*-Hopf algebroids
Edwin Beggs, Xiao Han, Shahn Majid

TL;DR
This paper develops a theory of $*$-structures for bialgebroids and Hopf algebroids, establishing conditions under which these structures form $*$-Hopf algebroids and exploring their applications in quantum groups and differential structures.
Contribution
It introduces a comprehensive $*$-structure framework for bialgebroids and Hopf algebroids, connecting them to quantum groups, Galois extensions, and differential calculus.
Findings
$*$-structures induce bar categories in module categories
Full $*$-Hopf algebroids arise from braided-commutative algebras in $H$-crossed modules
Ghobadi's bialgebroid with $*$-differential structure forms a $*$-bialgebroid pair
Abstract
We introduce a theory of -structures for bialgebroids and Hopf algebroids over a -algebra, defined in such a way that the relevant category of (co)modules is a bar category. We show that if is a Hopf -algebra then the action Hopf algebroid associated to a braided-commutative algebra in the category of -crossed modules is a full -Hopf algebroid and the Ehresmann-Schauenburg Hopf algebroid associated to a Hopf-Galois extension or quantum group principal bundle with fibre forms a -Hopf algebroid pair, when the relevant (co)action respects . We also show that Ghobadi's bialgebroid associated to a -differential structure on forms a -bialgebroid pair and its quotient in the pivotal case a -Hopf algebroid pair when the pivotal structure is compatible with . We show that when is…
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Taxonomy
TopicsAdvanced Topics in Algebra
