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Remark 2.15 As is well-known, the algebra of functions on noncommutative tori are, in fact, [MATH] -algebras and so are prototypical examples of quantum spaces according to Connes. However, we will not make use of [MATH] -algebra structures in the proceeding constructions nor in any further examples. |
Example 2.16 (Taft algebras) Similar to the previous examples, the Taft algebra [MATH] for any positive integer [MATH] is [MATH] and [MATH] , subject to the relations |
[EQUATION] where [MATH] is a primitive [MATH] -root of unity. Naturally, this algebra is [MATH] -graded and indeed an almost commutative algebra where the cocycle is given by |
[EQUATION] Remark 2.17 It is well-known that Taft algebras are examples of a Hopf algebra. Indeed, Taft algebras provide the first examples of neither commutative, nor cocommutative Hopf algebras. However, we will not make use of Hopf algebra structures in the proceeding constructions nor in any further examples. |
Definition 2.18 Let [MATH] be a [MATH] -graded algebra (not necessarily almost commutative). Then a [MATH] -derivation [MATH] of [MATH] of [MATH] -degree [MATH] is a linear map [MATH] of [MATH] -degree [MATH] , that satisfies the [MATH] -derivation rule |
[EQUATION] for all homogeneous elements [MATH] and [MATH] The reader can directly verify that the [MATH] -commutator of two [MATH] -derivation is again a [MATH] -derivation. Thus, the vector space of all [MATH] -derivations of [MATH] forms a [MATH] -Lie algebra of [MATH] -degree [MATH] . We denote this [MATH] -Lie alge... |
[EQUATION] where [MATH] and [MATH] . The reader can also quickly verify that we have a [MATH] -Leibniz rule [EQUATION] for any and all [MATH] and [MATH] |
Definition 2.19 Let [MATH] be an almost commutative algebra. Then a foliation in [MATH] is a subspace [MATH] such that (i) [MATH] is a left [MATH] -submodule, and |
(ii) [MATH] is a [MATH] -Lie subalgebra. The pair [MATH] is referred to as a foliated almost commutative algebra Remark 2.20 The notion of a foliated commutative algebra, as far as we know, goes back to Jan Kubarski in unpublished notes from the 6th International Algebraic Conference in Ukraine, July 1-7, 2007. This no... |
Because the bracket on [MATH] -derivations of [MATH] is the [MATH] -commutator, we have an almost commutative generalisation of a homological vector field, i.e., an odd vector field on a supermanifold that ‘squares to zero’ (see |
). Definition 2.21 Let [MATH] be an almost commutative algebra. A [MATH] -derivation [MATH] is said to be a homological [MATH] -derivation of [MATH] if and only if |
(i) [MATH] (ii) [MATH] For notational ease, we set [MATH] , which from the above definition, is equal to zero. The notion of closed and exact elements of [MATH] makes perfect sense. That is, [MATH] is [MATH] -closed if [MATH] , and [MATH] -exact if there exists some [MATH] such that [MATH] . Clearly, as [MATH] [MATH] -... |
who studied a much more general notion than which we propose. It is clear that the pair [MATH] is, in fact, a foliated almost commutative algebra, see Definition 2.19 . A quick calculation shows that for any [MATH] and [MATH] |
[EQUATION] and so we indeed have a [MATH] -Lie subalgebra of [MATH] Our attitude is that is that an almost commutative [MATH] -algebra is a ‘mild’ noncommutative generalisation of a Q-manifold. That is, a supermanifold equipped with a Grassmann odd vector field that squares to zero. Various classical structures in clas... |
. In physics, Q-manifolds are essential in the BV-BRST formalism of gauge theories, and in particular the AKSZ construction As noted by Bongaarts & Pijls |
almost commutativity is ‘close enough’ to commutativity to rather directly mimic the standard construction of the Cartan calculus on a smooth manifold. Moreover, the almost commutativity determines everything else: there are no real choices to be made here. We will briefly recall the key elements of the constructions h... |
We will write the left action of [MATH] on [MATH] as [MATH] . Similarly for [MATH] -linear maps [MATH] , we write [MATH] . Differential [MATH] -forms are then defined in ‘almost’ the same way as the classical case. First, the space of zero forms is defined as [MATH] . Then for [MATH] [MATH] are defined as the [MATH] -g... |
[EQUATION] and the [MATH] -antisymmetry [EQUATION] In this was we obtain a [MATH] -graded right [MATH] -module with [EQUATION] and the right action being |
[EQUATION] Naturally, the direct sum [MATH] is a [MATH] -graded [MATH] -module. Just as in the classical case, we have the de Rham differential , and for any [MATH] we have the interior product [MATH] and Lie derivative [MATH] all acting on [MATH] . The definitions of these operators are ‘almost’ the same as the classi... |
The de Rham differential is the linear map [MATH] , defined as [MATH] for [MATH] , and for [MATH] it is defined as [EQUATION] Given any [MATH] , the interior derivative is defined as |
[EQUATION] with [MATH] , and the Lie derivative is defined as [EQUATION] The space [MATH] carries a natural algebraic structure in ‘almost’ the same way as the differential forms on a manifold do. The most direct way to see this is to note that [MATH] is [MATH] -graded, where [MATH] . Any ( [MATH] -homogeneous) [MATH] ... |
Note that [MATH] and [MATH] are [MATH] -derivations on [MATH] of [MATH] -degrees [MATH] [MATH] and [MATH] . It can be shown that the Cartan identities generalise to the current situation in the expected way: |
[EQUATION] Observe that given any [MATH] -commutative algebra [MATH] , the de Rham complex [MATH] is an almost commutative Q-algebra (see Definition 2.21 ). This is clear as [MATH] is a [MATH] -commutative [MATH] -algebra, [MATH] , and the first of the Cartan identities can be written as [MATH] |
Remark 2.22 The notion of an almost Q-algebra and in particular the example [MATH] is more general than what one usually means by a first-order differential calculus (cf. Woronowicz |
). Specifically, there is no general condition that everything is [MATH] for [MATH] and [MATH] 3. Almost Commutative Non-negatively Graded Algebras |
Graded manifolds and particularly non-negatively graded manifolds provide an economic framework to encode various classical structures like Lie algebroids, Courant algebroids, Dirac structures and so on (see for example |
). In the current setting, will be interested in almost commutative algebras that have a compatible [MATH] -grading, which we refer to as weight . The basic idea is that we wish to algebraically capture the important features of the polynomial algebra on a graded bundle (see |
and for a review of the theory of graded bundles may consult ). Consider an [MATH] -bigraded algebra [EQUATION] As a bigraded algebra we have [MATH] . We will denote the [MATH] -degree of a homogeneous element [MATH] as before, i.e., [MATH] . The weight we will denote by [MATH] . Let [MATH] be a [MATH] -cocycle. We wil... |
Definition 3.1 An [MATH] -bigraded algebra [MATH] is said to be an almost commutative non-negatively graded algebra if and only if it is a [MATH] -commutative algebra. |
Remark 3.2 Naturally, we could allow a [MATH] -grading instead of just an [MATH] -grading. In physics, an example of such a grading would be ‘ghost number’. The important thing here is that the [MATH] or [MATH] -grading plays a very different rôle to the [MATH] -grading in the BV-BRST formalism. Likewise for almost com... |
We need to understand what it means for the ‘weight to be bounded’. In the classical setting of non-negatively graded manifolds this means that the weight of the local coordinates is bounded from above. We mimic this in the following way. First, we define |
[EQUATION] Then we define [MATH] as the [MATH] -commutative algebra [MATH] . Naturally, this is itself an [MATH] -bigraded algebra and an almost commutative subalgebra of [MATH] . We can then consider the following set |
[EQUATION] In words, this is the set of weight [MATH] elements of [MATH] that are built from elements of weight less that [MATH] . We then make the following definition. |
Definition 3.3 An almost commutative non-negatively graded algebra [MATH] is said to be of degree [MATH] if and only if [EQUATION] |
for all [MATH] As it stands, our definition of the degree means that if [MATH] is of degree [MATH] , then it is automatically also of degree [MATH] . However, by degree, we will be explicitly be referring to the minimal degree from this point on, i.e, the lowest possible degree. |
The space of [MATH] -derivations of an almost commutative non-negatively graded algebra is [MATH] -graded. That is, we naturally have [MATH] -derivations that carry a negative degree with respect to the weight of the algebra. |
Lemma 3.4 Let [MATH] be an almost commutative non-negatively graded algebra of degree [MATH] . Then the space of [MATH] -derivations of [MATH] is bounded from below at [MATH] , i.e., there are no non-zero [MATH] -derivations of [MATH] -degree less than [MATH] |
Proof. Suppose [MATH] is a [MATH] -derivation of [MATH] of [MATH] -degree [MATH] and that [MATH] . Clearly, such a derivation will annihilate any element of [MATH] of weight [MATH] . The only possible non-trivial action is on elements of weight [MATH] . However, as the algebra we are considering is of degree [MATH] , e... |
We can naturally decompose the [MATH] -Lie algebra of [MATH] -derivation of [MATH] (of degree [MATH] ) as the sum of two [MATH] -Lie subalgebras |
