The Study On Some Problems In Vertex Algebra Theory

Vertex algebra, or vertex operator algebra, is an important algebraic structure inconformal field theory and statistical mechanics. The coset construction is an importantmethod to construct new conformal field models, and has been researched deeply andapplied widely in conformal field theory. As a subalgebra of a vertex algebra, the com-mutant algebra is a mathematical generalization of the coset construction in conformalfield theory. In this thesis, we study the descriptions of certain commutant algebras invertex algebra theory.Given a Lie algebra g, there is a vertex algebra O(g,B), called the current algebra.Let V be a vector space. There is an even vertex algebra S(V ), calledβγ-system. If V isa representation of Lie algebra g, the current algebra O(g,B) can be realized as a subal-gebra of S(V ) for some symmetric invariant bilinear form B on g. In this thesis, for twodi?erent representations of Lie algebra sl(2,C), we study some commutant subalgebrasof the current algebra O(sl(2,C),B) in theβγ-system S(V ).In chapter 3, we describe the commutant algebra S(sl(2,C))^Θ+.Let S(sl(2,C))^Θ+ be the commutant algebra of the current algebra O(sl(2,C),?83K)in theβγ-system S(sl(2,C)). We get generators of the commutant algebra S(sl(2,C))^Θ+and their OPE relations. Our methods are as follows. Step 1, by the invariant theory, inparticular, theory of Hilbert series, we give the generators of the ?-ring gr(S(sl(2,C))^Θ+;Step 2, we show that ?-ring gr(S(sl(2,C))^Θ+ is isomorphic to the ?-ring gr(S(sl(2,C)^Θ+)of S(sl(2,C))^Θ+; Step 3, according to the reconstruction theorem of gr(S(sl(2,C))^Θ+),we get the generators of S(sl(2,C))^Θ+.In chapter 4, we describe the commutant algebra S(V₄)^Θ+.For the irreducible representation V₄ of sl(2,C) with the highest weight 4, let S(V₄)^Θ+be the commutant algebra of the current algebra O(sl(2,C),?85K) inβγ-system S(V₄).We prove that S(V₄)^Θ+ is a conformal vertex algebra generated strongly by finite elements,and give explicitly its generators and the OPE relations among these generators. We alsoshow that two canonical subalgebras Sβ(V₄)^Θ+, Sγ(V₄)^Θ+ of S(V₄)^Θ+ are both generatedstrongly by finite elements, and get their generators. Our methods are similar to themethods of describing S(sl(2,C))^Θ+. However, the case of the representation V₄ are muchmore complicated. In this case, using the theory of Hilbert series in classical invarianttheory, we first find generators of ?-ring gr(S(V₄)^Θ+). Secondly, carrying out a process ofquantum corrections on vertex operators, we get the vertex operators corresponding togenerators above obtained in S(V₄)^Θ+, then according to the reconstruction theorem of ?-ring gr(S(V₄)^Θ+), we get the generators of S(V₄)^Θ+. Moreover, through the calculationsof OPE relations of fields generated by free fields, we give the OPE relations among finitegenerators of S(V₄)^Θ+.