Maximal torus

Abstract: I Lie Groups and Lie Algebras.- II Elementary Representation Theory.- III Representative Functions.- IV The Maximal Torus of a Compact Lie Group.- V Root Systems.- VI Irreducible Characters and Weights.- Symbol Index.

Abstract: The converse was proved by A. Horn [5], so that all points in this convex hull occur as diagonals of some matrix A with the given eigenvalues. Kostant [7] generalized these results to any compact Lie group G in the following manner. We consider the adjoint action of G on its Lie algebra L(G). If T is a maximal torus of G and W its Weyl group, then it is well known that W-orbits in L(T) correspond to G-orbits in L(G). Now fix a G-invariant metric on L(G), so that we can define orthogonal projection. Then Kostant's result isf

Abstract: We construct projective unitary representations of (a) Map(S1;G), the group of smooth maps from the circle into a compact Lie groupG, and (b) the group of diffeomorphisms of the circle. We show that a class of representations of Map(S1;T), whereT is a maximal torus ofG, can be extended to representations of Map(S1;G),

Abstract: Recent developments in the understanding of $N=2$ supersymmetric Yang-Mills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining four-manifold invariants by counting $SU(2)$ instantons, one can define equivalent four-manifold invariants by counting solutions of a non-linear equation with an abelian gauge group. This is a ``dual'' equation in which the gauge group is the dual of the maximal torus of $SU(2)$. The new viewpoint suggests many new results about the Donaldson invariants.