<p style='text-indent:20px;'>We study a family of non-simple Lie conformal algebras <inline-formula><tex-math id="M2">$ \mathcal{W}(a,b,r) $</tex-math></inline-formula> (<inline-formula><tex-math id="M3">$ a,b,r\in {\mathbb{C}} $</tex-math></inline-formula>) of rank three with free <inline-formula><tex-math id="M4">$ {\mathbb{C}}[{\partial}] $</tex-math></inline-formula>-basis <inline-formula><tex-math id="M5">$ \{L, W,Y\} $</tex-math></inline-formula> and relations <inline-formula><tex-math id="M6">$ [L_{\lambda} L] = ({\partial}+2{\lambda})L,\ [L_{\lambda} W] = ({\partial}+ a{\lambda} +b)W,\ [L_{\lambda} Y] = ({\partial}+{\lambda})Y,\ [Y_{\lambda} W] = rW $</tex-math></inline-formula> and <inline-formula><tex-math id="M7">$ [Y_{\lambda} Y] = [W_{\lambda} W] = 0 $</tex-math></inline-formula>. In this paper, we investigate the irreducibility of all free nontrivial <inline-formula><tex-math id="M8">$ \mathcal{W}(a,b,r) $</tex-math></inline-formula>-modules of rank one over <inline-formula><tex-math id="M9">$ {\mathbb{C}}[{\partial}] $</tex-math></inline-formula> and classify all finite irreducible conformal modules over <inline-formula><tex-math id="M10">$ \mathcal{W}(a,b,r) $</tex-math></inline-formula>.</p>
In this paper, the structure of cocommutative vertex bialgebras is investigated.For a general vertex bialgebra V , it is proved that the set G(V ) of group-like elements is naturally an abelian semigroup, whereas the set P (V ) of primitive elements is a vertex Lie algebra. For g ∈ G(V ), denote by V g 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 ) V g , where V 1 is a vertex subbialgebra which is isomorphic to the vertex bialgebra V P (V ) associated to the vertex Lie algebra P (V ), and V g is a V 1 -module for g ∈ G(V ). In particular, this shows that every cocommutative connected vertex bialgebra V is isomorphic to V P (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 = V P (V ) ⊗ C[G(V )] as a coalgebra where the vertex algebra structure is explicitly determined.
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