Alphabeta MathLie Theory
Lie theory connects smooth groups with the infinitesimal algebra that governs their local behavior. The foundational pages construct Lie algebras from invariant vector fields, develop the exponential map and the Baker–Campbell–Hausdorff formula, and distinguish immersed subgroups from closed subgroups. Smooth actions, homogeneous spaces and covering groups provide the global geometric language.
The algebraic branch develops representations, universal enveloping algebras and the Poincaré–Birkhoff–Witt theorem before studying solvable, nilpotent and semisimple Lie algebras. Cartan subalgebras, root decompositions, root systems and Dynkin diagrams lead to classification and finite-dimensional highest-weight representations. Compact groups bring maximal tori, Weyl integration, root data and Peter–Weyl theory; real forms add Cartan and Iwasawa decompositions.
The advanced representation-theory track continues through central characters, Verma modules, category O, BGG reciprocity and resolutions, character formulas, geometric representation theory and Kac–Moody algebras. General smooth-manifold machinery remains in Differential Geometry. Page statuses distinguish the published foundational material from draft advanced pages, and all prerequisites remain explicit across category boundaries.
Pathway
The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.
Part 1 · Lie Groups, Lie Algebras and Representations
26 pagesThe published foundations begin with groups and their infinitesimal structure, then develop Lie-algebra structure and highest weights. Compact groups and real forms connect the algebraic and geometric branches. The advanced track develops central characters, category O and Kac–Moody representation theory; draft pages retain their status.
This page develops Lie algebras and their representations without a finite-dimensional hypothesis.
15 definitions, 6 lemmas, 10 propositions, 4 theorems, 3 corollaries, 6 false statements, 1 remarkExamples & counterexamples →A Lie group's tangent space at the identity acquires its bracket from left-invariant vector fields; right-invariant fields instead realize the opposite bracket.
16 definitions, 2 lemmas, 10 propositions, 11 theorems, 3 corollaries, 4 examples, 6 false statementsExamples & counterexamples →Lie subgroups carry an intrinsic manifold topology that need not be the ambient subspace topology.
12 definitions, 6 lemmas, 9 propositions, 17 theorems, 4 corollaries, 1 example, 6 false statementsExamples & counterexamples →- Solvable and Nilpotent Lie Algebras42 results
The derived series measures solvability, while the lower and upper central series measure nilpotence.
10 definitions, 4 lemmas, 7 propositions, 9 theorems, 6 corollaries, 6 false statementsExamples & counterexamples → Invariant trace forms connect representation theory to structure theory.
8 definitions, 2 lemmas, 4 propositions, 19 theorems, 7 corollaries, 3 examples, 6 false statementsExamples & counterexamples →This page develops Cartan subalgebras and root-space decompositions for finite-dimensional complex semisimple Lie algebras.
11 definitions, 3 lemmas, 7 propositions, 12 theorems, 4 corollaries, 1 example, 6 false statements, 2 remarksExamples & counterexamples →For a finite complex reflection group, Reynolds averaging first gives orbit separation and algebra generation.
7 definitions, 17 lemmas, 2 theoremsExamples & counterexamples →This page develops abstract root systems and the Cartan-Killing classification of complex semisimple Lie algebras.
17 definitions, 1 lemma, 13 propositions, 11 theorems, 1 corollary, 2 examples, 6 false statements, 1 remarkExamples & counterexamples →This page develops the Lie-theoretic interface for the center of the universal enveloping algebra: PBW and triangular decompositions, Killing forms and root spaces, Cartan…
15 definitions, 11 lemmas, 11 propositions, 11 theorems, 2 corollariesExamples & counterexamples →This page develops the finite-dimensional representation theory of a complex semisimple Lie algebra g up to the theorem of the highest weight: irreducible finite-dimensional…
7 definitions, 8 lemmas, 11 propositions, 4 theorems, 1 corollary, 2 examples, 6 false statements, 1 remarkExamples & counterexamples →This page develops the global structure and representation theory of compact connected Lie groups.
11 definitions, 5 lemmas, 8 propositions, 15 theorems, 9 corollaries, 6 false statementsExamples & counterexamples →Starting from a finite generalized Cartan matrix, this page constructs the minimal Cartan space, the universal contragredient algebra and its largest Cartan-disjoint quotient.
