Hodge Star as Braided Fourier Transform

We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ1 over a Hopf algebra A which are quotients of the augmentation ideal A+ as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation provides a bicovariant differential graded algebra on A. We introduce Λmax providing the maximal prolongation, while the canonical braided-exterior algebra Λmin = B−(Λ1) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ♯ by super-braided Fourier transform on B−(Λ1) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[S3] with its 3D calculus and obeying the q-Hecke relation ♯2 = 1 + (q − q−1)♯ in middle degree on kq[SL2] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra ℒ⊆A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}\subseteq A$\end{document} defines a sub-braided Lie algebra and Λ1⊆ℒ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\Lambda }^{1}\subseteq \mathcal {L}^{*}$\end{document} provides the required data A+ → Λ1.


Introduction
Differential exterior algebras on quantum groups were extensively studied in the 1990s since [39] and have a critical role as examples of noncommutative geometry more generally. However, one problem which has remained open since that era is the general construction of a Hodge star operator in noncommutative geometry, even in the quantum group case. Until now the Hodge operator has been treated mainly in an ad-hoc manner in particular examples, motivated typically by 2 = ±id as a requirement, e.g. [7,33]. We also note a framework [9] based on a pair of differential structures and contraction with a generalised metric, and [12] in another q-deformation framework. By contrast our new approach depends canonically on the braided-Hopf algebra structure of the exterior algebra which applies at least for bicovariant calculi on quantum groups and covariant calculi on quantum-braided planes. Moreover, this new approach gives very different, and we think more interesting, answers than the previous approaches. Specifically, Section 3.2 includes the example of k(S 3 ), the function algebra on the permutation group S 3 , with its 2-cycle calculus, where in middle degree obeys 3 = id so that as a whole has order 6. We also cover electromagnetism on this finite group as in [33] but using the new Hodge star and again achieving a full diagonalisation of the Laplace operator dδ + δd. Similarly, Section 4.2 computes our Hodge operator for k q [SL 2 ] with its standard 4D calculus [39] and in middle degree we find, rather unexpectedly, that it obeys the well-known q-Hecke relation when suitably normalised. Both of these examples are very different from requiring 2 = ±id. While the nicest version of the theory assumes a central bi-invariant metric and volume form, our Fourier approach is more general as illustrated in Section 3.1 on the quantum plane A 2 q . Conceptually, we adopt a novel point of view [17,29] on what the Hodge star is even classically. Namely, at every point of a manifold of dimension n the exterior algebra of differential forms has fibre the exterior algebra (R n ) with generators e i = dx i in local coordinates and the usual 'Grassmann algebra' relations e i e j + e j e i = 0. This is a finitedimensional super-Hopf algebra with super coproduct e i = e i ⊗ 1 + 1 ⊗ e i and dual super-Hopf algebra * of the same form with generators f i . It is well-known that e i 1 · · · e i m with i 1 < i 2 < · · · < i m , m ≤ n form a basis of and one can check that f i m · · · f i 1 provide a dual basis where the product of * is adjoint to the coproduct of . The reversal of order of the f i occurs here if we use categorical conventions whereby * ⊗ * ⊗ ⊗ → R is evaluated starting from the middle factors. The canonical 'identity' element of ⊗ * for this pairing is n k=0 j 1 <···<j k e j 1 · · · e j k ⊗ f j k · · · f j 1 = n k=0 1 k! e j 1 · · · e j k ⊗ f j k · · · f j 1 (1. 1) with summation of all repeated indices in the last expression. We also have the Berezin integral Vol = 1 where Vol = e 1 · · · e n , and zero on lower degrees. With these ingredients the super version of usual Hopf algebra Fourier transform F : → * is F(e i 1 · · · e i m ) = e i 1 · · · e i m 1 (n − m)! e j 1 · · · e j n−m ⊗ f j n−m · · · f j 1 = 1 (n − m)! i 1 ···i m j 1 ···j n−m f j n−m · · · f j 1 where only k = n − m in Eq. 1.1 contributes and we sum over repeated indices. The totally antisymmetric symbol defined by 12···n = 1 takes care of the reordering inside the integral. This construction extends to the whole manifold and in the presence of a metric gives the classical Hodge operator : m → n−m aside from a reversal of the order of products in the result, which amounts to a sign that depends on m (and is due to our conventions on the duality pairing). This point of view in [17,29] was also recently used for the Hodge star on supermanifolds [4]. The same approach will now be applied to bicovariant differential exterior algebras on Hopf algebras. These are parallelizable via an exterior algebra of leftinvariant differential forms forming a super-braided-Hopf algebra and we do super-braided Fourier transform on this.
In algebraic terms, in the bicovariant case is a super-Hopf algebra equipped with a split super-Hopf algebra projection H . Hence by a super-version of the Radford-Majid theorem [23,38] one knows that ∼ =A· < is the super-bosonisation of as a super-braided Hopf algebra in the braided category of crossed (or Drinfeld-Radford-Yetter) modules [5,38,40,41], where we assume that A has invertible antipode. Moreover, in the standard setting is generated by degrees 0,1 and hence has primitively generated by the subspace 1 of degree 1. As a result we can focus on such primitively-generated super-braided Hopf algebras and translate elements of noncommutative geometry in these terms. Particularly, the exterior derivative restricts to d : → to make into a differential graded algebra (but not within the category, d is not a morphism) and in Section 2 we give an explicit construction of this in the case = max corresponding to the maximal prolongation of 1 . Any other exterior algebra corresponds to a quotient of this and at the other extreme we revisit the more well-known case = min = B − ( 1 ) given by the canonical (super) braided-linear space associated to an object of an Abelian braided category. This B ± ( 1 ) construction appeared in the case where 1 is rigid in [27] while quadratic primitively generated braided-Hopf algebras appeared in [18,24]. The B + construction as an algebra is often called a 'Nichols algebra' cf [37] or 'Nichols-Woronowicz algebra' cf [39] but we note that neither of these works considered B ± as braided-Hopf algebras, that structure being introduced following the development of braided-Hopf algebras by the author. The super-braided Hopf algebra interpretation of the Woronowicz construction of bicovariant differential exterior algebras was in [28,29] among other works. In this context the universal property of B − corresponds in some sense to the minimal relations needed to ensure Poincaré duality, a remark that will be reflected in our approach to the Hodge star. It was also observed recently [35] that the exterior derivative d on a bicovariant exterior algebra is not only a super-derivation but also a super-coderivation in the sense where | | is the degree operator and is the super-coproduct. This turns out to be key to going the other way of building ( , d) from data d on . Although these matters are somewhat familiar, the braided approach to the differential structure requires proofs which we provide as part of a necessary systematic treatment. Section 2.4 similarly provides a canonical construction for the differential exterior algebra on B + ( 1 ) as a quantum-braided plane.
As part of this, and critical for Fourier transform on B ± ( 1 ), is the notion of braidedexponential in our approach to these algebras (being used notably in [27] to inductively build up the quasitriangular structure of quantum groups U q (g) as a succession of q-exponentials). Here [27] are defined in terms of braided factorials [m, ]! as in [18,24] and the latter generalises (1.1). We recall this theory in Section 2.2. In the B + case we have previously proposed braided Fourier theory on the Fomin-Kirillov algebra and its super-version as Hodge star in [29], but without a systematic treatment.
Our central results appear in Section 3. If we have a unique bi-invariant top degree (say of degree n) then super-braided-Hopf algebra Fourier transform gives us a map F : m → * (n−m) and in the presence of a quantum metric a Hodge star operator : m → n−m . This extends in the context above to the geometric : m → n−m . Proposition 3.8 establishes in some generality that commutes with the braided antipode S and is involutive in degrees 0, 1, n − 1, n. This in turn follows from some general results about the superbraided Fourier transform in Section 3.1 which build on our previous diagrammatic work, particularly [11]. Section 3.1 also covers the Hodge operator on the well-known quantum q , the fermionic quantum plane. In Section 4 we focus on the case of coquastriangular Hopf algebras (A, R) [5,18,21]. In line with the braided-Hopf algebra methods of the present paper, we first present a starting point for the construction of 1 itself, namely as the dual of a braided-Lie algebra. We show that every subcoalgebra L ⊆ A is a braided-Lie algebra in the sense introduced in [26]. This gives a significantly cleaner result than previous attempts such as [8] and builds on our recent work [35]. Everything is worked out in detail for C q [SL 2 ] recovering previous 'R-matrix' formulae when we take the standard matrix subcoalgebra, including the 4D braided-Lie algebra [26] and the Woronowicz 4D calculus [39] from = L * . We then compute the canonical braided-Fourier transform on the latter with results as described above.
We work over a general ground field k and q ∈ k × . In examples, we will assume characteristic zero for our calculations. We use the Sweedler notation a = a (1) ⊗ a (2) for coproducts and R v = v (0) ⊗ v (1) for right coactions (summations understood). We denote the kernel of the counit by A + and π : A → A + defined by π a = a − a the counit projection. We will make extensive use of the theory of braided-Hopf algebras [21] including the diagrammatic notation in [19,22].

