Root systems are probably the most symmetric patterns in the world. They are certain arrangements of vectors in finite dimensional Euclidean spaces. These patterns appeared in the classification of various mathematical objects, such as the singularities of differentiable mappings, finite simple groups, and simple Lie algebras.
The basic example of a simple Lie algebra is the vector space of all square matrices of fixed size with zero trace. The commutator of any two such matrices belongs to the same space. Another example is the vector space of all skew-symmetric matrices of fixed size. Together with the so-called symplectic Lie algebra, these are all classical simple Lie algebras.
The aim of the project is to study these Lie algebras, and the corresponding root systems. For the matrices of small size, the corresponding root systems have nice graphical presentations. While working on the project, it would be very helpful (but not a requisite) doing my course 059312 on the Lie Algebras in the Spring Term.
Initial Reading
Prerequisites
A Lie algebra is a vector space with a bilinear operation satisfying certain remarkable axioms. A typical example is the vector space of square matrices of fixed size, the bilinear operation being the usual matrix commutator [X,Y] = XY-YX . Lie algebras appear in various areas of modern Mathematics, including Differential Geometry and Mathematical Physics.
There are two most important classes of Lie algebras: the solvable Lie algebras and semisimple Lie algebras. The basic example of a solvable Lie algebra is the vector space of all upper triangular square matrices of fixed size. The basic example of a semisimple Lie algebra is the vector space of all those square matrices of fixed size which have zero trace. In either case the bilinear operation is the matrix commutator.
Given a Lie algebra, it is very important to find out whether it is solvable or semisimple. The first aim of the project is to learn how to do this, and what to do if the given Lie algebra is neither of these. While working on the project, it would be very helpful (but not a requisite) doing my course 059312 on the Lie Algebras in the Spring Term.
Initial Reading
Prerequisites
The symmetric group S(n) is the group of all permutations of the sequence 1, 2, ... , n . This group is one of the most fundamental objects in Algebra and Combinatorics. There is an elegant theory of representations of this group. This theory describes all possible homomorphisms from S(n) to the group of all linear transformations of any finite-dimensional vector space. This theory was launched independently by Alfred Young and Georg Frobenius about a century ago. There many surprising developments since then.
The aim of the project is to study various constructions of the "minimal" representations of the group S(n), called irreducible. These constructions can be given by explicit formulas, and in some particular cases can be checked by computing with MAPLE.
Initial Reading
Prerequisites
There are three families of groups which have been playing major roles in Mathematics for about a century. These groups are called the classical groups. The most important example is the so-called general linear group GL(n). It consists of all invertible square matrices of size n x n with complex coefficients, the group law being the matrix multiplication. Another example is the orthogonal group O(n). It consists of all invertible square matrices of size n x n with complex coefficients, such that the transposed matrix coincides with the inverse. Together with the so-called symplectic groups, the general linear and orthogonal groups are precisely the classical Lie groups.
These groups appear as "symmetries" of various mathematical problems, especially in Differential Geometry and Mathematical Physics. The intuitive notion of a symmetry can be axiomatized as a representation of a group G. This is any homomorphism from the group G to the general linear group GL(d) for some number d . The number d is called the dimension of the representation.
The aim of the project is to study various constructions of the "minimal" representations of the classical groups, called irreducible. These representations can be given by explicit formulas, and their dimensions are not at all arbitrary. It would be very helpful (but not a requisite) doing my course 054514 on Group Representation Theory in the Autumn Term.
Initial Reading
Prerequisites