nLab Takeuchi product

Given an associative algebra $A$, with enveloping algebra $A^e = A\otimes A^{op}$, the Takeuchi product $\times_A$ is a certain bifunctor in the category of $A^e$-rings. It generalizes a construction of M. E. Sweedler where $A$ is commutative; Sweedler’s article may be itself viewed in a sense a “generalization of the relative Brauer group and the associated theory”.

Takeuchi product is used in the theory of associative bialgebroids over noncommutative base.

• Mitsuhiro Takeuchi, Groups of algebras over $A \times \bar{A}$, J. Math. Soc. Japan 29, 459–492, 1977, MR0506407, euclid
• M. E. Sweedler, Groups of simple algebras, Publ. IHES 44, 79–189, MR51:587, numdam
• T. Brzeziński, G. Militaru, Bialgebroids, $\times_{R}$-bialgebras and duality, J. Algebra 251: 279-294, 2002, math.QA/0012164
• P. Schauenburg, Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules, Appl. Categ. Structures 6 (1998), 193–222, ps doi

category: algebra