On Iterated Twisted Tensor Products of Algebras

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. INTRODUCTION The difficulty of constructing concrete, nontrivial examples of noncommutative spaces starting from simpler ones is a common problem in all different descriptions of noncommutative geometry. If we think of the commutative situation, we have an easy procedure, the cartesian product, which allows us to generate spaces of dimension as big as we want from lower dimensional spaces. Thinking in terms of the existing dualities between the categories of spaces and the categories of (commutative) algebras, the natural replacement for the cartesian product of commutative spaces turns out to be the tensor product of commutative algebras. The tensor product has often been considered a replacement for the product of spaces represented by noncommutative algebras. As it was pointed out in [CSV95], this is a very restricted approach. If the “axiom” of noncommutative geometry consists in considering noncommutative algebras as the representatives for the algebras of functions over certain “quantum” spaces, hence assuming that two different measurements (or functions) on this kind of spaces do not commute to each other, then why should we assume that the measurements on the product commute to each other? There is no reason for imposing this artificial commutation, hence what we need is a “noncommutative” replacement of the tensor product of two algebras, which is supposed Pascual Jara and Javier López have been partially supported by projects MTM2004-08125 and FQM-266 (Junta de Andalucı́a Research Group). Javier López has also been supported by the Spanish MEC FPUgrant AP2003-4340 and the European Science Foundation Programme on NONCOMMUTATIVE GEOMETRY (NOG). Florin Panaite and Fred Van Oystaeyen have been partially supported by the EC programme LIEGRITS, RTN 2003, 505078, and by the bilateral project “New techniques in Hopf algebras and graded ring theory”, of the Flemish and Romanian Ministries of Research. Florin Panaite has also been partially supported by the CEEX programme of the Romanian Ministry of Education and Research, contract nr. CEx05-D11-11/2005 . 1 2 PASCUAL JARA MARTÍNEZ, JAVIER LÓPEZ PEÑA, FLORIN PANAITE, AND FRED VAN OYSTAEYEN to fit better as an analogue of the product of two noncommutative spaces and in the same time to be a useful tool for overcoming the lack of examples formerly mentioned. When we impose the natural restrictions a product should have, namely that it contains the factors in a natural way and having linear size equal to the product of the linear sizes of the factors, we arrive precisely at the definition of a twisted tensor product formerly studied by many people, either for the particular case of algebras (cf. [Tam90], [CSV95], [VDVK94]) or aiming to define similar structures for discrete groups, Lie groups, Lie algebras and Hopf algebras (as in [Tak81], [Maj90] and [Mic90]). Often, this structure appears in the so-called factorization problem of studying under what conditions we may write an object as a product of two subobjects having minimal intersection. From a purely algebraic point of view, twisted tensor products arise as a tool for building algebras starting with simpler ones, and also, as shown in [VDVK94], in close relation with certain nonlinear equations. Whatever the chosen approach to twisted tensor products is, a number of examples of both classical and recently defined objects fits into this construction. Ordinary and graded tensor products, crossed products, Ore extensions and skew group algebras are just some examples of well-known constructions in classical ring theory that can be described as twisted tensor products. In the Hopf algebras and quantum groups area we find smash products, Drinfeld and Heisenberg doubles, and diagonal crossed products. With a more geometrical flavour, quantum planes and tori may be realized as noncommutative products of commutative spaces. And last, but not least, we may also find some physical models for which this structure is particularly well suited, such as the Fock space representations of a particle system with generalized statistics, which is studied in [BM00] using techniques which arise directly from the realisation of certain crossed enveloping algebras as twisted tensor products. In the present work, our aim is to look at the twisted tensor product structure from a more geometrical point of view, regarding it as the natural representative for the cartesian product of noncommutative spaces. When we think of this construction geometrically, it becomes unnatural to restrict ourselves to take the product of only two spaces, so it appears the problem of finding suitable conditions that allow us to iterate the construction, and, whenever this is possible, to check that the obtained iterated product is “associative” in the same sense in which the usual tensor product is. Also, we will be interested in analyzing whether we may lift geometrical invariants that we are able to calculate on the single factors to the iterated twisted product and how to do this, if possible. Being such an ubiquitous construction, there are several equivalent definitions of the twisted tensor product appearing in the literature, often using different names and notation. In the Preliminaries we recall some of the results we will use later on, fixing a unified notation. Concretely, we introduce the definition of a twisted tensor product A ⊗R B of two algebras A and B by means of a twisting map R : B ⊗ A → A ⊗ B, whose existence is sufficient for the existence of a deformed product in the tensor product vector space A ⊗ B, and is also necessary when we impose unitality conditions. In Section 2, we deal with the problem of iterating the twisted tensor products, and the lifting of several structures to the iteration, finding that for three given algebras A, B and C, ON ITERATED TWISTED TENSOR PRODUCTS OF ALGEBRAS 3 and twisting maps R1 : B ⊗ A → A ⊗ B, R2 : C ⊗ B → B ⊗ C, R3 : C ⊗ A → A ⊗ C, a sufficient condition for being able to define twisting maps T1 : C⊗(A⊗R1B) → (A⊗R1B)⊗C and T2 : (B ⊗R2 C) ⊗ A → A ⊗ (B ⊗R2 C) associated to R1, R2 and R3 and ensuring that the algebras A ⊗T2 (B ⊗R2 C) and (A ⊗R1 B) ⊗T1 C are equal, can be given in terms of the twisting maps R1, R2 and R3 only. Namely, they have to satisfy the compatibility condition (A⊗ R2) ◦ (R3 ⊗ B) ◦ (C ⊗ R1) = (R1 ⊗ C) ◦ (B ⊗ R3) ◦ (R2 ⊗ A). This relation may be regarded as a “local” version of the hexagonal relation satisfied by the braiding of a (strict) braided monoidal category (in the same sense in which the twisted tensor product of algebras is the “local” version of the braided tensor product of algebras in a braided monoidal category). We also prove that whenever the algebras and the twisting maps are unital, the compatibility condition is also necessary. As it happens for the classical tensor product, and for the twisted tensor product, the iterated twisted tensor product also satisfies a Universal Property, which we will state formally in Theorem 2.7. Once the conditions needed to iterate the construction of the twisted tensor product are fulfilled, we will prove the Coherence Theorem, stating that whenever one can build the iterated twisted product of any three factors, it is possible to construct the iterated twisted product of any number of factors, and that all the ways one might do this are essentially the same. This result will allow us to lift to any iterated product every property that can be lifted to three-factors iterated products. As applications of the former results we will characterize the modules over an iterated twisted tensor product, also giving a method to build some of them from modules given over each factor. As a first step towards our aim of building geometrical invariants over these structures, we will show how to build the algebras of differential forms and how to lift the involutions of ∗–algebras to the iterated twisted tensor products. In Section 3, we illustrate our theory by presenting some examples of different structures that arose in different areas of mathematics and can be constructed using our method. Two of them (the generalized smash products and diagonal crossed products) come from Hopf algebra theory, while the other two (the noncommutative 2n–planes defined by Connes and Dubois– Violette, and the observable algebra A of Nill–Szlachányi) appear in a more geometrical or physical context. In particular, we show that the algebras defined by Connes and Dubois– Violette can be seen as (iterated) noncommutative products of commutative algebras (as it happens for the quantum planes and tori), and give a new proof of the fact that the algebra A is an AF–algebra, proof which does not imply calculating any representation of it. Section 4 (together with several results from Section 2), illustrates the fact that Hopf algebra theory represents not only a rich source of examples for the theory of twisted tensor products of algebras, but also a valuable source of inspiration for it. In this section we prove a result, called “invariance under twisting”, for a twisted tensor product of two algebras, which arose as a generalization of the invariance under twisting for the Hopf smash product (hence the name). It states that if we start with a twisted tensor product A ⊗R B together with a certain kind of datum corresponding to it, we can deform the multiplication of A to a new algebra structure A, we can deform R to a new twisting map R : B ⊗ A → A ⊗ B, so that the twisted tensor products A ⊗Rd B and A ⊗R B are isomorphic. It turns out that our 4 PASCUAL JARA MARTÍNEZ, JAVIER LÓPEZ PEÑA, FLORIN PANAITE, AND FRED VAN OYSTAEYEN result is general enough to include as particular cases some more independent results from Hopf algebra theory: the well-known theorem of Majid stating that the Drinfeld double of a quasitriangular Hopf algebra is isomorphic to an ordinary smash product, a recent result of Fiore–Steinacker–Wess from [FSW03] concerning a situation where a braided tensor product can be “unbraided”, and also a recent result of Fiore from [Fi02] concerning a situation where a smash product can be “decoupled” (this result in turn contains as a particular case the well– known fact that a smash product corresponding to a strongly inner action is isomorphic to the ordinary tensor product). We also prove that, under certain circumstances, our theorem can be iterated, containing thus, as a particular case, the invariance under twisting of the two-sided smash product from [BPVO]. Though we are mainly interested in results of geometrical nature, and hence most algebras we would like to work with are defined over the field C of complex numbers, most of the results can be stated with no change for algebras over a field or commutative ring k, that we assume fixed throughout all the paper. All algebras will be supposed to be associative, and usually unital, k–algebras. The term linear will always mean k–linear, and the unadorned tensor product ⊗ will stand for the usual tensor product over k. We will also identify every object with the identity map defined on it, so that A ⊗ f will mean IdA ⊗f . For an algebra A we will write μA to denote the product in A and uA : k → A its unit, and for an A–module M we will use λM to denote the action of A on M . For bialgebras and Hopf algebras we use the Sweedler-type notation ∆(h) = h1 ⊗ h2. It is worth noting that the proofs of most of our main results are still valid if instead of considering algebras over k we take algebras in an arbitrary monoidally closed category. 1. PRELIMINARIES 1.1. Twisted tensor products of algebras. The notion of twisted tensor product of algebras has been independently discovered a number of times, and can be found in the literature under different names and notation. In this section we collect some results that will be used later, fixing a unified notation. Main references for definitions and proofs are [CSV95] and [VDVK94]. When dealing with spaces that involve a number of tensor products, notation often becomes obscure and complex. In order to overcome this difficulty, especially when dealing with iterated products, we will use a graphical braiding notation in which tangle diagrams represent morphisms in monoidal categories. For this braiding notation, firstly appeared in [RT90], we refer to [Maj94] and [Kas95]. In this notation, a linear map f : A → B is simply represented by A f B . The composition of morphisms can be written simply by placing the boxes corresponding to each morphism along the same string, being the topmost box the corresponding to the map that is applied in the first place. Several strings placed aside will represent a tensor product of vector spaces (usually algebras), and a tensor product of two linear maps, f ⊗ g : A⊗B → C ⊗D will be written as A f