Classification of semisimple algebraic monoids

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a semisimple algebraic monoid with unit group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Associated with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is its polyhedral root system <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma normal upper Phi comma upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,\Phi ,C)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals upper X left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X = X(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the character group of the maximal torus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T subset-of-or-equal-to upper G"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo> ⊆ </mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">T \subseteq G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi subset-of-or-equal-to upper X left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:mo> ⊆ </mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Phi \subseteq X(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the set of roots, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C equals upper X left-parenthesis upper T overbar right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mover> <mml:mi>T</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C = X(\overline T )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the character monoid of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T overbar subset-of-or-equal-to upper E"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mi>T</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mo> ⊆ </mml:mo> <mml:mi>E</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline T \subseteq E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (Zariski closure). The correspondence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E right-arrow left-parenthesis upper X comma normal upper Phi comma upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E \to (X,\Phi ,C)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a complete and discriminating invariant of the semisimple monoid <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , extending the well-known classification of semisimple groups. In establishing this result, monoids with preassigned root data are first constructed from linear representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="applica