Finite-dimensional Representations of Yangians

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In this thesis, we study the cyclicity condition for an ordered tensor product of fundamental representations and the local Weyl modules of Yangians. We provide a sufficient condition for the cyclicity of an ordered tensor product L=Vₐ₁(ωb₁)⊗ Vₐ₂(ωb₂)⊗...⊗ Vₐₖ(ωbₖ) of fundamental representations of the Yangian Y(g). When g is a classical simple Lie algebra or the simple Lie algebra of type G, we make the cyclicity condition concrete, which leads to an irreducibility criterion for the ordered tensor product L. In the case when g=slₗ₊₁, a sufficient and necessary condition for the irreducibility of the ordered tensor product L is obtained. The cyclicity of the ordered tensor product L is closely related to the structure of the local Weyl modules of Y(g). We show that every local Weyl module is isomorphic to an ordered tensor product of fundamental representations of Y(g).