[EQUATION] where [EQUATION] Some direct observations here are the following: (i) The [MATH] -Lie algebra [MATH] is nilpotent. (ii) |
[MATH] -Lie algebra [MATH] is abelian. The situation here is exactly the same for vector fields on a non-negatively graded manifold, where the abelian group is simply [MATH] and cocycle is the standard sign factor, see Voronov |
4. Almost Commutative [MATH] -algebras and [MATH] -antibrackets Taking Vaĭntrob as our cue, we make the following definition. Definition 4.1 |
Fix some abelian group [MATH] and cocycle [MATH] . An almost commutative [MATH] -algebra of degree [MATH] is a pair [MATH] consisting of: |
(i) an almost commutative non-negatively graded algebra [MATH] of degree one; (ii) a homological [MATH] -derivation of [MATH] of [MATH] -degree one. |
morphism of almost commutative [MATH] -algebras of degree [MATH] is a bi-graded algebra morphism [EQUATION] that relates the respective [MATH] -derivations, i.e., |
[EQUATION] The idea is that an almost commutative [MATH] -algebra of degree [MATH] should be a noncommutative version of a Lie algebroid (or really a Lie antialgebroid ), or more carefully we have a noncommutative version of the Chevalley–Eilenberg complex associated with a Lie algebroid. Our attitude is that this comp... |
Remark 4.2 Schwarz defined a Q-algebra as a [MATH] (or [MATH] ) graded associative algebra [MATH] equipped with a derivation of degree [MATH] and an element [MATH] of degree [MATH] (the curvature) such that [MATH] . The algebras we define are closely related to Schwarz’s Q-algebras when we replace the commutator with t... |
We can modify the derived bracket construction of Kosmann-Schwarzbach (also see Voronov ) to the current setting of almost commutative algebras. For all of this section, we will specifically concentrate on the degree [MATH] case without further explicit reference. |
We will change nomenclature slightly in order to fit with classical vector bundles. By the sections of [MATH] , we mean the [MATH] -graded vector space [MATH] . Recall that we can also consider this space as an abelian [MATH] -Lie algebra, this will be important in the constructions. Moreover, [MATH] naturally has the ... |
Definition 4.3 Let [MATH] be an almost commutative [MATH] -algebra of degree [MATH] . Then the derived [MATH] -antibracket is the bilinear map |
[EQUATION] Let us now examine the properties of the derived [MATH] -antibracket. Theorem 4.4 Let [MATH] be an almost commutative [MATH] -algebra of degree [MATH] and let [MATH] denote the corresponding derived [MATH] -antibracket. The derived [MATH] -antibracket exhibits the following properties: |
(i) [MATH] (ii) [MATH] (iii) [MATH] In short, [MATH] , once equipped with the derived [MATH] -antibracket, is a [MATH] -Lie antialgebra (see Definition 2.3 ). |
Proof. These statements follow directly via calculation in almost exactly the same way as the classical [MATH] or [MATH] -graded case. |
(i) This follows directly from the definition and the properties of [MATH] -degree. (ii) This follows from the [MATH] -Jacobi identity and the [MATH] -skewsymmetry for the [MATH] -commutator. Specifically, it is easy to see that the [MATH] -Jacobi identity together with the [MATH] -skewsymmetry imply that |
[EQUATION] Because we are dealing with an abelian [MATH] -Lie subalgebra the left-hand side of the above vanishes and we can thus write |
[EQUATION] Now using the properties of the cocycle [MATH] we arrive at [EQUATION] and then as [MATH] , we obtain the required symmetry. |
(iii) Directly from the definitions we have [EQUATION] we then make use of the [MATH] -Jacobi identity for the [MATH] -commutator to obtain |
[EQUATION] and again using the [MATH] -Jacobi identity for the [MATH] -commutator we arrive at [EQUATION] Using the definition of the derived [MATH] -antibracket and once more the [MATH] -Jacobi identity for the [MATH] -commutator we finally obtain |
[EQUATION] Then as [MATH] we obtain the required result. Putting the above together, and using the fact that [MATH] we see that we do indeed have the structure of a [MATH] -Lie antialgebra. |
We also have a module left module structure of [MATH] to take into account. Following the classical picture of Lie algebroids, we define an anchor map in the following way. |
Definition 4.5 Let [MATH] be an almost commutative [MATH] -algebra of degree [MATH] . The associated anchor map is the morphism of (left) [MATH] -modules |
[EQUATION] defined by [EQUATION] for all [MATH] and [MATH] To be clear, we have that [EQUATION] for all [MATH] and [MATH] A direct calculation establishes the following result. |
