Vertex Representations, Partitions And Power Series Identities

In this thesis, the vertex representations for Vira,soro-Toroidal(including Alline case) Lie algebras are constructed in a unified view, and their irreducible decompositions are given(except G₂¹). Furthermore, we give a class of identities for power series of Lie type and some partition theorems.Through researching the correlation between alline Lie algebras of different classes, we can give the irreducible decompositions of vertex modules of type DCF. However, it is sufficient to deal with type C. Furtherly, this correlation makes it possible to express some power series in product form. Such results are very important in number theory.Main results:(1) The homogeneous vertex modules for Virasoro-toroidal (including toroidal and alline case) Lie algebras in a unified view. However, for A₂₁² , the vertex module is constructed exceptionally.(2) Give the irreducible decomposition for ABCDEF case.(3) A partition theorem.(4) The product forms of level one power series of Lie type and level two power series of A type.(5) Three "product-sum" expressions of (q, q⁶)_∞^-1(q⁵,q⁶)_∞^-1.