On Retract Rationality for Finite Connected Group Schemes

Abstract In the present paper, we prove the retract rationality of the classifying spaces BG for several types of finite connected group schemes G over algebraically closed fields of positive characteristic $$p&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . In particular, we prove the retract rationality for the finite simple group schemes G associated with the generalized Witt algebras $$W(m;\varvec{n})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>;</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in the case when $$\varvec{n}=\varvec{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:mrow> </mml:math> or $$m=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . To this end, we study the automorphism group schemes of the generalized Witt algebras and establish triangulations for them. Moreover, we extend the notion of Witt–Ree algebra to general base rings and discuss their properties.