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Register a new userSymmetric Invariant Bilinear Forms on Modular Vertex Algebras
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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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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 review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac--Moody (super)algebras are the most known examples.
In this paper we study the integral form of the lattice vertex algebra $V_L$. Based on the lattice structure, we introduce and study the associated modular vertex algebras $V_p$ and the quotient algebra $overline{V}_p$ as well as their irreducible modules over $mathbb Z_p$. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant $mathbb Z$-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras. We also prove that the irreducible modules (modulo Heisenberg generators of degree divisible by $p$) remain irreducible for the modular vertex algebra $overline{V}_p$.
As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2. We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or $0|1$, or $1|1$; it is equal to $1|1$ if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic distinct from 2, see arXiv:1806.05505). We give examples of NISes on deformations (with both even and odd parameters) of several simple finite-dimensional Lie superalgebras in characteristic 2. We also recall examples of multiple NISes on simple Lie algebras over non-closed fields.
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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