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***,*S*^{1}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*(2*n*,*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*(2*n*,**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
*S*^{2}, 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
*L*^{2}(*S*^{1}) and Fourier analysis - Example of
*L*^{2}(*S*^{2}) as a module over*SO*(3) and spherical harmonics

VII Finite Groups and Counting Principles

- Orders of elements and subgroups
- Groups of order
*p*are nilpotent^{n} - 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*(2*n*)

III Complex Semi-Simple Lie Groups and Lie Algebras

IV Irreducible Representations of Compact Groups

- Weight spaces, dominant weights
- Examples for
*SU*(*n*),*Sympl*(2*n*) 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