Università Roma Tre

Part I

Introduction: Physics and symmetries

Finite Groups: Definition of group; groups of order 2; the Cayley table; groups
of order 3; cyclic groups; subgroups; direct products; Lagrange's theorem; groups
of order 4; higher-order groups: the dihedral series, the quaternionic group; pre-
sentations; the symmetric group Sn; the alternating group An; cycles; Cayley's
theorem; representations: basic definitions; the regular representation; conjugation
and cosets; normal subgroups; factor groups; simple and semisimple groups; clas-
sification of simple groups (without proof); characters; review of finite-dim Hilbert
spaces; unitary representations; group-averaged inner product; equivalence of finite-
dim reps to unitary reps; reducible and fully reducible reps; reps of finite groups and
full reducibility; Schur's Lemmas; the orthogonality relations; the character table;
real, pseudoreal and complex reps; Kronecker products and Clebsch-Gordan series.
2 Lie Groups and Lie Algebras: group manifold; Lie groups; the classical ma-
trix groups; Lie algebras: definition and first properties; structure constants; ad-
joint representation; Cartan-Killing form; simple and semiimple Lie algebras; fully
antisymmetric structure constants; Casimir operators; quadratic Casimir; highest
(lowest)-weight representations; su(2); su(3); Gell-Mann matrices; roots; α-basis,
ω basis, Chevalley basis; general Lie algebras: Cartan subalgebra and structure
contants in the Cartan basis; roots; the map adH(X); α-chains; Weyl reflections;
root systems for semisimple Lie algebras: relative angles and lengths between roots;
positive roots; simple roots; Dynkin diagrams; rank-two algebras; the Cartan cata-
logue of simple Lie algebras; representation theory: highest weights and construction
of representation space; level vector; the adjoint representation and maximal roots;
Weyl dimension formula.

Part II

Partial Differential Equations: Generalities; linear, quasilinear, semilinear and
non linear PDE; well-posedness; initial data vs boundary data; types of boundary
conditions; the Cauchy-Kovalevskaya theorem; first-order equations; characteris-
tics: parametric solutions and Lagrange's method; second-order PDE: classifica-
tion of quasi-linear eqs: hyperbolic, parabolic and elliptic PDE; characteristics and
canonical forms; the wave equation: d'Alembert solution and Fourier solution by
separation of variables; normal modes; the heat equation: solution by separation of
variables for insulated ends and for ends at finite temperatures T1 and T2.

Referenze:

Parte I

- Ramond P, Group theory - a physicist’s survey (Cambridge University Press, 2010).
- Gilmore R, Lie Groups, Physics, and Geometry (Cambridge University Press, 2008).
- Fuchs J and Schweigert C, Symmetries, Lie algebras and representations (Cambridge University Press, 2003).
- Slansky R, Group theory for unified model building (Physics Reports 791, Elsevier, 1981).
- Hamermesh M, Group theory and its application to phyical problems (Dover, 1962).

Parte II

- Pradisi G, Lezioni di metodi matematici della fisica (Edizioni della Normale, Pisa, 2012).
- Evans L C, Partial differential equations (American Mathematical Society, 1998-2010).
- Courant R and Hilbert D, Methods of mathematical physics II (Wiley, 1962).