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Braided homology of quantum groups

We study braided Hochschild and cyclic homology of ribbon algebras in braided monoidal categories, as introduced by Baez and by Akrami and Majid. We compute this invariant for several examples coming from quantum groups and braided groups.


Introduction
Braided Hochschild homology HH Ψ * (A) was first defined by Baez [2] for an algebra A in a braided monoidal category C. Baez defined HH Ψ * (A) by an explicit chain complex, and showed that HH Ψ * (A) = Tor A⊗A op * (A, A), where A⊗A op is the braided enveloping algebra of A. However, his constructions relied on the assumption that A is weakly Ψ-commutative, meaning that µ • Ψ 2 = µ, where µ is the multiplication morphism of A. This was overcome by Akrami and Majid [1], who replaced weakly Ψ-commutative algebras by ribbon algebras, namely algebras A in C possessing an invertible morphism σ ∈ Mor C (A, A) that satisfies σ • µ = µ • (σ ⊗ σ) • Ψ 2 . They showed that there is a corresponding cyclic theory (in the sense of Connes [7]), which they called braided cyclic homology HC Ψ,σ * (A). If C is the category of C-vector spaces with braiding given by the flip v ⊗ w → w ⊗ v, then an algebra in C is simply an associative C-algebra, ribbon automorphisms are precisely algebra automorphisms, and the Akrami-Majid cyclic object reduces to the cyclic object defining twisted cyclic homology as studied by Kustermans, Murphy and Tuset [22].
In this paper we first show that the ribbon automorphism σ is also the missing ingredient required to make Baez's realisation of braided Hochschild homology as a derived functor work in full generality. We prove a braided analogue of a result of Feng and Tsygan [12], namely for Hopf algebras, Hochschild homology can be realised as a derived functor in the category of modules over the algebra itself. We apply this machinery to concrete examples of algebras and Hopf algebras in braided monoidal categories: the braided line, braided plane and braided quantum SL (2).
From the viewpoint of braided monoidal categories, braided Hochschild and cyclic homology are the natural abstractions of the original definitions of Hochschild and of Connes, and therefore natural objects to study. The computations carried out in this paper also show that braided Hochschild homology is (similarly to twisted Hochschild homology [4,14,15,16,22,27]) often less degenerate than classical Hochschild homology, for example in the sense that it overcomes the so-called dimension drop observed for quantisations of Poisson varieties [12]. To show whether the standard applications of cyclic homology in noncommutative geometry admit generalisations to the braided setting seems a promising direction for future research. Another interesting question is the relevance of these invariants for the study of conformal field theories that can be described in terms of ribbon algebras, see e.g. [26]. A summary of this paper is as follows. Throughout we work over C as ground field. In Section 2 we recall the definitions of a ribbon algebra and a Hopf algebra in a braided monoidal category C, with particular reference to the motivating example C = C(H), the category of comodules of a coquasitriangular Hopf algebra H. In Section 3 we define braided Hochschild homology H Ψ,σ * (A, M ) for a ribbon algebra (A, σ) and an A-bimodule M in C as the homology of a specific complex. We show that for C = C(H) this can be realised as a derived functor over an appropriate braided enveloping algebra A e (Theorem 3.7), thus generalising Baez's result to the case σ = id.
We discuss the precise relation to the Akrami-Majid cyclic object associated to (A, σ).
In Section 4 we prove a braided analogue of a result of Feng and Tsygan [12], namely that for a Hopf algebra A in C(H) there is an isomorphism of vector spaces H Ψ,σ n (A, M ) ≃ Tor A n (R(M ), C), for a suitable right A-module R(M ) associated to any A-bimodule M (Theorem 4.2). In Section 5 we apply this machinery to the braided line and braided plane.
In the final Section we consider braided Hopf algebras associated to coquasitriangular Hopf algebras via the process known as transmutation. For quantum SL(2), we obtain a no dimension drop type result (along the lines of [4,15,16] and η : Here and in the sequel we suppress Φ, ℓ, r. Dually, a coalgebra (or comonoid) in C is an object A of C with morphisms ∆ : Given an algebra A in C, we define the opposite algebra A op to be the object A equipped with multiplication µ op = µ • Ψ. Similarly we define the coopposite coalgebra A cop . Given algebras A and B in C, the braided tensor product algebra A⊗B is defined to be the object A ⊗ B with multiplication A bialgebra A in C is an algebra and coalgebra for which ∆ : A → A⊗A and ε : A → 1 are algebra morphisms (Figure 2), i.e. ∆•µ = (µ⊗µ)•(id⊗Ψ⊗id)•(∆⊗∆), and ε•µ = ε⊗ε. Further, a Hopf algebra (or braided group) in C is a bialgebra A together with a morphism S : Finally we need the notion of a ribbon algebra in a braided monoidal category, which is the basic data needed for Akrami and Majid's construction of braided cyclic (co)homology. Definition 2.1 ( Figure 3) A ribbon algebra in C is an algebra (A, µ, η) together with an invertible morphism σ : A → A (the ribbon automorphism) such that Note that if (A, σ) is a ribbon algebra in (C, Ψ), then (A, σ −1 ) is a ribbon algebra in (C, Ψ −1 ).

