Algebraic Groups (15)

Level

Graduate Course

University

Università degli Studi di Padova

Prerequisites

linear algebra, commutative algebra, topology

Duration

20 hours (1 lecture per week lasting 2 hours)

Next Lecture

Thursday (14/05) at 09:30 - 11:30 in 2BC30

This course is being given jointly with Dr Iulian Simion. My personal notes for the course are available here (last updated: 04/05/15).

Exam

You may return your solutions to the exam by email or in person any time before the deadline.

Start Time

Monday 09:00 on Monday 1st June 2015

End Time

Friday 17:00 on Friday 12th June 2015

The exam can be downloaded here.

Literature

There are a vast number of books on algebraic groups, the following list contains most of the standard references. We will closely follow Springer's text but one may find Humphrey's text an easier read. Malle and Testerman's book would be a concise place to read up on reductive groups as follow up material after the course.

• T. A. Springer, "Linear algebraic groups", Modern Bikhäuser Classics, Boston, MA: Birkhäuser Boston Inc., 2009.
• J. E. Humphreys, "Linear algebraic groups", vol. 21, Graduate Texts in Mathematics, New York: Springer-Verlag, 1975.
• A. Borel, "Linear algebraic groups", second edition, vol. 126, Graduate Texts in Mathematics, New York: Springer-Verlag, 1991.
• M. Geck, "An introduction to algebraic geometry and algebraic groups", vol. 10, Oxford Graduate Texts in Mathematics, Oxford: Oxford University Press, 2003.
• G. Malle and D. Testerman, "Linear algebraic groups and finite groups of Lie type", vol. 133, Cambridge Studies in Advanced Mathematics, Cambridge: Cambridge University Press, 2011.

Syllabus

• Affine algebraic geometry: algebraic sets, vanishing ideals, irreducible topological spaces, regular functions, prevarieties and varieties.
• Algebraic groups: definition, examples, G-spaces, the Jordan decomposition.
• Commutative algebraic groups: structure, classification of diagonalisable groups, classification of 1-dimensional algebraic groups.
• Differentials and Lie algebras: heuristics, derivations, differential of a morphism, the Lie algebra of an algebraic group.
• Topological properties of morphisms: dominant, finite and birational morphisms, homogeneous spaces, quotients
• Borel subgroups and maximal tori: complete varieties, parabolic subgroups, connected solvable groups, maximal tori.