Proposition 4.6 Let [MATH] be an almost commutative [MATH] -algebra of degree [MATH] . Furthermore let [MATH] and [MATH] denote the associated derived [MATH] -antibracket and anchor map, respectively. The derived [MATH] -antibracket satisfies the following [MATH] -Leibniz rule: |
[EQUATION] For Lie algebroids, it is well known that the compatibility of the anchor map and the bracket follow from the Jacobi identity. That is, the anchor map is a morphism of Lie algebras. The situation in the current context is essentially identical: the proof follows in exactly the same way as the classical case.... |
[EQUATION] given by [EQUATION] Naturally, the [MATH] -Jacobiator vanishes in our case as [MATH] Proposition 4.7 Let [MATH] be an almost commutative [MATH] -algebra of degree [MATH] . Furthermore let [MATH] and [MATH] denote the associated derived [MATH] -antibracket and anchor map, respectively. Then the derived [MATH]... |
[EQUATION] for all [MATH] and [MATH] Proof. After short computation, we see that [EQUATION] for all [MATH] and [MATH] and [MATH] . Now simply using the fact that the [MATH] -Jacobiator for the derived [MATH] -antibracket vanishes established the result. |
We see that in analogy with the classical theory of Lie algebroids, which provides a framework for foliations of manifolds, almost commutative Q-algebras of degree [MATH] provide a class of foliations in almost commutative algebras. |
Corollary 4.7.1 The image of the anchor map [MATH] , is a foliation in [MATH] , see Definition 2.19 Definition 4.8 The characteristic foliation of an almost commutative [MATH] -algebra of degree [MATH] is the foliated almost commutative algebra [MATH] |
Statement: Given the initial data of an almost commutative Q-algebra of degree [MATH] [MATH] , we canonically have the structure of a [MATH] -Lie antialgebra on the space of sections of [MATH] (Theorem 4.4 ), together with an anchor map that is compatible with the derived [MATH] -antibracket (Proposition 4.7 ). Up to o... |
who uses a different choice of grading). Example 4.9 (Trivial structures) Any almost commutative non-negatively graded algebra of degree [MATH] can be equipped with the trivial homological [MATH] -derivation, i.e., [MATH] . Geometrically, we think of a vector bundle equipped with the zero bracket and zero anchor. Moreo... |
Example 4.10 (The de Rham complex) Given any [MATH] -commutative algebra [MATH] , it is clear that the de Rham complex [MATH] is an almost commutative Q-algebra of degree [MATH] , where we take [MATH] and the cocycle to be one previously discussed, see Section |
Example 4.11 (The canonical action on the noncommutative [MATH] -torus) Continuing example 2.14 , it is well-known that the module of [MATH] -derivations on the noncommutative [MATH] -torus [MATH] is spanned by |
[EQUATION] Moreover, these [MATH] -derivations are the infinitesimal generators of the canonical action of the classical [MATH] -torus [MATH] on the noncommutative [MATH] -torus. The infinitesimal action is specified by two complex numbers [MATH] . Following the idea of BRST quantisation, we “promote” these complex num... |
[EQUATION] where the cocycle is the obvious one, i.e., [MATH] . The homological [MATH] -derivation we define to be [EQUATION] which is clearly of [MATH] -degree [MATH] . We thus interpret this construction as an almost commutative version of an action Lie algebroid. |
The [MATH] degree of the de Rham differential suggests a natural class of almost commutative Q-algebras. In particular, consider the algebra [MATH] , where we define the [MATH] degrees to be [MATH] and [MATH] . The [MATH] -commutation laws are then |
[EQUATION] The algebra [MATH] be take to be the algebra [MATH] , with possible further relations on the [MATH] , however, this will not affect what we write next. We understand the algebra to be a formal power series algebra in [MATH] . Clearly, by defining [MATH] and [MATH] , we obtain an almost commutative algebra of... |
[EQUATION] where we have assumed that [MATH] and [MATH] . Note that we have the [MATH] -skewsymmetry [MATH] . The reader should note that all of this is exactly as expected given the classical case (see Vaĭntrob |
). Next we assume that the [MATH] -derivation is homological, i.e., [MATH] . This imposes conditions on the components. After a direct, but not illuminating calculation, one can derive the structure equations |
[EQUATION] Again, these structure equations are formally identical to the classical structure equations for a Lie algebroid, but now we have to take into account extra factors due to the [MATH] -commutativity. |
5. First-order differential calculi and the Lie bracket We can construct a first-order differential calculi (in the sense of Woronowicz |
