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Hopf algebroids with bijective antipodes: axioms, integrals and duals

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arxiv math/0302325 v1 pith:ZGSJ63XM submitted 2003-02-26 math.QA

Gabriella B"ohm , Korn'el Szlach'anyi This is my paper
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keywords hopfalgebroidsexamplesalgebroidaxiomsbijectivedualsintegrals
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abstract

Motivated by the study of depth 2 Frobenius extensions we introduce a new notion of Hopf algebroid. It is a 2-sided bialgebroid with a bijective antipode which connects the two, left and right handed, structures. While all the interesting examples of the Hopf algebroid of J.H. Lu turn out to be Hopf algebroids in the sense of this paper, there exist simple examples showing that our definition is not a special case of Lu's. Our Hopf algebroids, however, belong to the class of $\times_L$-Hopf algebras proposed by P. Schauenburg. After discussing the axioms and some examples we study the theory of non-degenerate integrals in order to obtain duals of Hopf algebroids.

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  1. Examples of scalar extension Hopf algebroids over a universal enveloping algebra

    math.QA 2025-06 accept novelty 5.0 of 10

    For a finite-dimensional Lie algebra g, U(g) is a braided commutative Yetter-Drinfeld module algebra over any Hopf algebra H containing the adjoint matrix coefficients, making H smash U(g) a scalar extension Hopf alge...

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