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

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.

Albert algebras over and other rings

“…In the setting of nonassociative algebras, Albert algebras also arise naturally. Among commutative not-necessarily-associative algebras under additional mild hypotheses (the field has characteristic ≠ 2, 3, 5 and the algebra is metrized), every algebra satisfying a polynomial identity of degree ≤ 4 is a Jordan algebra (see [ChG,Proposition A.8]). Jordan algebras have an analogue of the Wedderburn-Artin theory for associative algebras [J68,p.…”

“…In the setting of nonassociative algebras, Albert algebras also arise naturally. Among commutative not-necessarily-associative algebras under additional mild hypotheses (the field has characteristic and the algebra is metrized), every algebra satisfying a polynomial identity of degree is a Jordan algebra (see [ChG, Proposition A.8]). Jordan algebras have an analogue of the Wedderburn-Artin theory for associative algebras [J68, p. 201, Corollary 2], and one finds that all the simple Jordan algebras are closely related to associative algebras (more precisely, “are special”) except for one kind, the Albert algebras (see, for example [J68, p. 210, Theorem 11] or [McCZ]).…”

Non-associative Frobenius algebras of type G2 and F4

“…Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic); see [CG21]. This construction is quite remarkable, because it applies to all simple linear algebraic groups up to isogeny, regardless of type and form.…”

Non-associative Frobenius algebras for simply laced Chevalley groups

“…Exactly one week after this paper was finished and posted on the arXiv, Maurice Chayet and Skip Garibaldi posted another paper on the arXiv about the exact same topic and also relying on the symmetric square of the Lie algebra. Their paper has now appeared; see [CG21]. Their construction is more general: it works for (almost) arbitrary base fields and is not restricted to the simply laced case.…”