) over any almost commutative algebra [MATH] in the following way (due to Bongaarts & Pijls ). First, as standard, we define [MATH] , and then we define |
[EQUATION] where the differential is the de Rham differential defined earlier. The [MATH] -commutative algebra of differential forms is then [MATH] and [MATH] . In general, we have a subalgebra of [MATH] . Clearly in this was we can interpret [MATH] as an almost commutative Q-algebra of degree [MATH] |
Proposition 5.1 Given any almost commutative algebra [MATH] , there is a natural bijection [EQUATION] Proof. Given any [MATH] , we can send it to a unique [MATH] . Recall that part of the definition of the interior derivative is that [MATH] |
In the other direction, consider an arbitrary [MATH] , of [MATH] -degree [MATH] . Just on weight grounds it is clear that [MATH] for any [MATH] . Thus, [MATH] . In particular, this implies that [MATH] is a unique [MATH] -derivation on [MATH] of [MATH] -degree [MATH] . Direct calculation show we have the required deriva... |
[EQUATION] Observing that [EQUATION] shows we have constructed mutual inverses. Thus we have the desired bijection. The above proposition states that - not surprisingly - that we can identify sections of [MATH] with [MATH] -derivations of [MATH] . This is of course exactly what have in the classical case that [MATH] wh... |
Statement: For any almost commutative algebra, the [MATH] -Lie bracket on [MATH] is identified as the derived [MATH] -antibracket with respect to the de Rham differential. That is, we make the identification (as sections of [MATH] |
[EQUATION] using the Cartan identities ( 2.2 ). Note that this is in complete agreement with the classical case (see Kosmann-Schwarzbach |
). Let us assume that the underlying almost commutative algebra [MATH] is finitely [MATH] . We define [MATH] and write the [MATH] -commutation rule as |
[EQUATION] To construct the first-order differential calculi we append the (free) generators [MATH] to the algebra and extend to a [MATH] grading as previously described. That is, we define the grading as |
[EQUATION] The [MATH] -commutation rules are thus [EQUATION] Any [MATH] -derivation can then be defined in terms of (algebraic) partial derivatives. Specifically, any [MATH] -derivation can be mapped to the interior derivative which is given by |
[EQUATION] The de Rham differential is given by [EQUATION] Clearly, we have an almost commutative Q-algebra of degree [MATH] . Following the definitions, we see that |
[EQUATION] as fully expected. 6. Q-modules over almost commutative Q-algebras Lie algebroid modules à la Vaĭntrob are important in the representation and deformation theory of Lie algebroids. Thus, it makes sense to carefully define the related notion for almost commutative Q-algebras of degree [MATH] |
Definition 6.1 Let [MATH] be an almost commutative Q-algebra of degree [MATH] . A (left) Q-module over [MATH] is a [MATH] -graded vector space [MATH] , equipped with the following structures: |
An action [EQUATION] that is [MATH] -degree preserving. By an action we mean that [MATH] A differential [EQUATION] of [MATH] -degree [MATH] and [MATH] -degree [MATH] , that satisfies the following [MATH] -Leibniz rule, |
[EQUATION] for all [MATH] and [MATH] . By a differential we naturally mean that [MATH] Naturally, it makes sense to refer to the map [MATH] as a flat connection . Moreover, one can speak of the cohomology of an almost commutative Q-algebra of degree [MATH] with values in the module [MATH] . A right Q-module over [MATH]... |
Example 6.2 (Adjoint module) It is clear that [MATH] is a [MATH] -graded module over [MATH] by considering weight. We claim that [MATH] provides the required flat connection. The action is the standard one. The bi-degree of the defined connection is obvious and the flatness follows from [MATH] . The Leibniz rule in the... |
Example 6.3 (Coadjoint module) Consider the module of differential forms on [MATH] which we bi-grade using [MATH] -degree and the weight rather than ‘form degree’; |
[EQUATION] Here it is convenient to consider the right module structure over [MATH] . Then using the Cartan calculus, we claim that [MATH] provides the structure of a flat connection. The condition in the bi-degree is clear and from the Cartan calculus we have that |
[EQUATION] Directly from the properties of the Lie derivative we obtain [EQUATION] Thus we have the structure of a right Q-module. |
7. Deformations and symmetries of almost commutative Q-algebras Again, we will stick to the degree [MATH] case explicitly. Example 6.2 shows that the space of [MATH] -derivations of an almost commutative algebra of degree [MATH] canonically comes equipped with a flat connection. More than this, the [MATH] -Jacobi ident... |
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