The alternative torus and the structure of elliptic quasi-simple Lie algebras of type 𝐴₂

Abstract

We present the complete classification of the tame irreducible elliptic quasi-simple Lie algebras of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{A_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and in particular, specialize on the case where the coordinates are not associative. Here the coordinates are Cayley-Dickson algebras over Laurent polynomial rings in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu greater-than-or-slanted-equals 3"> <mml:semantics> <mml:mrow> <mml:mi> ν </mml:mi> <mml:mo> ⩾ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\nu \geqslant 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> variables, which we call alternative tori. In giving our classification we need to present much information on these alternative tori and the Lie algebras coordinatized by them.