11 definitions, 12 lemmas, 6 propositions, 4 theoremsExamples & counterexamples →- Verma Modules and Shapovalov Forms19 results
Starting from the Verma-module construction and PBW model on the Harish–Chandra page, this page proves injectivity and uniqueness statements for homomorphisms, then uses the…
2 definitions, 10 lemmas, 3 propositions, 3 theorems, 1 corollaryExamples & counterexamples → The construction begins with finite Laurent sums and proves the residue identities before introducing a central bracket.
7 definitions, 3 lemmas, 4 propositions, 1 theoremExamples & counterexamples →Singular vectors control maps out of Verma modules. The PBW and Jantzen arguments here supply the ingredients for the directed strong-linkage criterion proved after finite…
2 definitions, 3 lemmas, 2 propositions, 3 theorems, 2 corollariesExamples & counterexamples →Verma universality and the unique simple quotient lead to the integrability criterion through explicit simple-root string calculations.
3 definitions, 8 lemmas, 4 propositions, 4 theorems, 1 corollaryExamples & counterexamples →- Primitive Ideals and Duflo Theorem15 results
This page develops the algebraic prefix of the primitive-ideal theory of the universal enveloping algebra: annihilators of modules, primitive ideals and their primeness, the…
4 definitions, 4 lemmas, 5 propositions, 1 corollary, 1 remarkExamples & counterexamples → This page develops the real forms of complex semisimple Lie algebras: complexification and the canonical conjugation, the bijection between real forms and conjugate-linear…
15 definitions, 1 lemma, 8 propositions, 20 theorems, 1 corollary, 6 false statements, 1 remarkExamples & counterexamples →- Category O Finiteness Duality and Blocks35 results
The classical BGG category O combines finite generation, a Cartan weight decomposition and local finiteness for the positive nilpotent algebra.
6 definitions, 13 lemmas, 8 propositions, 7 theorems, 1 corollaryExamples & counterexamples → - The Weyl Kac Character Formula16 results
The formal completion supplies finite coefficient operations before the denominator and character arguments.
4 definitions, 6 lemmas, 1 proposition, 2 theorems, 2 corollaries, 1 remarkExamples & counterexamples → This page develops the projective objects of a block of category O and their standard filtrations.
4 definitions, 17 lemmas, 2 propositions, 4 theorems, 2 corollariesExamples & counterexamples →- The BGG Resolution30 results
This page constructs and proves the BGG resolution of a finite-dimensional simple module — the resolution by Verma modules that computes its formal character and the dimensions of its n^--coinvariant Tor groups.
4 definitions, 20 lemmas, 1 proposition, 3 theorems, 2 corollariesExamples & counterexamples → This page develops Lie algebra cohomology as a representation-theoretic tool and culminates in Kostant's multiplicity-free description of the cohomology of the nilradical…
1 definition, 4 lemmas, 4 propositions, 2 theorems, 2 corollariesExamples & counterexamples →- Weyl Character and Multiplicity Formulas21 results
This page proves the denominator and character formulas rather than citing them.
4 definitions, 9 lemmas, 2 propositions, 5 theorems, 1 corollaryExamples & counterexamples → - Borel Weil and Borel Weil Bott12 results
This page computes the cohomology of the Borel-character equivariant line bundles L λ=G×^Bℂ -λ on the flag variety X=G/B of a connected simply connected complex semisimple affine algebraic group.
7 lemmas, 2 propositions, 2 theorems, 1 corollaryExamples & counterexamples → The page turns character multiplication into a general alternating tensor-multiplicity formula and then specializes it to type A, proving the Littlewood–Richardson and Pieri rules.
5 definitions, 4 lemmas, 4 propositions, 2 theorems, 4 corollariesExamples & counterexamples →
Part 2 · Algebra and representation interfaces
2 pages · after Part 1These draft extensions connect the existing foundations to algebraic structures and representation problems motivated by theoretical physics. Each companion supplies the planned examples and boundary cases; exact hypotheses, local proof obligations, and reuse holds remain in its page contract.
This page separates the finite-dimensional structure group G, its Lie algebra g=T eG, a principal G-bundle P→ M, its principal connection/curvature, and the gauge group G(P) of automorphisms over the identity of M.
3 definitions, 2 propositionsExamples & counterexamples →This page develops a characteristic-zero, complex Lie-superalgebra route with an explicit sign convention.
11 definitions, 2 lemmas, 6 theoremsExamples & counterexamples →