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\centerline{\bf Math 651 Syllabus, fall 2000} 

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{\bf Course=Topics in Algebra, Math 651} \par\noindent
Instructor=Sam Evens

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\par\noindent{\bf Text}=Representations and Invariants of the Classical Groups by Roe Goodman and Nolan R. Wallach. Cambridge University Press. \par\noindent
{\bf Alternate Text}=Introduction to Lie Algebras and Representation Theory, by James Humphreys, Springer Graduate Text, 9. \par\noindent{\bf Prerequisites}: First year graduate courses in mathematics. A good understanding of linear algebra will be helpful. 

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Lie groups and their representations are important in a wide range of areas, including algebraic geometry, combinatorics, 
topology, differential geometry
and mathematical physics. A Lie group can be thought of as the group of symmetries of an object, and as such the Lie group can be used to study many important properties related to the geometry of the object.

We will study compact Lie groups, semisimple Lie algebras and their representations, and applications of these ideas. The point of view will be mainly algebraic, although ideas from other areas will be introduced as needed. We will begin by discussing examples of Lie groups and their representations, develop basic structure theory for Lie groups and algebras including maximal tori, root systems, and Weyl groups. Then we will study basic structure of representation theory, including the classification of irreducible representations by the highest weight, and the Weyl character formula.

If enrollment permits, there will be a follow-up to this course in the spring of 2001 discussing
problems of current research interest, either on geometric constructions of representations or on (i) Lie algebra cohomology and Poisson geometry or (ii) Geometric constructions of representations. 

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