Braided Construction of Exterior Algebras on Hopf Algebras
In this preliminary section we give a self-contained braided-Hopf algebra approach to bicovariant exterior algebras on Hopf algebras building on our recent work [35]. We recall first that a differential graded algebra means a graded algebra = ⊕ n n equipped with a superderivation d increasing degree by 1 and squaring to 0. The standard setting is where 1 is spanned by elements of the form adb where a, b ∈ A = 0 , and is generated by degrees 0,1; in this case we say that we have an exterior algebra over A. Any first order differential structure ( 1 , d) over A can be extended to a 'maximal prolongation' exterior algebra.

Maximal Prolongation on a Hopf Algebra
When A is a Hopf algebra we can ask that left and right comultiplication extends to a bicomodule structure with coactions L , R on 1 and d a bicomodule map. In this case it is known that becomes a super-Hopf algebra with coproduct that of A on degree zero and L + R on degree 1, see [3,35]. In this case π : → A which sends all degrees > 0 to zero and is the identity on degree 0, forms a Hopf algebra projection split by the inclusion of A. As a result, assuming that the antipode of A is invertible and by a super version of [23,38], we have ∼ =A· < where is a super-braided Hopf algebra in the braided category of right A-crossed (or Drinfeld-Radford-Yetter) modules. We recall that a right Acrossed module means a vector space 1 which is both a right module and a right comodule such that (2) ⊗ (Sa (1) )v (1) a (3) , ∀v ∈ 1 , a ∈ A.
In this case there is an associated map : (1) . A similar map for any pair of crossed modules makes the category of these braided when the antipode S is invertible. Here A + = ker is itself a right crossed module by right multiplication and coaction Ad R (a) = a (2) ⊗ (Sa (1) )a (3) and the result in [39] that first order calculi ( 1 , d) are classified by ad-stable right-ideals can be viewed as saying that they are classified by surjective morphisms : A + → 1 .
Hence in order to construct ( , d) in the surjective case it suffices to take (2.1) as a definition d a := −( π a (1) )( π a (2) ) and show that this is well-defined and extends as a super-derivation of square zero. This will then make ( , d) itself into a differential graded algebra over k.

Proposition 2.2 Let
: A + 1 be a surjective morphism in the category of right crossed modules. Then max = T 1 / ( π ⊗ π ) ker together with d defined by the Maurer-Cartan equation gives a super-braided Hopf algebra in the category which is also a differential graded algebra obeying (2.2)-(2.3). Its bosonisation max = A· < max is the maximal prolongation differential calculus extending Proof We quotient by the minimal subspace in degree 2 for which d : 1 → 2 is welldefined by the Maurer-Cartan equation. Let be the usual cobar coboundary, where j denotes the coproduct in the i'th position, and define d on degree m by d.( π ) m = − · ( π ) ⊗(m+1) ∂. This is well-defined for the same reason as before because ∂ is the sum of terms each acting via the the coproduct. Clearly ∂ is a super-derivation and squares to 0, so d has the same features. We also need to check that we have a super-braided Hopf algebra. Since the algebra is quadratic, the main relation to check is that (( π a (1) )( π a (2) )) := ( π a (1) )( π a (2) ) ⊗ 1 + 1 ⊗ ( π a (1) )( π a (2) ) + π a (1) ⊗ π a (2) − ( π a (1) ⊗ π a (2) ) vanlshes whenever a ∈ I = ker π . This is clear for the first two terms and for the remaining two we use (2.4) to obtain ( π ⊗ π )Ad R (a) (much as in the proof of Lemma 2.1) which indeed vanishes as I is Ad-stable because was a morphism. Hence by the lemma we have the required data ( max , d) and obtain a bicovariant calculus after bosonisation, something one can also check directly from I an Ad-stable right ideal and the structure of A· < max . It is also clear from the construction, since we imposed the minimal relations compatible with the Maurer-Cartan equation, that our calculus is isomorphic to the maximal prolongation.

