A class of continuous non-associative algebras arising from algebraic groups including

Maurice Chayet,Skip Garibaldi

doi:10.1017/fms.2020.66

Abstract

Abstract We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field. Two attractions of this construction are that (1) when G has type $E_8$ , the algebra A is obtained by adjoining a unit to the 3875-dimensional representation; and (2) it is effective, in that the product operation on A can be implemented on a computer. A description of the algebra in the $E_8$ case has been requested for some time, and interest has been increased by the recent proof that $E_8$ is the full automorphism group of that algebra. The algebras obtained by our construction have an unusual Peirce spectrum.

Highlights

We present a construction that takes an absolutely simple linear algebraic group over a field and produces a commutative, unital non-associative algebra that we denote by ( )
We give an explicit formula (4.1) for the product on ( ), which makes our construction effective in the sense that one can perform computer calculations (Section 11), we do not rely on computer calculations for our results
This work may be viewed in the context of the general problem of describing exceptional groups as automorphism groups, which dates back to Killing’s 1889 paper [25]

Summary

Introduction

We present a construction that takes an absolutely simple linear algebraic group over a field and produces a commutative, unital non-associative algebra that we denote by ( ). Our original approach to the material in this paper was to focus on the case of 8 and leverage these tools In this way, we discovered the product formula on ( ), and only in hindsight did we see that it was a general construction that worked for all simple. Just before we intended to release this work on arXiv, the paper [11] appeared, which studies algebras that are almost the same, albeit restricted to the cases where the root system of is laced and is split and char = 0; see Remark 4.6 below Both that article and this one view the algebras as subspaces of S2 and provide an associative symmetric bilinear form (we say ( ) is metrized, whereas they say Frobenius), but from there our approaches and results diverge

Background material

Associativity of the bilinear form

11. Final remarks