A note on Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

In our previous work, Hopf cyclic cohomology in braided monoidal categories, we extended the formalism of Hopf cyclic cohomology due to Connes and Moscovici and the more general case of Hopf cyclic cohomology with coefficients to the context of abelian braided monoidal categories. In this paper we go one step further in reducing the restriction of the ambient category being symmetric. We let the ambient category to be non-symmetric but assume only the restriction on the braid map for the Hopf algebra object (in that category) which is the main player in the theory. In the case of Hopf cyclic cohomology with (nontrivial) coefficients we also need to have similar restrictions on the braid map for the object(s) providing the coefficients datum. We present a family of examples of non-symmetric categories in which many objects with such a restrictions on the braid map exist (anyonic vector spaces).


Introduction
In [2,3,4], Connes and Moscovici, motivated by transverse index theory for foliations, defined a cohomology theory of cyclic type for Hopf algebras endowed with a modular pair in involution (MPI). This theory was later extended in [5,6] to the more general case of Hopf cyclic cohomology with coefficients, by introducing the notion of stable anti Yetter-Drinfeld (SAYD) modules.
In [7] we extended all these formalisms of Hopf cyclic cohomology to the context of abelian braided monoidal categories. When the braiding is symmetric we associated a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution. When the braiding is non-symmetric we obtained a para-cocyclic object instead of a cocyclic object. To obtain a cocyclic object from a para-cocyclic object one has to work in an appropriate subspace.
In the present paper we reduce the restriction of the ambient category C being symmetric, i.e., ψ A⊗B ψ B⊗A = id B⊗A , for all A and B in C, to a less restrictive condition. We show that even in the non-symmetric case as long as we only have the relation ψ 2 H⊗H = ψ H⊗H ψ H⊗H = id H⊗H we will still obtain a cocyclic object. In the case of Hopf cyclic cohomology with coefficients in a H-module M we also need to have the relation ψ H⊗M ψ M ⊗H = id M ⊗H . In the most general case, for the triple (H, C, M ), we also need similar restrictions on ψ H⊗C and ψ M ⊗C . This is in fact because the main players in the theory are H, M and C and nowhere in any of the proofs in [7] we need more than the above relations. Therefore the proofs of the two theorems, Theorems 2.1 and 2.2, presented here are the same as the proofs of their analogous theorems, Theorems 3.6 and 7.1 in [7].
In the last section of this paper we provide a family of examples of such a situation where the ambient category C (category of anyonic vector spaces) is not symmetric but for many objects, A and B in C the property ψ A⊗B ψ B⊗A = id B⊗A holds.
We close the present paper by looking at a particular example of a braided Hopf algebra (in a non-symmetric category) for which ψ 2 = id. We will prove that in this particular case the braided Hopf cyclic cohomology (defied in Theorem 2.2) coincides with the (usual) Hopf cyclic cohomology in the sense of Connes and Moscovici [2,3,4]. This is expected because this braided Hopf algebra is also a Hopf algebra in the usual sense, i.e., within the category of vector spaces.
To keep this note short we don't include any preliminaries. We refer the reader to [7,8,9] for all the preliminary definitions and for our convention of the short notations often used in this paper.
The author would like to express his sincere appreciation to Masoud Khalkhali for illuminating discussions and encouragements.

Hopf cyclic cohomology in non-symmetric categories
In this section we first present an analogous of Theorem 3.6 in [7]. We assign a cocyclic object to a braided triple (H, C, M ) in an abelian braided monoidal category C but with the symmetric condition for C replaced by only restrictions on ψ H⊗M , ψ H⊗H , ψ H⊗C and ψ M ⊗C (Theorem 2.1). Then we let C = H and M = I, the identity object of C, and present an analogous of Theorem 3.6 in [7], but again with the symmetric condition for C replaced, in this case, by only one restriction on ψ H⊗H (Theorem 2.2). The latter case is in fact the braided version of Connes-Moscovici's Hopf cyclic cohomology in a non-symmetric category C in which only the relation ψ 2 H⊗H = id H holds.
The proofs of the two theorems presented here are the same as the proofs of their analogous theorems in [7].
We define faces δ i : C n−1 → C n , degeneracies σ i : C n+1 → C n and cyclic maps τ n : C n → C n by: We form the balanced tensor products (Here is when we need C to be abelian), with induced faces, degeneracies and cyclic maps denoted by δ i , σ i and τ n .
Proof. As in the proof of Theorem 3.6 in [7]. Now if we put M = I and C = H Theorem 2.1 reduces the braided version of Connes-Moscovici's Hopf cyclic theory [2,3,4], in non-symmetric monoidal categories, as follows.
Theorem 2.2. Let C be an abelian braided (not necessarily symmetric) monoidal category. Let H be a braided Hopf algebra in C for which, ψ H⊗H ψ H⊗H = id H⊗H (for short we show this by ψ 2 H⊗H = id). Then if H is endowed with a BMPI (δ, σ) the following data defines a cocyclic object in C: C 0 (H) = I and C n (H) = H n , n ≥ 1, Here by m n we mean, m 1 = m, and for n ≥ 2: (1 H j , ψ, ψ, ..., ψ n−j times , 1 H j ).
Proof. As in the proof of Theorem 7.1 in [7].
We will denote the braided Hopf cyclic cohomology of a braided Hopf algebra H in Theorem 2.2 by BHC * (H), as we denote the usual Hopf cyclic cohomology of a Hopf algebra H by HC * (H)

