Vertex Operator Algebras And Modules Associated To The Affine Nappi-witten Algebra (?)₄

In the theory of vertex (operator)algebras, one can construct a natural family of vertex (operator) algebras associated to the family of affine Lie algebras [50,94]. Furthermore, there is a one to one correspondence between the category of the modules for the vertex algebra and the category of the restricted modules for the affine Lie algebra. For the affine Lie algebra H4, we could study the restricted modules for the affine Lie algebra H4 to characterize the modules of VH4 (e,0) viewed as a vertex algebra. In chapter one, we classify the irreducible restricted modules for the affine Nappi-Witten Lie algebra H4 with some natural conditions. In the theory of vertex operator algebras, one of the most important problems is to construct new solvable vertex operator (super)algebras in the sense that irreducible modules and fusion rules can be completely determined and that intertwining operators can be explicitly constructed. To a certain extent, such algebras give rise to solvable physical models. One of many ways to get such vertex operator (super)algebras is to consider certain extensions of some well known algebras. It leads us to study the extension of the vertex operator algebra VH4(e,0) by the even lattice L. We give the structure of the extension VH4 (e,0) (?) C [L] and its irreducible modules via irreducible representations of VH4 (e,0) viewed as a vertex algebra.Since Nappi-Witten algebra H4 has many applications both in mathematics and physics, especially in two-dimensional conformal field theory(CFTS), it has been paid more attention by mathematicians and physicists. The associated vertex operator algebra VH4 (e,0) for a given complex number e and its representations were also studied in [7]. We know that orbifold vertex operator algebras play an important role in physics. In fact the moonshine module is the first example of so-called " orbifold conformal field theory ". For another thing, there is a one to one correspondence between the category of the admissible modules for the vertex operator algebra V and the category of the modules for the associative algebra A(V). So it is essential to characterize the structure of the associative algebra A(V). In chapter two, we classify all the automorphism groups for the vertex operator algebraâ…¤H4(e,0). And we give the generators for the associative algebra A(A(VH4 (e,0)+) of the fixed point vertex operator subalgebra VH4 (e,0)+under automorphism of the second orderThe supersymmetry, which plays an important role in two-dimensional con-formal field theories, is one of the main reasons to study vertex operator superal-gebras and their representations. Vertex operator superalgebras can be considered as natural generalizations of vertex operator algebras. In chapter three, we study automorphism groups of rank one lattice vertex operator superalgebras and give a complete description.