Automorphism-invariant Integral Forms in Griess Algebras.

nonassociative algebras;integral forms;lattices;Mathematics;Science;Mathematics

Simon, Gregory G.Hall, Jonathan I ;

University of Michigan

关键词: nonassociative algebras;    integral forms;    lattices;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/133314/ggsimon_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

Motivated by the existence of group-invariant integral forms in various vertex operator algebras, we classify maximal automorphism-invariant integral forms in some small-dimensional Griess algebras, which are certain finite-dimensional commutative, nonassociative algebras arising in the theory of vertex operator algebras. An integral form of a rational algebra is the integer span of a basis of the algebra that is closed under the algebra product. The main method is the development of ;;integral form detector functions;; and an investigation of their properties. Each of the small Griess algebras we analyzed - the eight Norton-Sakuma algebras and three others - have unique maximal automorphism-invariant integral forms. This provides a canonically defined lattice and subring inside these algebras.