A class of continuous non-associative algebras arising from algebraic groups including $E_8$. (14th January 2021)

• Record Type: Journal Article
• Authors: Chayet, Maurice; Garibaldi, Skip
• Is Part Of: Forum of mathematics. Volume 9(2021)
• Publication Date: 2021-01-14
• Subjects: 17B25, -- 17D99, -- 20G41; Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2020.66
• Languages: English
• ISSNs: 2050-5094

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.