Braided Linear Spaces
Here we take an aside to recall the theory of braided-linear spaces introduced in [18,24] but in a cleaner form as braided operators rather than braided matrices. Braided linear spaces was our term for primitively generated graded braided Hopf algebras, with particular emphasis in [27] on what have later been called 'Nichols-Woronowicz algebras' [1]. If V is an object in an Abelian braided category then it inherits a morphism = V ,V : V ⊗ V → V ⊗ V obeying the braid relations. Our setting is categorical but we use only the pair (V , ) and tensor powers of V in the following definition. [18,24] Let (V , ) be an object in an Abelian monoidal category and a braiding on it. The braided binomials here are defined recursively by
These are operator versions of binomial coefficients and generalise q-binomials when applied to the category of Z-graded vector spaces with braiding given by powers of q. Relevant to us, the braided factorials also generalise symmetrizers and antisymmetrizers. We need the following main theorem about them: The proof in [18] is written in matrix terms but immediately translates as operators in our setting. In fact, these results amount to identities in the group algebra of the braid group and are best done diagrammatically. The key observations are that ([r, ] ⊗ id) r · · · n−1 = r · · · n−1 ([r − 1, ] ⊗ id) + 1 · · · n−1 (2.5) since the first r − 1 terms in [r, ] commute with r · · · n−1 and by functoriality (since the braided-binomial is a morphism) or directly by induction using Definition 2.3 and repeated use of the braid relations. Using these properties, [18,24] then prove by induction on n that from which the theorem follows by repeated application. Also, by writing the above definitions as diagrams and turning the diagrams up-side down, we have co-binomial maps and co-integers defined by [n, ] = 1 + n−1 + n−2 n−1 + · · · + 1 · · · n−1 Moreover, where the factorials coincide by repeated use of the braid relations or because both cases can be written as σ ∈S n i 1 · · · i l(σ ) where σ = s i 1 · · · s i l(σ ) is a reduced expression in terms of simple transpositions s i = (i, i + 1). Next we consider the tensor algebra T V in an Abelian braided tensor category as a direct sum of different degrees T n V := V ⊗n and product given by concatenation of ⊗. Here T 0 V = k or more precisely the unit object of the category. The unit η of the algebra T V is the identity map from k → T 0 V . Thus is the identity map with suitable rebracketing (with an associator as necessary in the general case). We also consider the identity maps η n : V ⊗n → T n V with η 0 = η. Although all these maps are the identity, we are viewing them in different ways. We will consider two different braided Hopf algebra structures T ± V on T V , as a braided-Hopf algebra or as a super-braided Hopf algebra in the category. Proposition 2.5 [24] [18,Propn. 10.4.9] The tensor algebra has a braided Hopf algebra/super Hopf algebra structure T ± V with coproduct for the two cases. The counit is T n V = 0 for all n > 0.
Proof This is the content of [24] [18,Propn 10.4.9] in the free case where we impose no relations, but we rework the proof in the current more formal notations and we state the super case explicitly. We start with the linear coproduct T 1 V = id V ⊗ η + η ⊗ id V and for T + V we extend this as a Hopf algebra in the braided category, while for T − V we extend as a super-Hopf algebra in the braided category. We do the first case; the other is exactly the same by replacing by − . The proof is by induction assuming the formula for T n−1 V , where in the first line we split off the r = 0 part of T n−1 V and · is the braided tensor product. We then compute out the latter and for the 4th equality we renumber r + 1 → r in the second sum and absorb the otherwise missing r = 1 term. For the 5th equality we use Definition 2.3 and finally combine terms to obtain the desired expression for T n V . Again, this is really a result at the level of the braid group algebra and can be done with diagrams.
In this situation we are now ready to define the (super)-braided Hopf algebra quotients as the braided-symmetric algebra and braided exterior algebra on V respectively. That the coproduct descends to B ± (V ) follows immediately from Proposition 2.5 and Theorem 2.4. That ⊕ m [m, ± ]! is an ideal or equivalently that the product in T V descends to the quotient follows from the arrow-reversed version of Theorem 2.4 where the factorials are on the right. It is easy to see that when our construction is in a braided category and φ : V → W is a morphism then φ ⊗ (the relevant power) in each degree is a morphism B ± (V ) → B ± (W ) of (super)braided-Hopf algebras. This is because, by functoriality of the braidings, the braidings and braided factorials are intertwined by φ on each strand in the diagrammatic picture. The B + (V ) case is also called the Nichols-Woronowicz algebra of V due to the structure of the algebra, but the above description and the fact that it is a (super) braided-Hopf algebra is due to the author. The earliest examples were the braided-line and braided quantum-plane (see [18,24]) while other early examples were U q (n + ) in the work of Lusztig [14].
Our own motivation to consider (2.7) to all degrees of relations was in the case when V has a right dual V * . Recall that V * ⊗n is right-dual to V ⊗n by the nested use of ev V and we use the same nesting convention for a duality pairing , on tensor products. In this case [18,24] the tensor algebras T ± V * and T ± V are dually paired by and B ± (V * ), B ± (V ) are clearly the quotients by the kernel of the pairing. That the product on one side is the coproduct on the other follows immediately from Theorem 2.4 and Proposition 2.5. This means that B ± (V * ), B ± (V ) are nondegenerately paired (super) Hopf algebras in the braided category and the relations of B ± (V ) are the minimal relations compatible with this duality. The merit of this approach is that we also have an immediate result, which will need later: Corollary 2.6 [27] If V has a right dual and B ± (V ) has a finite top degree then it has a right dual via ev = , and the coevaluation map coev : is given by where the construction is independent of the choice of inverse image of [m, ± ]!.
This makes more precise the notion of braided-exponentials in [18,24] without formally assuming that the braided factorials are invertible. It was used explicitly in [27] to construct quasitriangular structures. If we take the well-known case V = kx in the braided category of

Minimal Prolongation on a Hopf Algebra
We now return to our setting of a Hopf algebra A with invertible antipode and a surjective morphism : A + → 1 in the braided category of right A-crossed modules. The following braided-Hopf algebra version of Woronowicz's construction [39] is largely known eg [28] but we provide a new direct construction for d on B − ( 1 ) going through (2.1)-(2.3) from [35]. This is a rather different from the approach in [39], which was to formally adjoin an inner element θ and define d = [θ, }.

Proposition 2.7 Let
: A + → 1 be a surjective morphism is the category of right crossed modules and min = B − ( 1 ). This is a quotient of max and inherits d obeying (2.1)-(2.3). Its super-bosonisation min = A· < min recovers the Woronowicz bicovariant calculus [39] on A associated to the Ad-stable right ideal ker .
Proof We start with the identity (2.4) and Ad-invariance of I implies now that ( π ⊗ π ) (I) ⊂ ker[2, − ] = id − , meaning that the relations ( π ⊗ π ) (I) = 0 of the maximal prolongation in Proposition 2.2 already hold among the quadratic relations in min = B − ( 1 ). The latter is therefore a quotient of max . Next we consider the coboundary ∂ m : A ⊗m → A ⊗(m+1) as in the proof of Proposition 2.2 and regard A as a right crossed module by Ad R and a b = π (a)b. Then π becomes a morphism. Lemma 2.8 below shows that ∂ descends to B − (A) → B − (A) in each degree as it respects the kernels of the relevant braided-factorials. Next, π being a crossed module morphism induces by π ⊗m in degree m a map spanned by elements where at least one of the tensor factors is 1. When we apply ∂ m then every term has at least one tensor factor 1 which is then killed by the final π ⊗(m+1) . This is the first cell of Similarly, given by ⊗n in degree n, andd descends to a map d : This is because the kernel of ⊗m consists of terms where at least one of the tensor factors is in I. When we compute thed of such terms using ∂ m , either a j does not act on this tensor factor, in which case this tensor factor is present in the output of j and the whole term is killed by the action of ( π ) ⊗(m+1) , or j does act on this element. But then · · · ⊗ ( π ⊗ π )( I) ⊗ · · · is in the kernel of id − in the relevant place as seen above, hence vanishes in B − ( ). We are using the fact that the kernel in each degree contains the degree 2 relations between adjacent tensor factors. In this way, d equips min = B − ( 1 ) with a differential as a quotient of the construction for max in Proposition 2.2. One can show that if 1 is inner by θ ∈ 1 then the same applies to , but we are not assuming this. The algebra structure of the bosonosation = A· < min is more well-known to be isomorphic to the one in [39].
The following lemma was needed to complete the proof. Here is the braiding for the crossed-module structure on A whereby π becomes a morphism. (2) ⊗ (Sb (1) )b (3) acting in the i, i + 1 position of a tensor power,

Lemma 2.8 Let A be a Hopf algebra and i = the induced braiding
Here Ad ⊗j denotes the tensor product right coaction on j copies (acting here in the first position).
Proof Clearly since the operators act on different tensor factors, just the numbering changes in the 2nd case. We also find by direct computation in the Hopf algebra that where Ad = Ad R and ⊗ Ad is the tensor product right coaction. As a warm-up, using these relations, we show Starting with this, we next prove by induction that Assuming this for m − 1 in the role of m, for the 2nd equality, where we picked out and computed the 1 · · · m−1 m−1 term from the sum in ∂ m−1 .
Looking now at the last expression, we compute where the Ad terms cancel between the sum and the displaced sum except for the top term of one sum and the bottom term of the other. In the sum all the indices of are two or more smaller than the index of so commute to the right. Combining with our previous calculation, we have Next we use this result as initial base for induction on i in a formula where [m] ≡ [m, ] for brevity and the nesting is rightmost as for braided factorials so The case i = 1 is (2.9) which we have already proven while the case i = m − 1 or i = m, suitably interpreted in the sense of absent sums or products when out of range, proves the lemma. We use identities where the commutation relation is due to acting in different spaces, with renumbering due to the notation. The first equation is a direct computation. One also has which we do not need right now, in the first case due to different tensor products and in the second case because is a morphism in the crossed module category and hence commutes with Ad applied to tensor powers that include those on which acts. Assuming Eq. 2.10 for i − 1 in the role of i, what we need to show to prove Eq. 2.10 for i is −2 as the i+1 and higher commute, reducing index by 1, while i computes as shown. Hence the middle terms gives the first term of which completes our previous sum to give the first desired term. Accordingly we need only show for the remaining term that But this is just the same identity (2.9) already proven but for [r]∂ r−1 , i.e. r = m − i + 1 in the role of m, for the r tensor factors numbered i, · · · , m. This completes our proof of Eq. 2.10 for all i and proves the lemma.
This fleshes out the braided-Hopf algebra interpretation of the Woronowicz exterior algebra on a Hopf algebra [39] using [35] for the direct treatment of d.

Differential Calculi on Braided Linear Spaces
For completeness, we give another braided construction namely the exterior algebra (B) on a Hopf algebras B in a braided Abelian category. This includes braided symmetric algebras B = B + ( 1 ) as above generated canonically by an object 1 . The further data we will need is a surjective morphism : B → 1 in the category such that This data arises naturally as follows: suppose B is a (possibly degenerately) dually paired braided-Hopf algebra from the right (so the pairing is ev : B ⊗ B → 1) and L a rigid primitive sub-object L ⊂ B (so that the coproduct restricted to L is the additive one). We view the duality pairing restricted to a map B ⊗ L → 1 as a map which then obeys (2.11). 1 the image of . In the case of B = B + ( 1 ) or any other graded braided Hopf algebra of the form B = 1 ⊕ 1 ⊕ B >1 generated in degree 1 by an object 1 , we simply take : B → 1 as the projection to degree 1.

Proposition 2.9
Let B be a Hopf algebra in an Abelian braided category and : B → 1 a surjective morphism obeying (2.11). Then is a differential exterior algebra on B in the category (one in which all structure maps are morphisms).
Proof The proof is done in Fig. 1 in a diagrammatic notation where diagrams are read as operations flowing down the page, with tensor products and the unit object 1 suppressed [19,22]. Two strands flowing down and merging denotes the product and one strand flowing down and splitting denotes the coproduct. Such proofs apply generally but for convenience of exposition we will also refer to concrete elements. The braided tensor product B⊗ 1 in concrete terms means (b ⊗ v)(c ⊗ w) = b (v ⊗ c)w and featured already in the definitiion of a braided-Hopf algebra. Part (a) computes d(bc) using the braided coproduct homomorphism property and Eq. 2.11. Using the counit axioms and a morphism we obtain bdc for the first term and (db)c for the second when we remember the braided tensor product. Part . 1 Diagrams in the proof of Proposition 2.9 for quantum differentials on braided planes (b) checks that d extends as a graded derivation with respect the braided tensor product. We compute hence vanishes in B ⊗ 2 which agrees with dω = 0 for all ω ∈ 1 . The same applies starting in ⊗ B.
This gives a differential structure on our braided-symmetric algebras B + (V ) regarded as noncommutative spaces. If the category is the comodules of a coquastriangular Hopf algebra, for examples, our construction is covariant in that all structure maps are comodule maps. Also note that if {e i } is a basis of V , we have explicitly They are given explicitly at the level of the tensor algebra by where the last tensor factor of the result is viewed in 1 .

Remark 2.10
If we take the quadratic version S(V ) := B quad + (V ) = T V / ker(id + ) then our above construction gives where ! denotes the Koszul dual. If a quadratic algebra on a vector space W has relations R ⊂ W ⊗ W as the subspace being set to to zero then its Koszul dual is the quadratic algebra on W with relations R ⊥ ⊂ W ⊗ W . This is normally done in the category of vector spaces but we do it here in a braided category using the right dual so that W = V .

Example 2.11
We let B = A 2 q = B + (V ) be the quantum plane associated to the standard corepresentation V = span{x, y} in the braided category of right k q [GL 2 ]-comodules with q 2 = 1. Here is given by a particular non-standard normalisation of the usual R on k q [GL 2 ] (one that does not descend to k q [SL 2 ]). The kernel of id + gives us the relations yx = qxy of the quantum plane since where this time the same basis is denoted {dx, dy} as a basis of 1 = V and one can check for example that (id + )(dx ⊗ dy) = dx ⊗ dx + qdy ⊗ dx from the stated braiding. Indeed, it known that A 2 q and A 0|2 q as Koszul dual as first pointed out by Manin [36]. The differential . This again comes from the same braiding as above but viewed now as defining the relations One can check for example that d(yx − qxy) = 0 as it should. By construction, this exterior algebra on the quantum plane is k q [GL 2 ]-covariant. The associated partial derivatives are using the braided coproduct on general monomials computed in [18] from the braidedintegers [n, ]. The partial derivatives here were first found by Wess and Zumino in another approach. They are naturally 'braided right derivations' with an extra q n in the first expression, in order that d is a left derivation, acting as braided q 2 -derivatives in each variable. One can check that ∂ 2 ∂ 1 = q∂ 1 ∂ 2 as operators, also as per the general theory in [18].
This reworks the treatment of quantum-braided planes and their differentials in [18, Section 10.4] [24] now as an example of our above canonical construction based on B + (V ), as opposed to a compatible pair of R-matrices R, R as previously.

Braided Fourier Transform and Application to Hodge Theory
Fourier transform on Hopf algebras is part of their classical literature. It was extended to braided-Hopf algebras in [15] and related works and applied to braided linear spaces in [11], though not the ones we consider here. We used diagram proofs and will do so again, while another work from that era is [16]. We first explain the general (super) formulation and then apply it to the Hodge operator, including k(S 3 ) as an example.

Super-Braided Fourier Theory
In any braided category C and B ∈ C a braided Hopf algebra dually paired with a braided Hopf algebra B , we have three actions [19] which we will consider and which we collect in Fig. 2 in the same diagrammatic notation as in the proof of Proposition 2.9, reading down the page. As in Section 2 when discussing duals, we assume a pairing ev : B ⊗ B → 1 which we can write diagrammatically as ∪ and with respect to which the product on one side is adjoint to the coproduct on the other. The counit of B is also adjoint to the unit η :  [19].
We also need the notion of a left integral and the simplest thing is to require a morphism : B → 1 in the sense (id ⊗ ) = η ⊗ . However, we do not want to be too strict about this. For example, for the finite anyonic braided line B = k[x]/(x n+1 ) in the braided category of Z/(n+1)-graded spaces with braiding given by an n+1-th root of 1 and |x| = 1, the obvious x m = δ m,n is not a morphism to 1. Our approach is to live with this and not necessarily assume any morphism properties; we can still use the diagrammatic notation but be careful not to pull the map through any braid crossings. A more formal approach is to view it as a morphism B → K where K = k taken with degree n in the case of the anyonic braided line. The uniqueness of the integral when it exists is similar to the Hopf algebra case (see [2] for a formal proof).
For Fourier transform we need not only that B is dually paired but that B is actually rigid with dual object B * . Again, this is a very strong assumption, analogous to finitedimensionality of B and amounting to this in the typical k-linear case. It means that there is a coevaluation map exp = coev : 1 → B ⊗ B * , denoted by ∩ in the diagrammatic notion, which obeys the well-known 'bend-straightenning axioms' with respect to ∪. We similarly require a right integral which is not necessarily a morphism : B → 1. We can live with this or suppose formally that : B → K * where K * ⊗ K = 1 = K ⊗ K * as objects. In our k-linear setting this will be by the identification with k. The theory below could be generalised to include some infinite-dimensional cases or else these could be treated formally eg in a graded case with B * a graded dual, each component rigid and the result a formal power series in a grading parameter.
Definition 3.1 [11,15] Let B be a Hopf algebra in a braided category with B rigid and a left integral as above. Then F : B → B defined in Fig. 2 is called the braided Fourier transform. We similarly define a dual Fourier transform F : B → B if B has a right integral.
These maps are no longer morphisms if the integrals are not, or one can say more formally that F : B → K ⊗ B and F : B → K * ⊗ B . The following extends and completes [11].

Proposition 3.2 In the setting of the definitions above
If the integrals are both unimodular and morphisms then FF = μS and [F , S] = 0 when μ is invertible (see Fig. 3 for the general case).
Proof Here FReg and F F are already covered in [11, so we do not repeat all the details here. We recall only the diagram proof for F * F using the lemma in [11,Fig. 2b] at the first equality in Fig. 3 and note that we did not need to assume that , * are morphisms to 1 as in [11] as long as we keep the integrals to the left. The second line now uses the same and ditto with x replaced by y. One can see that F F = μS using q n(n+1) 2 = (−1) n . Due to the nontrivial braidings of the integrals, however, the right hand side in Fig. 3 gives where D is the monomial degree operator. The same method as in the proof above now gives us FS = q 2D+1 SF or equivalently SF = FSq 2D+1 , which one may verify from the stated F, S.
There is up to scale a unique top degree in each case, namely Vol = e 1 e 2 and Vol * = f 1 f 2 and we find Vol We define integrals via Vol = 1 and * Vol * = 1 but note that these are not morphisms.
Rather we use braidings Vol * ⊗ Vol needed in the computation of F . We now read off from the diagrammatic definitions in Fig. 2, One can verify that F F = μS as it must by Proposition 3.2. We also have where D is the monomial degree and one can check that this agrees with the lower line in Fig. 3 where we use the above computations to read off the right hand side. In this case FS = q 2(D−1) SF or SF = FSq 2(D−1) as one can verify from the stated form of F, S.
A similar approach can be used for other quantum planes to express their differential exterior algebras as super-braided Hopf algebras with possibly a different underlying coquastriangular structure from one used for the coordinate algebra as a braided-Hopf algebra. In the presence of an invariant quantum metric we reproduce the otherwise ad-hoc approach to q-epsilon tensors and Hodge theory on braided-quantum planes in [18]. The f i generate antisymmetric vectors and , define an interior product connected to the exterior algebra product via F. This example should be seen as a warm-up to Section 3.2 where we look at bicovariant differentials on Hopf algebras themselves. As illustrated here, the actual theory is read off the diagrams with the appropriate braiding including signs. We could indeed shift all constructions to this new super-braided category and say that the above example is an ordinary braided-Hopf algebra there, but we not do so since there will normally be other (bosonic) objects also of interest in the original category. In our context the nicest case is where , * are morphisms to 1 when viewed in the original category but do not necessarily respect the super-degree, for example they could be odd maps in the super-sense in which case they are not morphisms in the super version with extra signs (so we need the slightly more general picture as above). We assume they have the same parity of support (both odd or both even maps). Then F, F also have this party. Proof For FF we have to compute the right hand side of the lower diagram in Fig. 3, which now has extra signs. We can still bring out μ = ( * ⊗ ) −1 sup exp since any signs from crossing the leg cancel with signs from crossing the leg by our assumptions. Next, we can lift through the crossing at the price of (−1) p in computing μ . We already have F F = μS from Proposition 3.2 and can then conclude the rest.
The behaviour of F with respect to Reg, has an unchanged form as these statements do not involve additional transpositions, except that the actions themselves are computed for the super-braided Hopf algebra eg with the super-braided coproduct and hence the superbraided Leibniz rule expressed in super-braided module algebra structures. The property (3.1) becomes SF • = ·((−1) pD ⊗ SF ) (3.2) due to the crossing of the first input on the right hand side with the integral in F.

Hodge Theory on Hopf Algebras
We are now going to compute our super-Fourier theory for B = min = B − ( 1 ) where 1 is a rigid object in the braided category of right A-crossed modules and A is a Hopf algebra with invertible antipode. Here B is a super-braided Hopf algebra in the category and we assume it has a top degree component K of dimension 1, i.e. up to scale a unique top form Vol ∈ B − ( 1 ). This gives us a unimodular integral B → k by Vol = 1 and zero for lower degrees. We have seen this in examples, while in general we obtain an integral because the formula in Proposition 2.5 ensures that Vol = Vol ⊗1+1⊗Vol plus terms of intermediate degree, so that we never reach the top degree on applying to lower degree. One can think of this more formally as a morphism B → K with some possibly non-trivial generator. We also have an identification B = B − ( 1 * ) by extending the duality pairing 1 * ⊗ 1 as a braided-Hopf algebra pairing, given that this is now non-degenerate after quotienting by the relations of B − as explained in Section 2.2. Hence we obtain a unimodular integral on this too. In the nicest case, the span of the top forms Vol, Vol * of degree n (say) are isomorphic to the trivial object 1 so that , * are morphisms to 1 but of parity n mod 2, so we are in the setting of Corollary 3.5. Next, in noncommutative geometry a metric is g ∈ 1 ⊗ A 1 with an inverse ( , ) : 1 ⊗ A 1 → A. One can show that in this case g must be central. Normally, one also requires the metric to be 'quantum symmetric' in the sense of the product ∧(g) = 0 in 2 . We are interested in left-invariant metrics where g ∈ 1 ⊗ 1 . Proof The metric being bi-invariant means that it is an element g ∈ 1 ⊗ 1 which is invariant under the coaction R on the tensor product. The existence of a bimodule map ( , ) requires g to be central which in turn requires that g is invariant under the crossed module right action (since this determines the cross product of A· < ). So a metric is equivalent to a morphism 1 → 1 ⊗ 1 in the crossed module category. Evaluation from the left makes this equivalent a morphism as stated, which we also denote g. Here * is again a right crossed module in the usual way (via the antipode). Clearly ∧(g) = 0 if and only if g ∈ ker[2, − ] = ker(id − ) according to the relations of B − ( 1 ).

