A class of vertex operator algebras generated by Virasoro vectors

Abstract: In this paper, we study a class of simple OZ-type vertex operator algebras $V$ generated by simple Virasoro vectors $\omega^{ij}=\omega^{ji}$, $1\leq i<j\leq n$, $n\geq 3$. We prove that $V$ is uniquely determined by its Griess algebra $V_2$. The automorphism group of $V$ is also determined. Furthermore, we give the necessary conditions for $V$ to be unitary.

• The paper demonstrates that the structure of the Griess algebra uniquely determines the VOA, employing iterative vertex operations on Virasoro generators for precise algebraic characterizations.
• The automorphism group is proven to be isomorphic to the symmetric group S_n, linking algebra symmetries with fundamental group theory insights.
• The study establishes specific unitarity conditions via positive-definite Hermitian forms, which are crucial for applications in quantum and conformal field theories.

A Class of Vertex Operator Algebras Generated by Virasoro Vectors

Introduction

This paper explores a specific class of vertex operator algebras (VOAs) classified as OZ-type, generated by simple Virasoro vectors $\omega^{ij} = \omega^{ji}$, indexed by pairs of integers satisfying $1 \leq i < j \leq n$ for $n \geq 3$. A significant result is that each vertex operator algebra $V$ in this class can be uniquely characterized by its Griess algebra $V_2$. Additionally, the paper determines the automorphism group for these algebras and establishes necessary conditions for them to be unitary.

Main Results and Implementation

Unique Determination by Griess Algebra

A vertex operator algebra $V$ in this category is simple of OZ-type, generated by its Virasoro vectors within its Griess algebra, $V_2$. The central finding is that the structure of $V_2$ uniquely determines the VOA. Specifically, the algebras are generated by iteratively applying the vertex operator on these vectors and their combinations. The relations among these vectors are encapsulated by specific commutation and fusion relations, further constraining the algebra's structure.

The implementation of these algebras stems from the definition of linear spanning sets. Practitioners can construct these algebras by defining linear combinations of the generators $\omega^{ij}$ and systematically applying vertex algebra axioms and commutation relations.

Automorphism Group

The automorphism group of these vertex operator algebras is determined as isomorphic to the symmetric group $\mathfrak{S}_n$. This stems from the natural action of $\mathfrak{S}_n$ on the set of indices $\{1, 2, \ldots, n\}$, reflecting the symmetric nature of the involutions defined on the set of generating Virasoro vectors. This result ties the symmetries of the algebra closely to well-understood group-theoretic structures, facilitating exploration of representation theory and applications in algebraic combinatorics.

Implementation involves identifying equivalent transformations under $\mathfrak{S}_n$ that map generating elements into each other while preserving the structure of the VOA.

Conditions for Unitarity

The paper delineates necessary conditions for the algebra to be unitary. It specifies that the Hermitian form on $V_2$ is positive-definite under particular conditions on the parameters $m$ and $n$. Specifically, this form is positive-definite only for $m \leq 3$ when $n = 3$, and for $m = 2$ when $n \geq 4$. This insight is crucial for applications requiring positive-definite structures, such as in certain quantum field theories and conformal field theories.

Practical Considerations

The construction and manipulation of these algebras are tied to computational and symbolic manipulation of the generators and relations defined. Researchers implementing these structures will leverage symbolic computation to handle the algebraic identities and calculations intrinsic to the VOA framework. Such systems often involve rigorous testing of algebraic properties and require efficient algorithms for symbolic expansion and reduction.

Conclusion

This exploration of OZ-type vertex operator algebras generated by Virasoro vectors delivers both theoretical insights and practical tools for the construction and application of such structures in mathematical physics and related domains. The results also open pathways for further research into related algebraic structures and their representation theories, inviting advances in both conceptual understanding and computational techniques.