Quasi-triangular decomposition and induced modules for vertex operator algebras

Abstract: In this paper, we introduce the notion of quasi-triangular decomposition for vertex operator algebras, which arise naturally in the lattice and affine VOAs and is a generalization of the triangular decomposition of a semisimple Lie algebra. The quasi-triangular decomposition leads to a new construction of Verma-type induced modules for the VOA embedding $U\hookrightarrow V$. We focus on a typical example of VOA embedding $V_P\hookrightarrow V_{A_2}$ given by a parabolic-type subVOA arising from a quasi-triangular decomposition of the lattice VOA $V_{A_2}$ and determine the induced modules. To achieve this goal, we prove that the Zhu's algebra $A(V_P)$ is a nilpotent extension of a skew-polynomial algebra. Using the generators and relations description of $A(V_P)$, we classify all the irreducible modules over $V_P$ and determine their inductions to the $V_{A_2}$-modules.