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(1) Lie algebras: nilpotent, solvable, semisimple, Cartan subalgebras, root systems.
(2) Classification theorem, Dynkin diagrams.
(3) Weyl groups, weight lattices, structure of representations, characters, Schur-Weyl duality.
(4) Lie Group representations. In particular, O(n) representations, Clifford algebras and the spin group. Homogeneous spaces.
(5) If time permits: affine algebras, the Virasoro, Heisenberg, and infinite dimensional Clifford algebras, quantum groups.
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Department of
Mathematics University of Illinois at Urbana-Champaign 273 Altgeld Hall, MC-382 1409 W. Green Street, Urbana, IL 61801 USA Telephone: (217) 333-3350 Fax (217) 333-9576 office@math.uiuc.edu |