An Introduction to Lie Theory (10)

Level

Graduate Course

University

University of Aberdeen

Prerequisites

linear algebra, commutative algebra, topology

Duration

8 hours (1 lecture per week lasting 1 hour)

My personal notes for the course are available here.

Literature

There are many good books covering the topics of the course; the following is merely a snapshot.

• J. E. Humphreys, "Reflection groups and Coxeter groups", no. 29, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1990.
• N. Bourbaki, "Lie groups and Lie algebras", Chapters 4-6, Elements of Mathematics (Berlin), Springer-Verlag, 2002.
• R. W. Carter, "Lie algebras of finite and affine type", no. 96, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2005.
• K. Erdmann and M. J. Wildon, "Introduction to Lie algebras", Springer Undegraduate Mathematics Series, Springer-Verlag, 2006.
• J. E. Humphreys, "Introduction to Lie algebras and representation theory", no. 9, Graduate Texts in Mathematics, Springer-Verlag, 1978.
• M. Geck, "An introduction to algebraic geometry and algebraic groups", vol. 10, Oxford Graduate Texts in Mathematics, Oxford: Oxford University Press, 2003.
• J. E. Humphreys, "Linear algebraic groups", vol. 21, Graduate Texts in Mathematics, New York: Springer-Verlag, 1975.

Syllabus

• Finite reflection groups: root systems, positive roots, simple roots, parabolic subgroups.
• Lie algebras: solvable and nilpotent Lie algebras, Killing form, classification of complex semisimple Lie algebras.
• Basic affine algebraic geometry: algebraic sets, vanishing ideals, irreducible topological spaces, regular functions, varieties.
• Algebraic groups: definition, examples, the Jordan decomposition, the Lie algebra, root data for reductive groups, classification of simple algebraic groups.