Cocommutative Vertex Bialgebras And Some Associative Algebras Associated To VOAs

In this paper,the structure theory of cocommutative vertex bialgebras and some associative algebras associated to vertex operator algebras are investigated.For a vertex bialgebra V,it is proved that the set G(V)of group-like elements is naturally an abelian semigroup with identity 1,whereas the set P(V)of primitive elements is a vertex Lie subalgebra.For g ∈ G(V),denote by Vg the connected component containing g.Among the main results,it is proved that if V is a cocommutative vertex bialgebra,then V=⊕g∈G(V)Vg,where V1 is a vertex subbialgebra which is isomorphic to the vertex bialgebra VP(v)associated to the vertex Lie algebra P(V),and Vg is a V1-module for g ∈ G(V).In particular,this shows that every cocommutative connected vertex bialgebra V is isomorphic to VP(V)and hence establishes the equivalence between the category of cocommutative connected vertex bialgebras and the category of vertex Lie algebras.Furthermore,under the condition that G(V)is a group and lies in the center of V,it is proved that V=VP(V)?C[G(V)]as a coalgebra where the vertex algebra structure is explicitly determined.For any vertex operator algebra V and any finite automorphism g of V,we construct a Z-graded associative algebra U(V[g]),which is called the universal enveloping algebra of V with respect to g.For n ∈(1/T)Z+,we prove that the twisted higher Zhu algebra Ag,n(V)is isomorphic to U(V[g])0/U(V[g])0-n-1/T,where U(V[g])0 is the homogeneous subspace of degree 0 of U(V[g])and U(V[g])0-n-1/T is some subspace of U(V[g])0.Thus,Ag,n(V)is a(sub)quotient of U(V[g]).