2024/12/10 by Keene, Alex, Soltermann, Christian, Yamskulna, Gaywalee
#17B69 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2412.07918
This paper investigates the algebraic structure of indecomposable ℕ-graded vertex algebras V = \bigoplusn=0∞ Vn, emphasizing the intricate interactions between the commutative associative algebra V0, the Leibniz algebra V1 and how non-degenerate bilinear forms on V0 influence their overall structure. We establish foundational properties for indecomposability and locality in ℕ-graded vertex algebras, with our main result demonstrating the equivalence of locality, indecomposability, and specific structural conditions on semiconformal-vertex algebras. The study of symmetric invariant bilinear forms of semiconformal-vertex algebra is investigated. We also examine the structural characteristics of V0 and V1, demonstrating conditions under which certain ℕ-graded vertex algebras cannot be quasi vertex operator algebras, semiconformal-vertex algebras, or vertex operator algebras, and explore ℕ-graded vertex algebras V=\bigoplusn=0∞Vn associated with Gorenstein algebras. Our analysis includes examining the socle, Poincaré duality properties, and invariant bilinear forms of V0 and their influence on V1, providing conditions for embedding rank-one Heisenberg vertex operator algebras within V. Supporting examples and detailed theoretical insights further illustrate these algebraic structures.