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Courses in Mathematics and Statistics Level 1 Modules Level 2 Modules Level 3 Modules Level 4 Modules Level 5 Modules Honours timetable 2010/2011 Sem. 1 2010/2011 Sem. 2 2011/2012 Sem. 1 & Sem. 2 2012/2013 Sem. 1 & Sem. 2

MT5827 LIE ALGEBRAS There are several reasons to study Lie algebras: Important applications to theoretical physics; used in the classification of finite simple groups; used in other group theory results such as the solution to the restricted Burnside problem; historical interest - many consider the classification by Killing in 1880 as the first real piece of modern algebra. Aims The aim of the course is to classify the semisimple Lie algebras over an algebraically closed field. Objectives To be able to calculate in finite dimensional Lie algebras L over the complex numbers, work with representations of L and L-modules, find the matrix of an adjoint map associated with an element of L, find the Killing form of L. Be able to work with star vectors and roots, and compute the Cartan matrix of an algebra given by a Dynkin diagram. Calculate the angle form of a connected graph and determine whether it is positive definite. Be able to calculate the Weyl group of a Lie algebra. Syllabus Definitions and examples; ideals and subalgebras; simple and semisimple algebras; representations; the adjoint representation and derivations; weights and roots; Cartan subalgebras; Cartan decomposition; Killing form; star vectors; fundamental roots; Cartan matrix; Dynkin diagrams Assessment 100% by two-and-a-half hour examination Prerequisites MT3501 and (MT4003 or MT4517) Availability Academic year 2010/11 in semester 2 at 11 Lecturer Dr M Neunhöffer Click here for access to past examination papers for this module. Click here to see the University Course Catalogue entry. Revised: JOC (September 2010)