nLab Takeuchi product

Overview

Given an associative unital kk-algebra AA, with enveloping algebra A e=AA opA^e = A\otimes A^{op}, the Takeuchi product × A\times_A is a certain bifunctor in the category of A eA^e-rings defined as an end of a coend.

It generalizes a construction of M. E. Sweedler where AA 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 MM be an AA-bimodule and NN an A opA^{op}-bimodule.

M× AN= b a a opM b op aN b 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 b\int_b and coend a\int^a (while most Hopf algebraic references interchange the notion upside down, writing b a\int^b\int_a instead).

This calculates to

M× AN={ im in iM AN|(bA) im i.b opn i= im in i.b}M AN 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 MM is in fact A eA^e-bimodule we define the AA-bimodule structure on M× ANM\times_A N by acting on MM factor; likewise if NN is an A eA^e-bimodule we define the A opA^{op}-bimodule structure on M× ANM\times_A N by acting on NN factor. Thus if both are A eA^e-bimodules, M× ANM\times_A N is canonically an A eA^e-bimodule.

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

Define

M× AP× AN= b,d a,c a opM b op a,c opP b,d op cN d 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

α:(M× AP)× ANM× AP× AN \alpha : (M\times_A P)\times_A N \to M\times_A P\times_A N
α:M× A(P× AN)M× AP× AN \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 α\alpha and α\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

category: algebra

Last revised on September 10, 2024 at 20:10:58. See the history of this page for a list of all contributions to it.