Example 2.2
If we take C = C−Vec to be the category of complex vector spaces, then an algebra A in C is simply an ordinary algebra over C. If we take the braiding Ψ to be the flip v ⊗ w → w ⊗ v, then (1) becomes σ(ab) = σ(a)σ(b), i.e. σ is an ordinary algebra automorphism of A.
Here we use Sweedler's notation ∆(f ) = f (1) ⊗ f (2) (summation suppressed) for the coproduct. As a consequence r satisfies r(1, f ) = r(f, 1) = ε(f ), and the quantum Yang-Baxter equation r 12 r 13 r 23 = r 23 r 13 r 12 . Here, and in the sequel, we use the convolution product of multilinear maps from a coalgebra to an algebra, and lower indices refer to the components in tensor products where these are applied. Thus explicitly the quantum Yang-Baxter relation reads r(f (1) , g (1) )r(f (2) , h (1) )r(g (2) , h (2) ) = r(g (1) , h (1) )r(f (1) , h (2) )r(f (2) , g (2) ) ∀ f, g, h ∈ H The bilinear formr 21 Note that coquasitriangularity is the notion dual to the more commonly used concept of quasitriangularity, which refers to the existence of an element R ∈ H ⊗ H (the "universal Rmatrix") with certain properties. While in our case the multiplication and opposite multiplication are linked via r, a universal R-matrix links the coproduct and the coopposite one. The disadvantage of quasitriangularity is that it often can only be applied rigorously when working either with finitedimensional Hopf algebras or with topological ones and suitable completions of tensor products. Working with the dual notion of coquasitriangularity offers a way out in many examples. An alternative elegant and purely algebraic option is to pass to multiplier Hopf algebras, see for example the recent work [25] by Neshveyev and Tuset and the references therein.
For a bialgebra H, let C(H) denote the category of right H-comodules, and C f (H) the category of finite-dimensional right H-comodules. The algebra structure of H corresponds to a monoidal structure on C(H), C f (H) with tensor product that of underlying C-vector spaces, and coaction The unit object is C with trivial coaction C ∋ 1 → 1 ⊗ 1 ∈ C ⊗ H. A coquasitriangular structure on H turns C(H), C f (H) into braided monoidal categories with braiding s(f g) = s(f (1) )s(g (1) )q(f (2) , g (2) ), s(1) = 1, s(S(f )) = s(f ) ∀ f, g ∈ H Note that r(a, ·), r(·, a) ∈ H • . Thus sq(a, ·) = q(a, ·)s, and sq(·, a) = q(·, a)s, for all a ∈ H. For coribbon Hopf algebras, C f (H) becomes a ribbon category [19] with ribbon structure Since s is cocentral, this is a morphism in C f (H). It is also well-defined for objects of C(H), and turns algebras in C(H) into ribbon algebras in the sense of Definition 2.1. However C(H) is in general not a ribbon category, due to the lack of duals. For a full discussion see [1].
Example 2.5 The trivial Hopf algebra H = C is coquasitriangular, via r(1, 1) = 1, and coribbon via s(1) = 1. Hence C(H) is the braided monoidal category of Example 2.2, however the only ribbon automorphism arising from s is σ = id.
Example 2.6 Any coribbon Hopf algebra (H, r) becomes a ribbon algebra in C(H) with coaction ∆. Then (1) is a direct translation of the defining property of σ. We call the resulting braiding the canonical braiding on (H, r).
. The resulting braiding on H is By the definition of coquasitriangularity, we have µ • Ψ = µ. So H is Ψ-commutative, in the terminology of [2]. In particular σ = id is a ribbon automorphism.
Example 2.8 Let G be a complex simple Lie group, with Lie algebra g. Let q ∈ C \ {0} be not a root of unity, and H = C q [G] and U q (g) be the standard quantised coordinate ring and enveloping algebra of G and g respectively. Then C (f ) (H) consists of (finite) direct sums of the so-called type I representations of U q (g) that are obtained by deformation of finite-dimensional representations of G. Both are braided monoidal categories. The corresponding universal r-form on C q [G] can be given explicitly in terms of the so-called Rosso form, see e.g. [18]. For the classical matrix Lie groups G the ribbon functionals on C q [G] were classified by Hayashi [17]. The Hopf algebra structure on H is given by There are two useful vector space bases of H: One is of Poincaré-Birkhoff-Witt type and is given by the monomials {e i,j,k := a i b j c k , i ∈ Z, j, k ∈ N}, where a i := d −i for i < 0, and by convention x 0 = 1, for x ∈ H, x = 0. The second relies on the fact that H is isomorphic as coalgebra to C[SL (2)] and hence cosemisimple (any comodule is the direct sum of its irreducible subcomodules). Therefore, it admits a Peter-Weyl type basis consisting of the matrix coefficients C (m) rs , m ∈ N, r, s = 0, . . . , m, of the irreducible corepresentations of H. In particular, C (1) 00 The universal r-form r of Example 2.8 is given on generators by We refer to [20] for more details and proofs.

Gauge transformations and cochain twists
There is a natural notion of equivalence of braided monoidal categories, provided by functors implementing an equivalence of categories that transforms the braided monoidal structures into each other. We recall this explicitly for the case C = C(H), for H a coquasitriangular Hopf algebra. We refer to [1,3,19] for the precise definition in abstract braided monoidal categories. Let H be a Hopf algebra, and f a convolution invertible bilinear form on H, with inversef . The multiplication of H can be twisted by f as follows: In general the result is a nonassociative algebra H f (a so-called quasi-Hopf algebra) since [10,11,24] and H f is associative. We say that f is a 2-coboundary if f (x, y) = ϕ(x (1) )ϕ(y (1) )φ(x (2) y (2) ) for some linear map ϕ : H → k. A 2-coboundary is automatically a 2cocycle, but not conversely. For H f to be associative it is sufficient that ∂f is cocentral, meaning and not necessarily a 2-cocycle. Since H = H f as coalgebras, C(H) and C(H f ) coincide. The two algebra products correspond to different monoidal structures on this category. Both are given by tensor product of vector spaces, but the twist is encoded in the isomorphism of vector spaces which transforms V ⊗ W as an object of C(H) into an object of C(H f ). Following [11] we refer to F as a gauge transformation. This transforms an H-comodule algebra A and A f coincide as vector spaces, but the multiplication in A f is given in terms of the multiplication in A by µ f = µ • F . For this to be associative f must be a 2-cocycle. Further, the r-form r on H transforms to f 21 rf , which is an r-form on H f provided f is a 2-cocycle. For general f the formulae have to be modified by incorporating ∂f , the Drinfeld associator. Twisting Hopf algebras and comodule algebras by coboundaries yields isomorphic structures: Proof. The isomorphisms are given by h →φ(h (1) )h (2) ✷ Finally we note that ribbon functionals and automorphisms are preserved under gauge transformations.

Example 2.11
Let (H, r) be a coquasitriangular Hopf algebra, and q = r 21 r as in Section 2.2. Then an invertible central element s ∈ H • is a ribbon functional if and only if q = ∂s. In particular q is then a 2-cocycle.
Taking this into account, we get: Proposition 2.12 Let (H, r) be a coquasitriangular Hopf algebra such that r = ∂ϕ. Suppose further that ϕ 2 is cocentral (i.e. in the centre of the dual Hopf algebra). Thenφ 2 is a ribbon functional.
Proof. Indeed, if r = ∂ϕ, then q = ∂ϕ 2 : is a 2-coboundary. This is essentially shown in [21] Corollary 4.1.7, where the authors work with formal deformation quantisations. Concretely, one can prove directly that for C q [SL (2)] the (convolution invertible) linear functional ϕ defined by satisfies r = ∂ϕ, for the r-form r defined by (7). For β = q −1/4 , γ = −q −5/4 one obtains ϕ for which ϕ 2 is cocentral, i.e., is the inverse of the ribbon functional. In terms of the Peter-Weyl basis, this ϕ is given by otherwise , see [6] p.262. The inverse of the ribbon functional is thus given by Note that this is cocentral. The functional ϕ gives rise to the extension of the quantised universal enveloping algebra dual to C q [SL (2)] known as the quantum Weyl group. Usually, the fact that the coboundary of ϕ is r is considered the other way round, the aim being to use the r-form to compute the coproduct of ϕ as in Proposition 8.2.3 in [6]. But as we see here, the inverse point of view is relevant as well.
Example 2.14 [9] Let G be a finite group and ω : Then the tensor product A ⊗ A • of the commutative algebra A = C[G] of functions on G with the group algebra A • = CG can be turned into a generalised Drinfeld double with product and coproduct where x, y are identified with the corresponding basis elements of CG, and δ g : G → C is the Dirac function. This gives a quasi Hopf algebra D ω (G) (the product is associative, but ∆ is only quasi coassociative in general). Let H := (D ω (G)) • be the dual. Then, H depends up to a 2-cochain twist only on the cohomology class [ω] ∈ H 3 (G, U (1)), see [9], and it is always coquasitriangular. The comodule category describes certain rational conformal field theories that arose in the work of Dijkgraaf and Witten, and this link to mathematical physics is another main motivation for the current interest in ribbon categories, see e.g. [26].
3 Braided Hochschild and cyclic homology

Braided Hochschild homology
Let (A, σ) be a ribbon algebra in a braided monoidal Ab-category C, which we assume possesses kernels and cokernels. This is in particular true for C = C(H), with H coquasitriangular. There are obvious notions of left and right A-module and A-bimodule in C. For example, by an A-bimodule, we mean an object M of C, together with morphisms (left and right actions) and similarly replacing Ψ by Ψ −1 . From this point for convenience we drop the notation "•" for composition of morphisms. Given such M , define In a diagram representing a morphism with source A ⊗m ⊗ M ⊗ A ⊗n , we number the strands 0, 1,. . ., m + n (with M appearing as the strand labelled m, and represented graphically in bold). We write where we define, for m ≤ n < p,    The braided enveloping algebra A e is the object A ⊗ A equipped with the multiplication morphism ( Figure 6) That is, A e is A⊗A op taken in (C, Ψ −1 ) rather than (C, Ψ). We use this convention in order to be compatible with [1]. Baez works with the opposite convention, taking A⊗A op in (C, Ψ), resulting in a graphical calculus where braidings are replaced by inverse braidings.  Proof. We check that where we used the ribbon property in the form µΨ(σ ⊗ σ) = σµΨ −1 . ✷ So if A is a ribbon algebra, we can make any A-bimodule M into both a left and a right A e -module. Baez performed these constructions under the assumption that A is weakly Ψcommutative, meaning µΨ 2 = µ. The ribbon automorphism σ is the missing ingredient needed to make Baez's constructions work in full generality (the weakly Ψ-commutative case corresponds to σ = id). Conversely, it is easy to check that: As for any simplicial object, {C * (A, A)} gives rise to a resolution of A in C, called the bar resolution, when considered with the differential b ′ n+1 : we number the strands 0,1, . . . , n, and µ i,i+1 = id ⊗i ⊗ µ ⊗ id ⊗(n−i−1) ). In general, projectivity (in C) of this resolution is a subtle question. However, if we take C = C(H) for a coquasitriangular Hopf algebra H (over C) as in Section 2.2, then we have a forgetful functor C → C−Vec, and we can consider A e simply as a C-algebra. Since C is a field A e is projective as a C-module, and the bar resolution will then be a projective resolution in the category of modules over the C-algebra A e (in the usual sense of ring theory). Tensoring the bar resolution over A e on the left by M (with right A e -module structure given by ◭) gives, up to isomorphism of complexes of C-vector spaces, the complex defining braided Hochschild homology. Hence we have:  We give a graphical illustration. Figure 8 shows the relation ∼ for which M ⊗ A e A ⊗n ∼ = M ⊗A ⊗(n+2) / ∼. Figure 9 shows that Tor

Braided cyclic homology
Suppose now that M = A, and for each n ≥ 0 define, as in Figure 10, This makes {C * (A, A)} into a paracyclic object (see [13] for this notion). Passing to the coinvariants of T n := t n+1 n , that is considering the cokernels C Ψ,σ n := coker(id − T n ) we obtain the cyclic object of Akrami and Majid [1].  Note that the simplicial homology of this cyclic object, which we will denote by HH Ψ,σ * * is not exact, as we see from the following simple example.
Example 3.10 Let A be the 3-dimensional unital C-algebra generated by x, y with relations x 2 = 0 = y 2 = xy = yx. Let Ψ be the flip, and σ the automorphism defined by σ(x) = x, σ(y) = x + y.
i.e. as an algebra, Af ⊗f is isomorphic to the twisted quasi-Hopf algebra H f . Apply this with H = C q [G], r the standard universal r-form, and f the inverse of the Drinfeld twist. This gives the braiding Ψ as in (4)

Braided Hochschild homology of Hopf algebras in braided categories
In this Section we extend a result of Feng and Tsygan [12], giving a simpler description of braided Hochschild homology as a derived functor in the case when A is a Hopf algebra. Throughout we gives a right action of A on M ( Figure 11). We write R(M ) for the corresponding right A-module.
Proof. Writing ∆ m for id ⊗m ⊗ ∆ ⊗ id ⊗n (m ≥ 0), we check that where we used the identity µΨ(σS −1 ⊗ σS −1 ) = σS −1 µ. ✷ Working with the coopposite Hopf algebra yields the straightforward generalisation of the action considered in [12]. However the above variant is more convenient for graphical calculations.
As in Theorem 3.7, we now restrict to C = C(H), for H a coquasitriangular Hopf algebra. We note that for A a Hopf algebra in C, then 1 C(H) = C is a left A-module via the counit ε. We now have the following generalisation of [12], Corollary 2.5: Proof. Define maps ξ n , ξ ′ n : M ⊗ A ⊗n → M ⊗ A ⊗n , n = 1, 2, . . . by ξ 1 and ξ ′ 1 are shown in Figure 11. We have: Proof. We prove this by induction. First of all (14), The proof that ξ n • ξ ′ n = id is completely analogous. ✷ Proof. First of all, Proof. We have Finally it is straightforward to check that: Therefore ξ n , ξ ′ n define isomorphisms of complexes between the braided Hochschild complex and the complex {M ⊗A ⊗n ,b n }, whereb n+1 =d 0 + n j=1 (−1) j d j +d n+1 : M ⊗A ⊗(n+1) → M ⊗A ⊗n . That the homology of the latter complex is Tor A * (R(M ), C) follows from [5] Cor. IX.4.4, applied with K, Λ, A, B being C, A, C, R(M ) respectively. ✷ Figure 12 shows a graphical proof in degree zero.

The braided line and the braided plane
In this section we consider the braided line and braided plane. We observe that braided Hochschild homology is less degenerate than ordinary Hochschild homology of the classical counterparts. Throughout this section q ∈ C will denote a nonzero parameter which is not a root of unity.

