Speaker：Haisheng Li ( Rutgers University-Camden )

Time：2024-5-17, 8:00

Location：Conference Room 105 at Experiment Building at Haiyun Campus

Abstract:

This talk is about the structure of cocommutative vertex bialgebras. For a general vertex bialgebra $V$, we show that the set $G(V)$ of group-like elements is an abelian semigroup, whereas the set $P(V)$ of primitive elements is a vertex Lie algebra. For $g\in G(V)$, denote by $V_g$ the connected component containing $g$. Among the main results, we show that if $V$ is a cocommutative vertex bialgebra, then $V = \oplus_{g\in G(V)}V_g$, where $V_{\bf 1}$ is a vertex subbialgebra which is isomorphic to the vertex bialgebra $\mathcal{V}_{P(V)}$ associated to the vertex Lie algebra $P(V)$, and $V_g$ is a $V_{\bf 1}$-module for $g\in G(V)$. In particular, this shows that every cocommutative connected vertex bialgebra $V$ is isomorphic to $\mathcal{V}_{P(V)}$ 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 $V$, we prove that $V = \mathcal{V}_{P(V)}\otimes\C[G(V)]$ as a coalgebra where the vertex algebra structure is explicitly determined. This talk is based on a joint work with Jianzhi Han and Yukun Xiao.