Abstract: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 $E_8$ acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.

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中文翻译：

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简单结合 Chevalley 群的非结合 Frobenius 代数

摘要：我们为简单花边类型的每个简单 Chevalley 群的一类交换非结合代数提供了一个显式构造。此外，我们为这些代数配备了关联双线性形式，将它们变成了 Frobenius 代数。此类包括一个 3876 维的代数，类型 $E_8$ 的 Chevalley 群通过自同构作用于该代数。我们还证明这些代数承认（轴向）分解代数的结构。

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