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Lie Groups and Representations

This course is taken in sequence, part 1 in the fall, and part 2 in the spring.

Lie Groups and Representations I

I Basic Notions

  • Abstract groups, algebraic groups over a field, topological groups, Lie groups
  • Subgroups, normal subgroups, quotient groups
  • Homomorphisms of groups – image, kernel, exact sequences
  • Cyclic groups, abelian groups, nilpotent groups
  • Conjugacy classes, left and right cosets of a subgroup

II Algebraic Examples

  • Units of a ring, k* for k a field, roots of unity in a commutative ring, R*, S1 in C*
  • GL(n, R) as the group of units of n x n-matrices over a commutative ring
  • The determinant and SL(n, R), O(n, R), Sympl(2n, R) when there is (-1) in R
  • Algebraic groups of the above types over a field, definition of linear algebraic groups
  • Group structure on an elliptic curve
  • Group of p-adic integers, and its multiplicative group of units

III Geometric Examples and Symmetry

  • Permutation groups
  • Symmetries of regular plane figures and of Platonic solids
  • The Lie groups SL(n, R), SO(n, R), SO(p, q), Sympl(2n, R)
  • Isometries of the line, the plane, and higher dimensional Euclidean spaces
  • Isometries of spheres and of Minkowski space. The PoincarĂ© group
  • Isometries of the hyperbolic plane, conformal isomorphisms of S2, relation with SL(2, R) and SL(2, C)
  • Clifford algebras and the spin groups
  • The Heisenberg group

IV Lie Algebras

  • Definition, examples of the Lie algebra of an associative algebra
  • The Lie algebra of a Lie group. The universal enveloping algebra and the PoincarĂ©-Birkhoff-Witt theorem

V Representations

  • Definition in the various categories of groups, representations of a Lie algebra
  • Infinitesimal generators for the action of a Lie group
  • The infinitesimal representation associated to a linear representation of a Lie group
  • Turning actions into linear representations on the functions
  • Classification of the (finite dimensional) representations of sl(2, C), SU(2), and SO(3)
  • Representations of the Heisenberg algebra

VI Representations of Finite and Compact Lie Groups

  • Complete reducibility, Schur’s lemma, characters, orthogonality relations for characters of a finite group
  • Dimension of the space of characters of a finite group
  • The decomposition of the regular representation of a finite group
  • Characters of a compact group – complete reducibility, Schur’s lemma, orthogonality of characters
  • Peter-Weyl theorem (except the proof of the decomposibility of a Hilbert space representation into finite dimensional sub representations)
  • Example of L2(S1) and Fourier analysis
  • Example of L2(S2) as a module over SO(3) and spherical harmonics

VII Finite Groups and Counting Principles

  • Orders of elements and subgroups
  • Groups of order pn are nilpotent
  • Subgroups of index 2 are normal
  • The Sylow theorems
  • Classification of groups of order pq for p, q distinct primes. Groups of order 12

Lie Groups and Representations II

I Lie Groups and Lie Algebras: the Exponential Mapping

  • Baker-Campbell-Hausdorff formula
  • A Lie group is determined by its Lie algebra up to covering
  • Action of a Lie group is determined by its infinitesimal action

II Maximal Tori of a Compact Lie Group

  • Existence and uniqueness up to conjugation
  • Every element is contained in a maximal torus
  • Regular elements
  • The Weyl group
  • Weyl group action on the maximal torus and on corresponding abelian Lie algebra
  • Decomposition of the adjoint representation root spaces. Weyl chambers
  • Groups generated by reflection
  • Positive roots, dominant root and alcove
  • Dynkin diagrams
  • The classical examples SU(n), SO(n), Sympl(2n)

III Complex Semi-Simple Lie Groups and Lie Algebras
IV Irreducible Representations of Compact Groups

  • Weight spaces, dominant weights
  • Examples for SU(n), Sympl(2n) and SO(n)

V Selected Topics, chosen from

  • Borel-Weil-Bott theory
  • Infinite-dimensional representations of SL(2, R)
  • Kac-Moody algebras
  • The Virasoro algebra
  • Supersymmetry
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