Enveloping algebra is a Yetter–Drinfeld module algebra over Hopf algebra of regular functions on the automorphism group of a Lie algebra

Abstract We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>U</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝔤</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {U(\mathfrak{g})} of a finite-dimensional Lie algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔤</m:mi> </m:math> {\mathfrak{g}} over arbitrary field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒌</m:mi> </m:math> {{\boldsymbol{k}}} , and the Hopf algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒪</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝔤</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} of regular functions on the automorphism group of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔤</m:mi> </m:math> {\mathfrak{g}} . This pairing induces a Hopf action of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒪</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝔤</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>U</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝔤</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {U(\mathfrak{g})} , which, together with an explicitly given coaction, makes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>U</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝔤</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {U(\mathfrak{g})} into a braided commutative Yetter–Drinfeld <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒪</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝔤</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} -module algebra. From these data one constructs a Hopf algebroid structure on the smash product algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="normal">♯</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mi mathvariant="script">𝒪</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo>