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

  • Zoran Škoda ORCID logo EMAIL logo and Martina Stojić ORCID logo
Published/Copyright: October 28, 2025
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Abstract

We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra U ( 𝔤 ) of a finite-dimensional Lie algebra 𝔤 over arbitrary field 𝒌 , and the Hopf algebra 𝒪 ( Aut ( 𝔤 ) ) of regular functions on the automorphism group of 𝔤 . This pairing induces a Hopf action of 𝒪 ( Aut ( 𝔤 ) ) on U ( 𝔤 ) , which, together with an explicitly given coaction, makes U ( 𝔤 ) into a braided commutative Yetter–Drinfeld 𝒪 ( Aut ( 𝔤 ) ) -module algebra. From these data one constructs a Hopf algebroid structure on the smash product algebra 𝒪 ( Aut ( 𝔤 ) ) U ( 𝔤 ) , retaining essential features from earlier constructions of a Hopf algebroid structure on infinite-dimensional versions of the Heisenberg double of U ( 𝔤 ) , including a noncommutative phase space of Lie algebra type, while avoiding the need of completed tensor products. We prove a slightly more general result, where the algebra 𝒪 ( Aut ( 𝔤 ) ) is replaced by 𝒪 ( Aut ( 𝔥 ) ) and where 𝔥 is any finite-dimensional Leibniz algebra having 𝔤 as its maximal Lie algebra quotient.

MSC 2020: 16T05; 16S40

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Received: 2023-09-13
Revised: 2025-06-04
Accepted: 2025-06-23
Published Online: 2025-10-28

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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