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Title:
Reductions of KP type hierarchies, related to conjugacy classes of the Weyl group of classical Lie algebras
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Abstract:
Victor Kac and Dale Peterson showed in 1985 that one can associate to each conjugacy class of the Weyl group of the Lie algebras of type A, D and E a vertex-operator construction of the basic module of the corresponding affine Lie algebra. This gives, for instance, 112 different realizations for the Lie algebra of type E_8.It is also well-known that a hierarchy of differential equations describes the loop group orbit of the highest weight vector. For the Lie algebra of type A_1, there are two conjugacy classes. One is related to the KdV hierarchy and the other to the AKNS hierarchy.We will show that each conjugacy class of the Weyl group leads to a (different) hierarchy of differential equations. We achieve this as follows:We embed all loop algebras of a classical Lie algebra in an infinite matrix algebra of a certain type. Using some representation theory of these matrix algebras, we can describe the corresponding group orbit of the highest weight vector, which are parametrized by a fermionic (bosonic for type C) version of the KP hierarchy of type A, B, C or D. The restriction to the loop algebra/group gives a reduction of this hierarchy. And finally, each conjugacy class of the Weyl group gives a vertex-operator realization of the module and as such a different hierarchy of differential equations for each conjugacy class. This is based on joint work with Victor Kac.
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