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In this paper, we study a notion of what we call vertex Leibniz algebra. This notion naturally extends that of vertex algebra without vacuum, which was previously introduced by Huang and Lepowsky. We show that every vertex algebra without vacuum can be naturally extended to a vertex algebra. On the other hand, we show that a vertex Leibniz algebra can be embedded into a vertex algebra if and only if it admits a faithful module. To each vertex Leibniz algebra we associate a vertex algebra without vacuum which is universal to the forgetful functor. Furthermore, from any Leibniz algebra $g$ we construct a vertex Leibniz algebra $V_{g}$ and show that $V_{g}$ can be embedded into a vertex algebra if and only if $g$ is a Lie algebra.
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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 $sl_2$. Under suitable conditions, we construct an indecomposable non-simple $mathbb{N}$-graded vertex algebra $overline{V_B}$ from the $mathbb{N}$-graded vertex algebra $V_B$ associated with the vertex $A$-algebroid $B$. We show that this indecomposable non-simple $mathbb{N}$-graded vertex algebra $overline{V_B}$ is $C_2$-cofinite and has only two irreducible modules.
In this paper, we study Virasoro vertex algebras and affine vertex algebras over a general field of characteristic $p>2$. More specifically, we study certain quotients of the universal Virasoro and affine vertex algebras by ideals related to the $p$-centers of the Virasoro algebra and affine Lie algebras. Among the main results, we classify their irreducible $mathbb{N}$-graded modules by explicitly determining their Zhu algebras and show that these vertex algebras have only finitely many irreducible $mathbb{N}$-graded modules and they are $C_2$-cofinite.
We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra $mathcal{M}(p)$, $pgeq 2$. The first category consists of all finite-length $mathcal{M}(p)$-modules with atypical composition factors, while the second is the subcategory of modules that induce to local modules for the triplet vertex operator algebra $mathcal{W}(p)$. We show that every irreducible module has a projective cover in the second of these categories, although not in the first, and we compute all fusion products involving atypical irreducible modules and their projective covers.
A theory of quasi modules at infinity for (weak) quantum vertex algebras including vertex algebras was previously developed in cite{li-infinity}. In this current paper, quasi modules at infinity for vertex algebras are revisited. Among the main results, we extend some technical results, to fill in a gap in the proof of a theorem therein, and we obtain a commutator formula for general quasi modules at infinity and establish a version of the converse of the aforementioned theorem.
In this paper, we study contragredient duals and invariant bilinear forms for modular vertex algebras (in characteristic $p$). We first introduce a bialgebra $mathcal{H}$ and we then introduce a notion of $mathcal{H}$-module vertex algebra and a notion of $(V,mathcal{H})$-module for an $mathcal{H}$-module vertex algebra $V$. Then we give a modular version of Frenkel-Huang-Lepowskys theory and study invariant bilinear forms on an $mathcal{H}$-module vertex algebra. As the main results, we obtain an explicit description of the space of invariant bilinear forms on a general $mathcal{H}$-module vertex algebra, and we apply our results to affine vertex algebras and Virasoro vertex algebras.
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