SYSTEMATIC MATHEMATICS

Lie Groups, Lie Algebras & Representation Theory

Use 106 visible bilingual chapters in twelve content-sized units to move from familiar reversible symmetries and classical matrix groups through intrinsic Lie groups, closed subgroups, quotients, coverings, semidirect products, invariant vector fields, Lie brackets, exponentials, adjoint actions, BCH, homogeneous spaces, Maurer–Cartan geometry, finite-dimensional representations, Schur and averaging, characters, tensor constructions, solvable and semisimple structure, Cartan subalgebras, roots, Weyl groups, Dynkin classification, highest weights, Verma modules, Weyl formulas, compact-group Haar and Peter–Weyl theory, induced and unitary representations, coadjoint orbits, Borel–Weil, and eight original reproducible dossiers. Every chapter begins in ordinary language, reads every symbol with its ambient space and field, states one exact theorem or deliberately bounded claim, exposes the complete proof route at the length the argument requires, names the nearest failed hypothesis, reconstructs an exact example, supplies matched practice with a complete answer, and ends with a misconception check. Local Lie-algebra data are never promoted to a unique global group; reducibility is never confused with complete reducibility; a formal character is never called a constructed module; compact Haar arguments are never copied to noncompact groups; and orbit or localization heuristics are never called universal classifications. This is a Stage 5 doctoral specialization core and research preparation, not a complete construction of real reductive representation theory, Harish-Chandra modules, Langlands classification, category O, D-modules, Kazhdan–Lusztig theory, affine or quantum groups, geometric Satake, automorphic forms, trace formulas, or geometric Langlands.

Before this course: Completed Linear Algebra; Abstract Algebra; Real Analysis; Measure & Lebesgue Integration; Topology; Functional Analysis; Differential Geometry & Manifolds; Algebraic Topology; Differential Topology, Vector Bundles & Morse Theory; Differential Forms, de Rham Cohomology & Hodge Theory; Harmonic Analysis & Wavelets; Algebraic Geometry; and Symplectic Geometry & Hamiltonian Dynamics. No prior complete course in Lie theory, finite- or infinite-dimensional representation theory, compact-group harmonic analysis, orbit methods, category O, or geometric representation theory is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

A symmetry is a reversible transformation preserving declared structure

Objective: Does preserving area alone make a plane map a Euclidean symmetry?

A square can be rotated or reflected without changing distances, but “looks unchanged” is not a definition. We must name the set being transformed, the allowed maps, and the feature that is preserved. Start with the transformation and the quantity it preserves; only then replace the picture by a group, manifold, algebra, or representation. State whether the scalar field is R or C and whether each map is linear, smooth, continuous, unitary, algebraic, or merely set-theoretic.

Read the notation aloud before manipulating it: Sym(X,s)={g:X→X | g is bijective and s(gx,gy)=s(x,y)}; e is the identity; g^{-1} reverses g. The objects and their ambient spaces are: A group is a set G with an associative product, identity e, and inverse g^{-1} for every g. A symmetry group is a group under composition. A product such as gh means group multiplication, while XY may mean composition or matrix multiplication; neither is automatically commutative.

The exact theorem or bounded claim is: The bijections preserving a fixed structure are closed under composition and inverse, hence form a group. Its complete proof route is: 1. Compose two preserving maps and substitute twice into the preservation equation. 2. Use the identity map to verify the neutral element. 3. Set x=g^{-1}u and y=g^{-1}v to prove the inverse also preserves the structure. Keep connectedness, compactness, closedness, simply connectedness, characteristic, finite dimension, and choice of real or complex form visible at the step where each is used.

Reconstruct the exact model “On R², verify that r(x,y)=(-y,x) preserves x²+y² and find r⁴.”. The calculation proceeds as 1. Compute (-y)²+x²=x²+y². 2. Apply r twice to obtain r²(x,y)=(-x,-y). 3. Apply twice more to recover (x,y). The checked result is r preserves the Euclidean norm and r⁴=e; it generates the four rotations of a square. The nearest failure boundary is Noninvertible structure-preserving maps form at most a monoid; for example a constant map may preserve a zero-valued relation but has no inverse.

A matrix table, weight diagram, character value, truncated tensor product, sampled orbit, or numerical integral is evidence for the declared example only. It does not by itself prove global classification, convergence, irreducibility, completeness, geometric realization, or an infinite-dimensional Plancherel theorem. The diagnostic question is “Does preserving area alone make a plane map a Euclidean symmetry?”; the answer is No. The shear (x,y)↦(x+y,y) has determinant one and preserves area, but changes most lengths and angles.

The bijections preserving a fixed structure are closed under composition and inverse, hence form a group.

Compose two preserving maps and substitute twice into the preservation equation.

Use the identity map to verify the neutral element.

Set x=g^{-1}u and y=g^{-1}v to prove the inverse also preserves the structure.

On R², verify that r(x,y)=(-y,x) preserves x²+y² and find r⁴.

  1. Compute (-y)²+x²=x²+y².
  2. Apply r twice to obtain r²(x,y)=(-x,-y).
  3. Apply twice more to recover (x,y).

Result: r preserves the Euclidean norm and r⁴=e; it generates the four rotations of a square.