7 research outputs found
Non-associative Frobenius algebras for simply laced Chevalley groups
We provide an explicit construction for a class of commutative,
non-associative algebras for each of the simple Chevalley groups of simply
laced type. Moreover, we equip these algebras with an associating bilinear
form, which turns them into Frobenius algebras. This class includes a
3876-dimensional algebra on which the Chevalley group of type E8 acts by
automorphisms. We also prove that these algebras admit the structure of (axial)
decomposition algebras
Modules over axial algebras
We introduce axial representations and modules over axial algebras as new tools to study axial algebras. All known interesting examples of axial algebras fall into this setting, in particular the Griess algebra whose automorphism group is the Monster group. Our results become especially interesting for Matsuo algebras. We vitalize the connection between Matsuo algebras and 3-transposition groups by relating modules over Matsuo algebras with representations of 3-transposition groups. As a by-product, we define, given a Fischer space, a group that can fulfill the role of a universal 3-transposition group
Decomposition algebras and axial algebras
We introduce decomposition algebras as a natural generalization of axial
algebras, Majorana algebras and the Griess algebra. They remedy three
limitations of axial algebras: (1) They separate fusion laws from specific
values in a field, thereby allowing repetition of eigenvalues; (2) They allow
for decompositions that do not arise from multiplication by idempotents; (3)
They admit a natural notion of homomorphisms, making them into a nice category.
We exploit these facts to strengthen the connection between axial algebras and
groups. In particular, we provide a definition of a universal Miyamoto group
which makes this connection functorial under some mild assumptions. We
illustrate our theory by explaining how representation theory and association
schemes can help to build a decomposition algebra for a given (permutation)
group. This construction leads to a large number of examples. We also take the
opportunity to fix some terminology in this rapidly expanding subject.Comment: 23 page
Non-associative Frobenius algebras for simply laced Chevalley groups
We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras