Associative bialgebroid

In mathematics, an associative L-bialgebroid where L is an associative algebra over some ground field k is another associative k-algebra H together with a number of additional structure maps involving H, L and various tensor products of those. Associative bialgebroids are a generalization of a k-bialgebra where a ground ring k is replaced by a possibly noncommutative k-algebra L. Hopf algebroids are associative bialgebroids with an additional antipode map which is an antiautomorphism of H satisfying additional axioms.

The term bialgebroid for this notion has been first proposed by J-H. Lu. The modifier associative is often dropped from the name, and retained mainly only when we wants to distinguish it from the notion of a Lie bialgebroid, often also referred just as a bialgebroid. Associative bialgebroids come in two chiral versions, left and right. A dual notion is the notion of a cobialgebroid.

There is a generalization, an internal bialgebroid which abstracts the structure of an associative bialgebroid to the setup where the category of vector spaces is replaced by an abstract symmetric monoidal category admitting coequalizers commuting with the tensor product.