The braided line
Let H be the commutative Hopf algebra CZ = C[t, t −1 ], with coproduct ∆(t) = t ⊗ t. Then H is coquasitriangular, via r(t m , t n ) = q mn for all m, n ∈ Z. Let C[x] be the unital C-algebra in a single indeterminate x. Then C[x] is a right H-comodule algebra, via x m → x m ⊗ t m , for all m ≥ 1. The braiding (2) is Ψ(x m ⊗ x n ) = q mn x n ⊗ x m . The braided line A is the braided Hopf algebra given by It follows from the ribbon relation µ • (σ ⊗ σ) • Ψ 2 = σ • µ that any ribbon automorphism satisfies σ(1) = 1, σ(x n ) = q n(n−1) σ(x) n for all n ≥ 1. Taking σ(x) = n≥0 λ n x n , compatibility with the braiding forces λ n = 0 for n = 1, so σ(x) = λx, σ(x n ) = (λq n−1 ) n x n for all n ≥ 1. Since we need σ to be invertible, λ = 0. Every elementary tensor x m0 ⊗. . .⊗x mn is an eigenvector of T n , hence Proposition 3.11 applies and HH Ψ,σ n (A) ∼ = H Ψ,σ n (A, A) for all n.

For
Proof. We use Theorem 4.2. Consider the resolution of C by free left A-modules: 0 → A → ϕ A → ε C → 0, where ϕ(x n ) = x n+1 . Tensoring on the left by R(A) ⊗ A −, where R(A) has underlying space A and right action gives HH Ψ,σ * (A) as the homology of 0 → R(A) It follows that if λ / ∈ q −N , then HH Ψ,σ 0 (A) = C [1], HH Ψ,σ 1 (A) = 0, whereas if λ = q −N , then ker(φ) is one-dimensional, and coker(φ) is two-dimensional, with generators as above. ✷ Using standard spectral sequence arguments [15] we calculate the braided cyclic homology. We note that for the map B 0 :

The braided plane
In this section, A denotes Manin's quantum plane, that is, the unital algebra (over C) generated by indeterminates x, y satisfying yx = qxy. This is a right H = C q [SL (2)]-comodule algebra via One can identify A with the subalgebra of H generated by a, b. Under this identification, y becomes a, x becomes b, and the coaction becomes ∆ (so A is an "embeddable quantum space" of H). Obviously, the monomials x m y n form a vector space basis of A. Under the embedding into H these monomials become proportional to the Peter-Weyl basis elements C (m+n) 0m . The precise factor is given e.g. in [20], but it is irrelevant for us.
The braiding (2) induced from the coquasitriangular structure of H is determined by: There is a braided Hopf algebra structure on A given by [24]: The ribbon automorphisms of A are parametrised by λ = 0 and given by σ(x m y n ) = λ m+n q (m+n)(m+n−1)/2 x m y n .

Braided Hopf algebras associated to coquasitriangular Hopf algebras
Let (H, r) be a coquasitriangular Hopf algebra. Let A ∈ Ob(C(H)) be H equipped with the right adjoint coaction Ad R (a) = a (2) ⊗ S(a (1) )a (3) . In general A is not a right H-comodule algebra. Now define B = B(H) to be the algebra with underlying vector space H and (associative) product Then B is via Ad R a right H-comodule algebra and in fact a Hopf algebra in C(H) with coproduct and antipode given by ∆(a) = a (1) ⊗ a (2) , S(a) = S(a (2) )r(S 2 (a (3) )S(a (1) ), a (4) ) (see [24] and [20] Section 10.3.2). We call B the braided Hopf algebra associated to H, alternatively the transmutation of H. The coaction Ad R gives a braiding 6.2 Braided homology of quantum SL(2) Since in the classical limit q = 1 we would like σ = id, we will restrict attention to σ + . The corresponding s ∈ A • is defined by s(a) = q 3/2 = s(d), s(b) = 0 = s(c) (see [19], p366). Then: Proposition 6.2 For the braiding (20) and σ = σ + , HH Ψ,σ 0 (A) = 0.