Crossed products and inner actions of Hopf algebras

Abstract

This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include "inner" group gradings of algebras. We prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi colon upper H right-arrow upper H overbar"> <mml:semantics> <mml:mrow> <mml:mi>π</mml:mi> <mml:mo>:</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mover> <mml:mi>H</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi :H \to \overline H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Hopf algebra epimorphism which is split as a coalgebra map, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is algebra isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A number-sign Subscript sigma Baseline upper H"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">#</mml:mi> <mml:mi>σ</mml:mi> </mml:msub> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A{\# _\sigma }H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a crossed product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the left Hopf kernel <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π</mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Given any crossed product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A number-sign Subscript sigma Baseline upper H"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">#</mml:mi> <mml:mi>σ</mml:mi> </mml:msub> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A{\# _\sigma }H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (weakly) inner on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A number-sign Subscript sigma Baseline upper H"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">#</mml:mi> <mml:mi>σ</mml:mi> </mml:msub> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A{\# _\sigma }H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to a twisted product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript tau Baseline left-bracket upper H right-bracket"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{A_\tau }[H]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with trivial action. Finally, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies that of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A number-sign Subscript sigma Baseline upper H"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">#</mml:mi> <mml:mi>σ</mml:mi> </mml:msub> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A{\# _\sigma }H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; in particular this is true if the (weak) action of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is inner.