On Iterated Twisted Tensor Products of Algebras

Abstract

We introduce and study the definition, main properties and applications of iterated twisted tensor products of algebras, motivated by the problem of defining a suitable representative for the product of spaces in noncommutative geometry. We find conditions for constructing an iterated product of three factors, and prove that they are enough for building an iterated product of any number of factors. As an example of the geometrical aspects of our construction, we show how to construct differential forms and involutions on iterated products starting from the corresponding structures on the factors, and give some examples of algebras that can be described within our theory. We prove a certain result (called ``invariance under twisting'') for a twisted tensor product of two algebras, stating that the twisted tensor product does not change when we apply certain kind of deformation. Under certain conditions, this invariance can be iterated, containing as particular cases a number of independent and previously unrelated results from Hopf algebra theory.

44 pages, 21 figures. More minor typos corrections, one more example and some references added

References in corpus (4)

• Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples
• Generalized diagonal crossed products and smash products for quasi-Hopf algebras. Applications
• General twisting of algebras
• On the Decoupling of the Homogeneous and Inhomogeneous Parts in Inhomogeneous Quantum Groups

Cited by in corpus (9)

• General twisting of algebras
• Extending structures I: the level of groups
• Connections over twisted tensor products of algebras
• A dichotomy between twisted tensor products of bialgebras and Frobenius algebras
• Iterated weak crossed products
• Double crossed biproducts and related structures
• Braid group representations from twisted tensor products of algebras
• A Primer on Twists in the Noncommutative Realm Focusing on Algebra, Representation Theory, and Geometry
• The construction of observable algebra in field algebra of $G$-spin models determined by a normal subgroup