Lemma 3.6 A bi-invariant metric on a Hopf algebra A with bicovariant calculus is equivalent to an
Given a bi-invariant metric we therefore have B ± ( 1 * ) ∼ =B±( 1 ) hence combined with the above remarks in the finite-dimensional case, an isomorphism which we also denote g : B ± ( 1 ) * → B ± ( 1 ). We are now ready to define the Hodge operator, using the B − version. We do it in the nicest case but the same ideas can be used more generally as we have seen in Section 3.1. By construction our is a morphism in the crossed-module category. In geometric terms this means that it extends as a bimodule map and is bicovariant under the quantum group action on . We also define = (−1) D • F • g −1 where D is the degree operator. Proof Here Vol, Vol is non-zero since otherwise Vol would be zero in B − ( 1 ), and its inverse supplies the coefficient of the top component of exp, which is μ. Since μ = 0 we can apply Corollary 3.5 to see in particular that , S graded-commute. That S| 0,1,n−1,n = (−1) D i.e. on the outer degrees is clear on degrees 0,1 and then holds on degrees n, n − 1 due to , S graded-commuting. Next, in terms of with the metric identification, the result in on all degrees, giving | 0,1,n−1,n = μ −1 on the outer degrees. On the other hand, we have exp = 1 ⊗ 1 + g + · · · + g (n−1) + μVol ⊗ Vol (for some element g (n−1) ∈ n−1 ⊗ n−1 ), while the definition of is such that it is given by integration agains −1 exp without any signs. Since g (by the quantum symmetry assumption) and 1 ⊗ 1 are invariant under , these terms are the same, and hence = on degrees n − 1, n and hence 2 = μ on these degrees. In that case ( ω) = μω = 2 ω on all ω of degree n − 1, n tells us that = on degrees 0,1 also, and hence that 2 = μ on these degrees also. This means that −1 sup exp = 1 ⊗ 1 − g + · · · + (−1) n−1 g (n−1) + (−1) n μVol ⊗ Vol for the computation of F and similarly without the signs for .
We similarly define left and right interior products by restricting the left and right actions in Section 3.1 (these are the left and right braidedpartial derivatives in the sense of [18,24]). We then extend these to bicovariant bimodule maps given by (aη) (bω) = (aη, bω (1) )ω (2) , (bω) (aη) = bω (1) (ω (2) , aη), ∀a, b ∈ A, η ∈ 1 , ω ∈ where we underline the braided-coproduct of . In other words, we extend the braided coproduct as a bimodule map → ⊗ A (not to be confused with the super-coproduct of as a super-Hopf algebra) and then use the quantum metric pairing to evaluate, taken as zero when degrees do not match. We can now interpret our Fourier theory in Section 3.1 as where S is the super-braided antipode of B − ( 1 ) extended as a bimodule map to . It should not be confused with the super-coproduct of . We also define a left Lie derivative by and associated codifferential and Hodge Laplacian The use of (S ) −1 here is adapted to the left handed and left-handed partial derivatives defined by df = a (∂ a f )e a for any choice of basis {e a } of 1 . One could equally well use but this would be adapted to and right-handed partial derivatives. We also define the Leibnizator L δ (ω, η) = δ(ωη) − (δω)η − (−1) |ω| ωδη as in [32].

Corollary 3.9 The codifferential and Hodge Laplacian in Eq. 3.4 obey
for all f ∈ A and ω ∈ . Moreover, δα = α a δe a + g ab ∂ a α b , f = (∂ a f )δe a + g ab ∂ a ∂ b f α = α a e a +( α a )e a +∂ a α b (L δ (e a , e b )+L e a e b )+∂ a ∂ b α c ((e a (e b e c ))+g bc e a −g ab e c ) where α = α a e a in a basis and g ab = (e a , e b ) (summation understood). If δα = 0 then α = α a δde a + ∂ a α b (δ(e a e b ) + e a de b ) + ∂ a ∂ b α c (e a (e b e c )) Proof The formula for δ(f ω) follows immediately from the derivation property of d and the first interior product property in Eq. 3.3. The formula for (f ω) then follows from this and the Leibniz rule for d as in [32]. These results then give the explicit formulae for α = α a e a .
Note concerning α that (e a (e b e c )) + g bc e a − g ab e c = (e a e b ) e c in the classical case, which is antisymmetric in a, b, while L δ (df, ω) + L df ω = 2∇ df ω in the classical case as shown in [32]. Here ∇ is the classical Levi-Civita connection referred back to a derivative along 1-forms via the metric. The special case shown in Corollary 3.9 is relevant to 'Maxwell theory' where F = dα and Maxwell's equation δF = J has a degree of freedom to change α by an exact form, which freedom can be reduced by fixing δα = 0. Maxwell's equation then becomes α = J where J is required to be a coexact 'source'.

Finite Group Case
To give a concrete example we recall that any ad-stable subset of a finite group not containing the group identity defines an ad-stable ideal in k(G) + and hence a bicovariant calculus. There is a canonical basis of 1-forms {e a } labelled by the subset and relations e a f = R a (f )e a where R a (f ) = f (( )a) is right translation. This corresponds to crossed module action e a f = f (a)e a . There is a natural choice of bi-invariant metric namely g = a e a ⊗ e a −1 provided our subset is closed under inversion. Here the left coaction is trivial on 1 and the right coaction is the crossed module one, namely R e a = g∈G e gag −1 ⊗δ g . The element θ = a e a is similarly bi-invariant and makes the calculus inner (so d = [θ, } as a graded commutator). The above results therefore give us a Hodge star on any finite group with bicovariant calculus stable under inversion and for which B − ( 1 ) has (up to scale) a unique top form which is bi-invariant. The braiding is (e a ⊗ e b ) = e aba −1 ⊗ e a , from which the braided factorials can be computed. The metric sends g(f a ) = e a −1 where {f a } is a dual basis. The interior products given by the braided-derivatives are e a (e b e c ) = g ab e c − g a,bcb −1 e b , (e a e b ) e c = e a g bc − e aba −1 g ac on evaluating against the super-braided coproduct via the metric. One also has dθ = 0 and in nice cases (when the associated quandle is locally skew and the field is characteristic zero) it was shown in [34] that H 1 (G) = kθ. If one similarly has H n−1 (G) = kθ as an expression of Poincaré duality (which is often the case, including in the following example) then a source J is coexact if and only if δJ = 0, G (J, θ ) = 0 by elementary arguments using Eq. 3.3, where G means a sum over the group. Here : n → k is defined as the tensor product of G and on B − ( 1 ) and dβ = 0 on any n − 1-form β under our assumption of a unique bi-invariant top form Vol of degree n.

Example 3.10
The standard 3D calculus on the permuation group S 3 on 3 elements is given by the conjugacy class of 2-cycles. We recall that 1 = k − span{e u , e v , e w } is a k(S 3 )crossed module as above, where a = u, v, w are the 2-cycles u = (12), v = (23), w = uvu = vuv = (13). The minimal exterior algebra in this case is known to be a super version of the Fomin-Kirillov algebra [6,29] with relations and exterior derivative e u e v + e v e w + e w e u = 0, e v e u + e w e v + e u e w = 0, e 2 u = 0, de u + e v e w + e w e v = 0 and the two cylic rotations of these where u → v → w → u. The dimensions in the different degrees are dim( ) = 1 : 3 : 4 : 3 : 1 so there is a unique top form up to scale, which we take as Vol = e u e v e u e w . This is clearly central and one can check that it is also bi-invariant. This can be done noting that Vol = e u e v e u θ and computing R (e u e v e u ) = e u e v e u ⊗ (δ e + δ w ) + e w e u e w ⊗ (δ u + δ vu ) + e v e w e v ⊗ (δ uv + δ v ). Hence we have a canonical Hodge star. The coevaluation element exp is a computation from Proposition 2.6 which in a basis reads where {e a } is our basis of 1 and {f a } is a dual one and we sum repeated indices. This is in general but in our case and using the metric identification comes out as (e a e b ) = e aba e a = · (e a ⊗ e b ) We see that on the different degrees, 2 | 0,1,3,4 = −id, 3 | 2 = id so that has order 6. The first of these illustrates Proposition 3.8 while we note that on degree 2 coincides with minus the braided-antipode S of B − ( 2 ) (because this is braided-multiplicative along with an extra sign for the super case, and S| 1 = −id). The cohomology for this calculus in characteristic zero is known to be H 0 = k, H 1 = k, H 2 = 0, H 3 = k, H 4 = k and one can see that is an isomorphism H m∼ =H 4−m as expected, the cohomologies being spanned by 1, θ = e u +e v +e w in degrees 0,1 and their in degrees 3,4.
Next we compute S on degree 3 as S(e u e v e w ) = − · (Se u ⊗ S(e v e w )) = · (e u ⊗ e u e v ) = e u e w e u = −e u e v e w . A similar computation gives SVol = Vol so that S = (−1) D on degree D = 2 as we saw in Proposition 3.8, while we have already observed that S = − on degree 2.
We now compute := F •g as a check of our theory. The inverse braiding is −1 (e a ⊗ e b ) = e b ⊗e b −1 ab in general and extends similarly to products, with the metric identification, we have which is (-2 times) the standard graph Laplacian for the corresponding Cayley graph on S 3 . It is fully diagonalised as usual by the matrix elements of irreducible representations (the eigenvalues are 0, 6, 12 with eigenspaces of dimensions 1,4,1 respectively). We also have δα = g ab ∂ a α b and if this vanishes then We note that the last term here only has contributions from a = b. The above expression is a short computation from Corollary 3.9 using Solving Maxwell theory in the form α = J , the source J has to be coexact. From the remarks above and H 3 (S 3 ) = kθ , this is equivalent to It is a useful check of our formula for on 1 to see directly that when restricted to coexact forms its image indeed is again coexact. Moreover, by computer one finds the same eigenspaces (each 4-dimensional) as in [33] with eigenvalues 3,6,9, so that up to an overall constant the Laplacian restricted to coexact 1-forms is the same in spite of the Hodge operators being rather different. Explicitly, e u − e v , e u − a (δ au + δ uau )e a and their cyclic rotations under u → v → w → u have eigenvalue 3 (and along with their cyclic rotations add up to zero). Multiplying these by the sign function on S 3 gives eigenvectors of eigenvalue 9 while for the eigenvectors of eigenvalue 6 we can use the 'point sources' in [33], where three points that share a common node in the graph have a zero sum of their sources. This solves the 'Maxwell theory' on S 3 for this calculus by diagonalising on coexact 1-forms.
In [33] and all other such models until now it has been assumed that the Hodge operator should be designed to square to ±1, whereas our canonical Hodge operator in this example is order 6 and looks very different, but nevertheless gives the same reasonable Laplacians in degrees 0,1. Our construction also works for S 4 and S 5 with their 2-cycles calculus and can be analysed similarly, while higher S n , n > 5 are conjectured [29] to have infinitedimensional B − ( 1 ).

Calculus and Hodge Operator on Coquasitriangular Hopf Algebras
Here we start with a new, braided-Lie algebra, approach to the construction of bicovariant ( 1 , d) on quantum groups such as k q [G] for G a complex semisimple Lie group. We recall that these are all coquasitriangular in that they come with a convolution-invertible map for all a, b, c ∈ A. This is just dual to Drinfeld's theory in [5], see [18,21]. We will need the 'quantum Killing form' The construction of differential calculi on a coquasitriangular Hopf algebra A has its roots in R-matrix constructions from the 1990s but the following general construction builds on our recent treatment in [35]. It is shown there that A is a left A-crossed module by Ad L (a) = a (1) (1) )R(a (2) ⊗ b (3) ). (4.1) Then any subcoalgebra L becomes a left A-crossed module by restriction and its dualisation L * in the finite-dimensional subcoalgebra case becomes a right A-crossed module.
It is shown that the quantum Killing form regarded by evaluation on its first input as a map Q : A + → L * is a morphism of crossed modules. This gives:

Proposition 4.1 cf [35] Let A be a coquasitriangular Hopf algebra and L ⊆ A a nonzero finite-dimensional subcoalgebra. Then 1 = image(Q) and
= Q defines a bicovariant differential calculus 1 on A.
In [35] we used a version of this to naturally construct possibly non-surjective differential calculi with 1 = L * , but we also see from this result that L itself is the more fundamental object as starting point.

Braided-Lie Algebras
Our new approach is to start with a Hopf algebra B in a braided category C and find a 'Lie algebra' for it. We then take its dual to define a calculus.

Definition 4.2 [26]
A left braided-Lie algebra is a coalgebra L in a braided category, together with a morphism [ , ] : L ⊗ L → L subject to the axioms shown diagrammatically in Fig. 4. The associated braiding˜ and braided-Killing form are also shown.
A principal result in the case of an Abelian braided category is the construction of the braided-enveloping algebra U(L) as a bialgebra. This is defined by the relations of commutativity with respect to the associated braiding˜ . In the category of sets a braided-Lie algebra reduces to a quandle and this was used recently to prove the cohomology theorem for finite group bicovariant calculi [34]. Nondegeneracy of the Killing form also turns out to be an interesting characteristic related at one extreme to the Roth property of a finite group [13]. The axioms themselves, however, were inspired by the properties of the braided adjoint action of a braided-Hopf algebra on itself as also recalled in Fig. 4. The first part was done in [26]. Braided cocommutativity with respect to a B-module is just the axiom (L2) when specialised to [ , ] = Ad and the proof that the adjoint action then obeys (L3) appeared in [25,Prop A.2] in dual form (turn the diagrams there up-side-down). Clearly: Corollary 4.4 Any subcoalgebra L ⊆ B of an Ad-cocommutative braided-Hopf algebra closed under Ad is a braided-Lie algebra by restriction.

Proposition 4.6 When L ⊆ A is finite-dimensional (1) U(L) is Koszul dual to a righthanded quadratic version of the bicovariant calculus in Proposition 4.1. (2) The braided-Killing form is
where {e i } is a basis of L and {f i } a dual basis and u(a) = R(a (2) ⊗ Sa (1) ). as stated. For the remark about the dualisation we note that A has a right crossed-module structure given by Ad R and a b = a (2) (1) )R(a (3) ⊗ b (2) ), ∀a, b ∈ A and its crossed module braiding turns out, by similarly using the properties of R as above, to coincide with the fundamental˜ for the braided-Lie algebra (as computed in the proof of Theorem 4.5). On the other hand this crossed module is the right handed version of Eq. 4.1 which dualized to give the crossed module structure defining the calculus in Proposition 4.1. This means that U(L) is the Koszul or quadratic algebra dual of quad (where we impose only the degree 2 relations of B − ( 1 )). The braided-Lie bracket and exterior derivative can also be related as part of a general theory of 'quantum Lie algebras' in [39] when 1 / ∈ L. Here every bicovariant calculus gives a quantum Lie algebra in the sense of [39] and meanwhile (one can show that) every non-unital braided-Lie algebra L gives a quantum Lie algebra by extending by 1 and then taking the kernel of the counit.
The above theorem is a new result and is needed to complete the picture. In the special case where L has a matrix coalgebra form on a basis {t i j } (such data defines a matrix corepresentation of A) we recover the R-matrix braided-Lie algebra construction introduced in [26] but now as a corollary of the above.

Corollary 4.7 cf [26] Let A be a coquasitriangular Hopf algebra and t ∈ M n (A) a matrix corepresentation. Then the matrix subcoalgebra L = {t i j } has braided-Lie bracket, categorical braiding and braided Killing form
The braided enveloping algebra U(L) is generated by the {t i j } with new relations We sum over repeated indices in these expressions.
Proof We expand out Q using the properties of R, then the above bracket can also be written explicitly as (1) )R(b (4) ⊗ a (2) )R(a (3) ⊗ b (5) )R(a (4) ⊗ Sb (1) ) (4.6) and the categorical braiding in M A is (2) )R(Sa (2) ⊗ b (4) )R(a (4) ⊗ b (5) )R(a (5) ⊗ Sb (1) ) (4.7) From these we immediately read off the expressions stated, where R i is the 'second inverse'. We likewise read off the relations of U(L) from˜ or from Eq. 4.3 to give the result stated. In all cases we can moveR and another R to the left hand side, for example the relations can be written compactly as  [18,20], the braided analog of the more familiar FRT bialgebra A(R).
This derives the explicit R-matrix formulae needed to compute examples. This in turn recovers the 4D braided-Lie algebra of k q [SL 2 ] found in [26]: Example 4.8 [26] For A = k q [SL 2 ] with q 2 = ±1, its standard matrix coalgebra and rescaled generators where λ = 1 − q −2 (we use different symbols for the entries of t i j to avoid confusion with the quantum group), the nonzero braided-Lie brackets are Here (n) q = (q n − q −n )/(q − q −1 ) and we used Corollary 4.7 and the standard R-matrix for SL 2 with nonzero entries R 1 2 2 The braided Killing form is (4) q /q 13 times the nonzero values The enveloping algebra U(L) = B q [M 2 ] is generated by α, β, γ , δ with relations at this point denotes commutator not Lie bracket. This is the algebra of qdeformed 2 × 2 braided hermitian matrices which means that geometrically it should be thought of as q-Minkowski space [18,20]. There are two natural central elements, the braided determinant det q = αδ − q 2 γβ which should be thought of as the q-Lorentzian distance from the origin and q-trace tr q = q −1 α + qδ = t which should be thought of as the 'time' direction. In these variables (as opposed to the rescaled 'Lie algebra' variables) the classical limit is commutative allowing us to think of this as a noncommutative geometry. Over C our braided-Lie algebra has a natural real form or * -involution α * = α, β * = γ, δ * = δ for real q, which fits with the mentioned geometric picture.

Calculus and Hodge Operator on k q [SL 2 ]
In the case of a coquasitriangular Hopf algebra A with a generating matrix subscoalgebra {t i j }, Proposition 4.1 or dualization of Corollary 4.7 recovers a version of a known R-matrix construction [10] The associated right crossed-module braiding on 1 will be denoted˜ also (it is adjoint to the one for the braided-Lie algebra) and is computed from the right crossed module structure as and expands out as We now focus on A = k q [SL 2 ] where the smallest nontrivial irreducible is 2dimensional, giving us 1 = M 2 (C). We write basis E 1 1 = e a , E 1 2 = e b , E 2 1 = e c , E 2 2 = e d and use the standard SL 2 R-matrix as in Example 4.8 to give the bimodule relations of the well-known 4D calculus first found in [39], where [x, y] q := xy − qyx and λ = 1 − q −2 = 0. The exterior differential is necessarily inner with θ = e a + e d which implies that Note that we should scale d or θ by λ −1 in order to have the right classical limit but we have not done this in order to follow the general construction. The right coaction on left-invariant 1-forms is As in degree 0, we note that λ −1 d has the right classical limit not d itself. The dimensions here in each degree are dim( ) = 1 : 4 : 6 : 4 : 1. The following is mostly known e.g. [30] but we give a short proof as it is critical for us.

Proposition 4.9
For the above 4D calculus on k q [SL 2 ] with q 2 = ±1 there is a unique bi-invariant central metric The inverse metric is . We can add to this a multiple of θ ⊗ θ since this is also invariant [30], which we have now fixed so that g is central. Thus for example, (e c ⊗ e b +q 2 e b ⊗ e c )a = e c ⊗ ae b + q 3 λe b ⊗ be a + q 2 e b ⊗ ae c = a(e c ⊗ e b + q 2 e b ⊗ e c ) + qλbe a ⊗ e b + q 3 λbe b ⊗ e a + q 4 λ 2 ae a ⊗ e a (e z ⊗ e z − θ ⊗ θ)a = −(1 + q −2 )(λe a ⊗ e a + e a ⊗ e d + e d ⊗ e a )a = a(e z ⊗ e z −θ ⊗ θ)−(1−q −4 )(q 2 λae a ⊗ e a +qbe b ⊗ e a +q −1 be b ⊗ e a ) using the above commutation relations. Comparing these we see that We now use g to identify 1 * ∼ = 1 and compute using Vol = 1. where e i , 1 ≤ i ≤ 4 refer in order to e a , e b , e c , e d and I = (i 1 , j 2 , · · · , i m ) with i 1 < i 2 · · · < i m labels of a basis of m and m B I J = e i 1 · · · e i m , e j 1 · · · e j m = ev(e i 1 ⊗ e i 2 · · · ⊗ e i m , [m, −˜ ]!(e j 1 ⊗ e j 2 · · · e j m )) = g i 1 p 1 · · · g i m p m [m, −˜ ]! in the basis enumerations 12,13,14,23,24,34 and 123,124,134,234 respectively for the middle cases here. In particular, we see that μ = q 6 . The matrix 1 B −1 here is the inverse of the matrix g ij in our basis and necessarily gives the coefficients of the metric g ∈ 1 ⊗ 1 , and we note also that μ = 1/ det(g) in this basis. We then carefully integrate against this exp, for example where we read from the 2nd row of 2 B −1 for the terms in exp of the form e a e c ⊗ · · · and from the last row for terms of the form e c e d ⊗ · · · . The other possibilities in our basis for the first tensor factor of exp have zero integral. We then evaluate the first displayed integral as −q 2 and the second integral as λ on using the relations of the exterior algebra, to give q 4 e b e d + (1 − q 4 )e a e b as stated. Integrating against g is easier and gives on degree 3. We could now deduce on degree 1 using Proposition 3.8 that 2 = μ on degrees D = 2 but one can also compute it similarly by integrating against 3 B −1 for a direct calculation and then verify 2 . The polynomial identify for on degree 2 is a direct calculation.
We see that on degree 2 is not of finite order for generic q but is a deformation of order 2. Indeed, μ −1 = det(g) (see the proof above) suggests a geometric normalisation to = μ − 1 2 = q −3 in our case, then is involutive on degree D = 2 and obeys the standard q-Hecke relation 2 = id + (q − q −1 ) on degree 2. This is the same relation as obeyed by the braiding in the defining representation of the quantum group, which is also the braiding on the generators of the associated quantum plane. One can also compute directly and verify that it is given on degree 2 by One may similarly compute S on degrees 3,4 to find S = (−1) D id on all degrees D = 2 as must be the case by Proposition 3.8. Note that this feature of the antipode is not true for the outer degrees of all braided exterior algebras, see Example 3.4. Finally, one can check that [ , S] = 0 on all degrees as it must by Corollary 3.5, which in degree 2 provides a very good cross-check of both the displayed S and computations.
The cohomology for this calculus in characteristic zero and for generic q is known to be H 0 = k, H 1 = k, H 2 = 0, H 3 = k, H 4 = k, which is the same as in Example 3.10. One can see that is again an isomorphism H m∼ =H 4−m , the cohomologies being spanned by 1, θ = e a + e d in degrees 0,1 and their in degrees 3,4, where θ = −q 4 e b e c e z deforms the classical 3-volume. Applications of this theory to q-electromagnetism will be considered elsewhere. Here we consider only the general result in Section 3.2 for the Laplacian on functions. We first recall from [30]  is the naturally arising q-deformed Laplace-Beltrami operator on SL 2 . There are general arguments that ∂ 0 relates to the action of the quadratic q-Casimir of the quantum enveloping algebra, but we will need the direct calculation as stated.

Proposition 4.11
For q 2 = ±1 and the quantum metric in Proposition 4.9 on the 4D calculus on k q [SL 2 ], the Hodge Laplacian on degree 0 is where g ij = (e i , e j ) in our basis {e b , e c , e z , e 0 } and we sum over i, j .
Proof Writing e a = (θ + e z )/(1 + q −2 ) and e d = (θ − q 2 e z )/(1 + q 2 ) we have in terms of the partial derivatives in our original basis read off from Eq. 4.9. The former comes out on c k b n d m as ∂ z = λ (2) q (q m+n+2 (k) q − q −k (m + n) q ) + λ 2 (2) q q 2−m (k) q (n) q S − c S − b