An alternative notion of Hopf algebroid

Details An alternative notion of Hopf algebroid An alternative notion of Hopf algebroid

2003-01-16

arxiv.org/abs/math/0301169v2

Lect. Notes in Pure and Appl. Math. 239, 31-54 Dekker 2004

Mathematics

Quantum Algebra

19 pages LaTeX 2e file, notations explained in more details, references updated

Scientific paper

In [1] a new notion of Hopf algebroid has been introduced. It was shown to be inequivalent to the structure introduced under the same name in [17]. We review this new notion of Hopf algebroid. We prove that two Hopf algebroids are isomorphic as bialgebroids if and only if their antipodes are related by a `twist' i.e. are deformed by the analogue of a character. A precise relation to weak Hopf algebras is given. After the review of the integral theory of Hopf algebroids we show how a right bialgebroid can be made a Hopf algebroid in the presence of a non-degenerate left integral. This can be interpreted as the `half of the Larson-Sweedler theorem'. As an application we construct the Hopf algebroid symmetry of an abstract depth 2 Frobenius extension [2].

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