Examples
In this section we provide examples of non-symmetric categories (anyonic vector spaces, [10,11]) in which many objects A with the property ψ 2 A⊗A = id A⊗A exist. In fact we prove that, more generally, there are many objects A and B for which ψ B⊗A ψ A⊗B = id A⊗B .
We need to recall that, [1,10,11], if (H, R = R 1 ⊗R 2 ) is a quasitriangular Hopf algebra and C the category of all left H-modules, then C is a braided monoidal abelian category. Here the monoidal structure is defined by for any V and W in C, where ✄ denotes the action of H.
Let H = CZ n , the group (Hopf) algebra of the finite cyclic group, Z n , of order n. In addition to the trivial one, R = 1 ⊗ 1, there exists a nontrivial quasitriangular structure for H = CZ n defined by [11]: where g is the generator of Z n . The category of all left H-modules, denoted here by C, is known as the category of anyonic vector spaces. The objects of C are of the form V = n−1 i=0 V i . They are Z n -graded representations of CZ n and the action of Z n on V is given by, where |v| = k is the degree of the homogeneous elements v in V k . The morphisms of C are linear maps that preserve the grading. Applying formulas (3.2) and (3.3) to (3.1) will give the formula for the braiding map in C as: where |v| and |w| are the degrees of homogeneous elements v and w in objects V and W , respectively. This category is not symmetric when n > 2. But we will now prove that for many values of n one can always find objects A and B in C such that, A set of examples is as follows. Let n = 2m 2 , for some integer m ≥ 2, and let A = n−1 i=0 A i and B = n−1 i=0 B i be objects which are focused only in degrees, km, for integers k ≥ 0, i.e., degrees zero, m, 2m, 3m, 4m and so on. By A being focused only in these degrees we mean, A i = 0 when i = km, for integers k ≥ 0. Thus for any two homogeneous elements x in A and y in B, 2πi|x||y| 2m 2 = klπi when, |x| = km and |y| = lm, for integers k, l ≥ 0. This implies that, by formula (3.4), ψ A⊗B (x ⊗ y) is equal to either y ⊗ x or −y ⊗ x. Therefore ψ B⊗A ψ A⊗B (x ⊗ y) = x ⊗ y, for all homogeneous elements x in A and y in B.
An example of above case is when n = 18 and A = 17 i=0 A i and B = Relative to examples provided above a somewhat trivial case is when objects of interest, A = n−1 i=0 A i , are focused only in degree zero, i.e., A i = 0 when i = 0. But for us this case is still interesting because we have the following example which is a concrete example of a braided Hopf algebra H in a non-symmetric category C for which ψ 2 H⊗H = id. For the following example we recall that [1,11], referring to what we have recalled in the beginning of this section, H itself could be turned into a braided Hopf algebra H in C the category of left H-modules. This braided Hopf algebra H has the following structure. As an algebra H = H, with H-module structure given by conjugation, a ✄ h = a (1) hS(a (2) ).
The rest of Hopf algebra structure on H is given by: with counit ε = ε, and antipode Example 3.1. Let H = CZ n with the nontrivial quasitriangular structure defined by the formula (3.2). Notice that, since ∆(g) = g⊗g and S(g) = g −1 , the conjugation action of g on all elements of H = CZ n is trivial, i.e., g a ✄ g m = g a g m g −a = g m , for all integers a and m. Then using formulas (3.1) and (3.2) we have, for all v and w in H. Here we have used the fact that (1/n) n−1 a,b=0 e (−2πiab)/n = 1. Therefore ψ H⊗H is the usual flip and ψ 2 H⊗H = id. Also this, in light of formula (3.3), means that H as a Z n -graded Hopf algebra in C, is focused only in degree zero. which is equal to 1 if b = 0 and 0 otherwise [11]. Also by formulas ( What we have shown implies that BHC * (H) = HC * (H), i.e., the braided Hopf cyclic cohomology (defied in Theorem 2.2) coincides with the Hopf cyclic cohomology for H = H = CZ n . This was expected as CZ n is also a Hopf algebra in the usual sense, i.e., within the category of vector spaces. The cohomology HC * (CZ n ) is well known to be C in even degrees and zero in odd degrees.
We refer to [12] for further examples of braided Hopf algebras and computing their braided Hopf cyclic cohomology (BHC) in symmetric and nonsymmetric categories.