[Paper Review] Heisenberg VOAs over Fields of Prime Characteristic and Their Representations

Haisheng Li, Qiang Mu|arXiv (Cornell University)|Jan 18, 2015

Algebraic structures and combinatorial models21 references6 citations

TL;DR

This paper studies Heisenberg vertex algebras over fields of prime characteristic, where they are no longer simple, and constructs a family of simple quotient vertex algebras $L_{\widehat{\mathfrak{h}}}(\ell,0,\lambda)$. It proves that every module over these algebras is completely reducible and that the adjoint module is the unique irreducible module up to isomorphism, establishing a complete reducibility theorem for a category of modules over affine Heisenberg Lie algebras in positive characteristic.

ABSTRACT

In this paper, we study Heisenberg vertex algebras over fields of prime characteristic. The new feature is that the Heisenberg vertex algebras are no longer simple unlike in the case of characteristic zero. We then study a family of simple quotient vertex algebras and we show that for each such simple quotient vertex algebra, irreducible modules are unique up to isomorphism and every module is completely reducible. To achieve our goal, we also establish a complete reducibility theorem for a certain category of modules over Heisenberg algebras.

Motivation & Objective

• To investigate the structure of Heisenberg vertex algebras over fields of prime characteristic, where they fail to be simple, unlike in characteristic zero.
• To construct and classify simple quotient vertex algebras $L_{\widehat{\mathfrak{h}}}(\ell,0,\lambda)$ from $V_{\widehat{\mathfrak{h}}}(\ell,0)$ by quotienting by maximal $\widehat{\mathfrak{h}}$-submodules.
• To establish a complete reducibility theorem for a certain category of modules over affine Heisenberg Lie algebras in positive characteristic.
• To analyze the representation theory of the simple quotient vertex algebras, particularly the uniqueness and complete reducibility of their modules.

Proposed method

• Define the vertex algebra $V_{\widehat{\mathfrak{h}}}(\ell,0)$ as the $\widehat{\mathfrak{h}}$-module generated by a vacuum vector $\mathbf{1}$, with $\mathbf{k} \cdot \mathbf{1} = \ell \mathbf{1}$ and $(\mathfrak{h} \otimes \mathbb{F}[t]) \mathbf{1} = 0$.
• Construct maximal $\widehat{\mathfrak{h}}$-submodules $J(\lambda)$ using functionals $\lambda \in (\widehat{\mathfrak{h}}_+)^*$, generated by vectors $u(-np) - \lambda(u(-np))$ and $(u(-n) - \lambda(u(-n)))^p$ for $u \in \mathfrak{h}, n \geq 1$.
• Prove that $J(\lambda)$ is a maximal $\widehat{\mathfrak{h}}$-submodule and that it is a vertex algebra ideal if and only if $\lambda(u(-n)) = 0$ for all $n \geq 2$, leading to the simple quotient $L_{\widehat{\mathfrak{h}}}(\ell,0,\lambda)$.
• Use the Segal-Sugawara construction to define a conformal vector $\omega = \frac{1}{2\ell} \sum_{i=1}^d u^{(i)}(-1)u^{(i)}(-1)\mathbf{1}$, giving $V_{\widehat{\mathfrak{h}}}(\ell,0)$ the structure of a vertex operator algebra of central charge $d = \dim \mathfrak{h}$.
• Establish a complete reducibility theorem for $\widehat{\mathfrak{h}}$-modules by analyzing the action of the Heisenberg subalgebra $\widehat{\mathfrak{h}}'' = \sum_{n \notin p\mathbb{Z}} \mathfrak{h} \otimes \mathbb{F}t^n + \mathbb{F}\mathbf{k}$.
• Leverage the $\mathbb{Z}$-grading and $L(0)$-eigenvalue decomposition to analyze the grading structure of $L_{\widehat{\mathfrak{h}}}(\ell,0,\lambda)$, showing that $J(\ell,\lambda)$ is $L(0)$-stable, leading to $\mathbb{Z}_p$-grading for $\lambda \neq 0$.

Experimental results

Research questions

• RQ1 How do Heisenberg vertex algebras over fields of prime characteristic differ from their characteristic zero counterparts in terms of simplicity and structure?
• RQ2 What are the maximal $\widehat{\mathfrak{h}}$-submodules of $V_{\widehat{\mathfrak{h}}}(\ell,0)$, and when are they vertex algebra ideals?
• RQ3 What is the representation theory of the simple quotient vertex algebras $L_{\widehat{\mathfrak{h}}}(\ell,0,\lambda)$, particularly regarding complete reducibility and uniqueness of irreducible modules?
• RQ4 How does the presence of an infinite-dimensional center in the affine Heisenberg algebra $\widehat{\mathfrak{h}}'$ affect the representation theory in positive characteristic?
• RQ5 Can a complete reducibility theorem be established for modules over affine Heisenberg Lie algebras in prime characteristic, and how does it support the structure of vertex algebra representations?

Key findings

• The vertex algebra $V_{\widehat{\mathfrak{h}}}(\ell,0)$ is not simple in prime characteristic, as it contains infinitely many maximal ideals due to the infinite-dimensional center of $\widehat{\mathfrak{h}}'$.
• The maximal $\widehat{\mathfrak{h}}$-submodules $J(\lambda)$ are generated by $u(-np) - \lambda(u(-np))$ and $(u(-n) - \lambda(u(-n)))^p$ for $u \in \mathfrak{h}, n \geq 1$, and they exhaust all maximal $\widehat{\mathfrak{h}}$-submodules.
• The quotient $L_{\widehat{\mathfrak{h}}}(\ell,0,\lambda)$ is a simple vertex algebra if and only if $\lambda(u(-n)) = 0$ for all $u \in \mathfrak{h}$ and $n \geq 2$, ensuring $J(\lambda)$ is a vertex algebra ideal.
• Every $L_{\widehat{\mathfrak{h}}}(\ell,0,\lambda)$-module is completely reducible, as established via a complete reducibility theorem for the associated category of $\widehat{\mathfrak{h}}$-modules.
• The adjoint module $L_{\widehat{\mathfrak{h}}}(\ell,0,\lambda)$ is the unique irreducible module up to isomorphism, implying that all irreducible modules are equivalent to the adjoint one.
• For $\ell \in \mathbb{F}^\times$, the quotient $L_{\widehat{\mathfrak{h}}}(\ell,0,0)$ is a simple vertex operator algebra of central charge $d = \dim \mathfrak{h}$, while $L_{\widehat{\mathfrak{h}}}(\ell,0,\lambda)$ with $\lambda \neq 0$ is not a vertex operator algebra due to non-grading-invariance of $J(\ell,\lambda)$.