noetherianity with polynomial-ring lemma, direct-sum and triangular multiplication, characteristic-zero equivariant symmetrization and centre-as-vector-space.", "supplementary_results": [] }, { "lesson": "RT-LIE-14", "proved_core": "Integral/dominant weights, general Verma construction/unique simple quotient, finite-dimensional highest-weight classification with finite bound and local nilpotence, Weyl multiplicities/saturation, explicit small representations.", "supplementary_results": [] }, { "lesson": "RT-LIE-15", "proved_core": "Central characters/projection, shifted W invariance, full Chevalley restriction and Harish-Chandra isomorphisms, invariant orbit separation/linkage, Casimir scalar and moment identity. Current §9 proves homogeneous Weyl invariant polynomiality by characters and completions.", "supplementary_results": [ { "name": "invariant polynomial generators", "statement": "For a rank-r Weyl group, S(h)^W is a polynomial algebra C[p1,...,pr] on r homogeneous algebraically independent generators.", "proof_status": "proved_in_owned_current_lesson", "internal_proof": "RT-LIE-15 §9, Theorem9.1 and Lemmas9.1–9.11", "proved_components": [], "remaining_component": null } ] }, { "lesson": "RT-LIE-16", "proved_core": "Finite virtual Verma-character expansion, algebraic Weyl character formula, denominator, dimension Taylor limit, Kostant/Freudenthal multiplicities, classical determinant/dimension formulas, G2 examples and signed tensor folding.", "supplementary_results": [] }, { "lesson": "RT-LIE-17", "proved_core": "Real forms/conjugations, complex versus real-linear action equivalence, compact-real-form bracket signs and negative Killing form conditional on the admitted basis, compact adjoint group conditional on closed-subgroup foundations, compact element conjugacy, representation integration/path independence, Haar construction/averaging, classical real forms/signatures and Lorentz map. Current §§2–3 prove compact-form uniqueness and finite compact cover; §8 proves Cartan-involution existence, compact-involution classification reduction and maximal split Cartans.", "supplementary_results": [ { "name": "Chevalley opposite root signs", "statement": "There exist root vectors with [e_alpha,e_-alpha]=h_alpha, real root constants and N_-alpha,-beta=-N_alpha,beta simultaneously; integral constants may also be chosen.", "proof_status": "proved_in_owned_current_lesson", "internal_proof": "RT-LIE-12 §5, Theorem5.1 and (5.14); RT-LIE-17 §2", "proved_components": [], "remaining_component": null }, { "name": "compact form uniqueness", "statement": "Any two compact real forms of a complex semisimple Lie algebra are conjugate by Int(g).", "proof_status": "proved_in_owned_current_lesson", "internal_proof": "RT-LIE-17 §2.2, Theorem2.3", "proved_components": [], "remaining_component": null }, { "name": "Weyl compact cover", "statement": "A compact connected Lie group with semisimple Lie algebra has finite fundamental group and compact simply connected covering group.", "proof_status": "proved_in_owned_current_lesson", "internal_proof": "RT-LIE-17 §§3.1–3.4, Theorem3.4 and Corollary3.5", "proved_components": [], "remaining_component": null }, { "name": "Satake classification", "statement": "Real isomorphism classes of real semisimple Lie algebras are classified by admissible Satake diagrams up to diagram isomorphism; the underlying complex Dynkin diagram specifies the complexification.", "proof_status": "proved_in_owned_current_lesson", "internal_proof": "RT-LIE-17 §8.5, Theorem8.4, Definition8.4 and Lemmas/Propositions8.5–8.12", "proved_components": [], "remaining_component": null } ] }, { "lesson": "RT-LIE-18", "proved_core": "All-weight category O finiteness and generalized central-character decomposition, simples/finite length, enough projectives/projective covers, graded freeness/Verma filtrations, restricted duality, BGG reciprocity, every sl2 central-character category and regular-block algebra equivalence. Section8.5 proves the full finite KL inverse-polynomial identity, including both longest-element conventions. Section8.6 supplies conditional rank-one dual-standard character, IC recognition/parity/positivity, signed perverse Grothendieck relation and dominant Verma conversion, with all general geometric inputs exposed. Sections8.7–8.11 also fully prove finite-Weyl Hodge theory, indecomposable canonical characters, KL coefficient positivity and nonnegative Laurent structure constants. Sections8.12–8.15 prove the actual finite complex regular integral category-O comparison and general Verma/KL multiplicity formula, Theorem8.13.", "supplementary_results": [ { "name": "Hecke standard basis", "statement": "For the Hecke algebra over Z[q^(1/2),q^(-1/2)] with braid relations and (Ti+1)(Ti-q)=0, reduced products Tw are well-defined and form a standard basis.", "proof_status": "proved_in_owned_current_lesson", "internal_proof": "RT-LIE-18 §8.1, Proposition8.1", "proved_components": [], "remaining_component": null }, { "name": "KL canonical basis", "statement": "Unique P_x,y(q) in Z[q] satisfy P_y,y=1, P_x,y=0 for x not <= y, degree at most (length(y)-length(x)-1)/2 for x