Abstract

Let A be a finite dimensional unital commutative associative algebra and let B be a finite dimensional vertex A-algebroid such that its Levi factor is isomorphic to sl<inf>2</inf>. Under suitable conditions, we construct an indecomposable non-simple N-graded vertex algebra V<inf>B</inf>‾ from the N-graded vertex algebra V<inf>B</inf> associated with the vertex A-algebroid B. We show that this indecomposable non-simple N-graded vertex algebra V<inf>B</inf>‾ is C<inf>2</inf>-cofinite and has only two irreducible modules.