nLab Takeuchi product

Overview

Given an associative unital $k$-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 defined as an end of a coend.

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”.

Definition

Let $M$ be an $A$-bimodule and $N$ an $A^{op}$-bimodule.

$$M\times_A N = \int_b \int^a {}_{a^{op}}M_{b^{op}} \otimes {}_a N_{b}$$

where we use MacLane’s conventions for end $\int_b$ and coend $\int^a$ (while most Hopf algebraic references interchange the notion upside down, writing $\int^b\int_a$ instead).

This calculates to

$$M\times_A N = \{\sum_i m_i\otimes n_i\in M\otimes_A N \,|\, (\forall b\in A)\sum_i m_i.b^{op} \otimes n_i = \sum_i m_i\otimes n_i.b \}\subset M\otimes_A N$$

If $M$ is in fact $A^e$-bimodule we define the $A$-bimodule structure on $M\times_A N$ by acting on $M$ factor; likewise if $N$ is an $A^e$-bimodule we define the $A^{op}$-bimodule structure on $M\times_A N$ by acting on $N$ factor. Thus if both are $A^e$-bimodules, $M\times_A N$ is canonically an $A^e$-bimodule.

Now if $M= R,N = S$ are moreover $A^e$-rings, then they are in particular $A^e$-bimodules. While $R\otimes_A S$ is not carrying a well defined algebra structure, $R\times_A S$ is an associative unital algebra by factorwise multiplication.

Define

$$M\times_A P\times_A N = \int_{b,d}\int^{a,c} {}_{a^{op}}M_{b^{op}} \otimes {}_{a,c^{op}} P_{b,d^{op}}\otimes {}_c N_d$$

Then there are canonical maps

$$\alpha : (M\times_A P)\times_A N \to M\times_A P\times_A N$$

$$\alpha':M\times_A (P\times_A N) \to M\times_A P\times_A N$$

which are not isomorphisms in general.

Properties

As it involves both a limit and a colimit construction, limits commute with limits, colimits with colimits, but not limit with colimits in general, Takeuchi’s product is not associative up to isomorphism, hence it does not provide a monoidal category in general.

Takeuchi proves some sufficient conditions singling out a class of bimodules for which the morphisms $\alpha$ and $\alpha'$ above have the same image, effectively having the associativity in this class.

Applications

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

Literature

• 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
• Tomasz Brzeziński, Gigel Militaru, Bialgebroids, $\times_{R}$-bialgebras and duality, J. Algebra 251: 279-294, 2002, math.QA/0012164
• Peter Schauenburg, Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules, Appl. Categ. Structures 6 (1998), 193–222, ps doi

category: algebra