Bialgebroids, ×A-Bialgebras and Duality☆

By Tomasz Brzezinski, Gigel Militaru. 2002

99 citations recorded by the source corpus.

An equivalence between Lu's bialgebroids, Xu's bialgebroids with an anchor, and Takeuchi's ×A-bialgebras is explicitly proven. A new class of examples of bialgebroids is constructed. A (formal) dual of a bialgebroid, termed bicoalgebroid, is defined. A weak Hopf algebra is shown to be an example of such a bicoalgebroid.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/88c2c5ac8a30ef760cffa2b1b1c013295c0da0b1

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In Algebra and Number Theory

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In Mathematical Physics

Citations in the corpus

The cyclic theory of Hopf algebroids (2009, background): “ (noncommutative) base algebra. First formulated in a still commutative setting in [S], given in fullgeneralityin[T]andindependentlydevelopedin[Lu,Sch1,X2],alldeﬁnitions are shown to be equivalent in [BrzMi]. Approaches begin to differ when adding the antipode. The ﬁrst general deﬁnitionappearedin[L”
Monoidal Functors, Species, and Hopf Algebras (2010, background): “There is a more general notion of bialgebroid that allows for the base algebra K to be noncommutative and requires only that the images of s and t in H commute elementwise [57, 231]; see [139] for a survey of related notions.”
Plethystic algebra (2004, background): “Thus, the structure of a twisted k-bialgebra on A is the same as the structure of a kbialgebroid on A where the structure map k⊗Zk → A factors through multiplication k ⊗Z k → k. (Several equivalent formulations of the notion of bialgebroid are discussed in Brzezinski–Militaru [3].)”