This article is about the twisted tensor product of associative algebras yielding a new algebra containing the original algebras as subalgebras. For another notion of twisted tensor product of a dg-algebra with a dg-coalgebra yielding a chain complex see there. While twisting tensor products of dg-algebra with a dg-coalgebra involves a twisting cochain, the twist for algebras here is rather of different nature, boiling down to some variant of distributive laws.

This is the definition for the relative case over a possibly noncommutative ring $R$.

$R$-ring $D$ is a twisted tensor product of an $R$-ring $A$ and $R$-ring $B$ if there are $R$-ring morphisms $i_A:A\to D$ and $i_B:B\to D$ such that the canonical map $A\otimes_R B\to D$ given by $a\otimes b\mapsto i(a)i(b)$ is an isomorphism of $R$-bimodules.

(Suppose $R$ is in the center and bimodules are central.) If $\tau:B\otimes A\to A\otimes B$ is an $R$-linear map such that $\tau(b\otimes 1) = 1\otimes b$ and $\tau(1\otimes a) = a\otimes 1$. This implies that the mixed triangle $\tau\circ(B\otimes\eta_A) = A\otimes\eta_B$ holds; clearly also the distributive laws triangles $\mu_B\circ(B\otimes\eta_A) = B$ and $\mu_A\circ (\eta_B\otimes A)=A$. A hexagon formula

$\mu_{A\otimes B} = (\mu_A\otimes \mu_B)\circ(A\otimes_R\tau\otimes_R B) : (A\otimes_R B)\otimes (A\otimes_R B)\to
(A\otimes_R B)$

defines an associative multiplication iff

$\tau\circ(\mu_B\otimes_R\mu_A)= (\mu_A\otimes_R\mu_B)
\circ(A\otimes\tau\otimes B)\circ(\tau\otimes\tau)\circ(B\otimes\tau\otimes A)$

The following distributive law pentagons are needed and are equivalent to the above condition (we skip adorning the relative tensor products):

$(A\otimes\mu_B)\circ (\tau\otimes B)\circ(B\otimes \tau)= \tau\circ(\mu_B\otimes A): B\otimes B\otimes A\to A\otimes B$

$(\mu_A\otimes B)\circ (A\otimes\tau)\circ(\tau\otimes A)= \tau\circ(B\otimes\mu_A): B\otimes A\otimes A\to A\otimes B$

The present notion is related to factorizations of algebras, hence also to distributive laws in categorical setup.

Original reference with one hexagon identity instead of two pentagons for associative algebras

- A. Čap, H. Schichl, J. Vanžura,
*On twisted tensor products of algebras*, Commun. Alg.**23**(1995) 4701 doi

The construction for (concrete) twisted algebras with the explicit two pentagon identities for twist appear e.g. (see Eqs. (4.4), (4.5)) in

- Piotr Podleś, S. L. Woronowicz,
*Quantum deformation of Lorentz group*, Comm. Math. Phys.**130**(2) (1990) 381–431 doi

and the dual version (C) for coalgebras at page 1004 of

- Georges Skandalis,
*Operator algebras and duality*, in: Proceedings of the ICM Kyoto (1990) 997–1009 pdf

More elaborate versions for matched products and bicrossproducts in Hopf algebra theory were earlier in

- W. M. Singer,
*Extension theory for connected Hopf algebras*, J. Algebra 21 (1972) 1-16 - S. H. Majid,
*Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction*, J. Algebra**130**:1 (1990) 17-64 doi

In some abstract setups (factorization algebras, factorization monads etc.) more general construction was known beforehands, using distributive laws and further generalizations appear later.

For dg-algebras including a relative version, $R$-dg-rings,

- Dmitri Orlov,
*Twisted tensor products of DG algebras*, Russ. Math. Surv. 76 (2021) 1146;*Smooth DG algebras and twisted tensor product*, arXiv:2305.19799

category: algebra

Last revised on June 12, 2023 at 11:38:59. See the history of this page for a list of all contributions to it.