The Hopf modules category and the Hopf equation

Let (A,∆) be a Hopf-von Neumann algebra and R be the unitary fundamental operator on A defined by Takesaki in [28]: R(a⊗ b) = ∆(b)(a⊗ 1). Then R12R23 = R23R13R12 (see lemma 4.9 of [28]). This operator R plays a vital role in the theory of duality for von Neumann algebras (see [28] or [2]). If V is a vector space over an arbitrary field k, we shall study what we have called the Hopf equation: R12R23 = R23R13R12 in End k(V ⊗ V ⊗ V ). Taking W := τRτ , the Hopf equation is equivalent with the pentagonal equation: W 12W 13W 23 = W 23W 12 from the theory of operator algebras (see [2]), where W are viewed as map in L(K ⊗ K), for a Hilbert space K. For a bialgebra H, we shall prove that the classic category of Hopf modules HM H plays a decisive role in describing all solutions of the Hopf equation. More precisely, if H is a bialgebra over k and (M, ·, ρ) ∈ HM H is an H-Hopf module, then the natural map R = R(M,·,ρ) is a solution for the Hopf equation. Conversely, the main result of this paper is a FRT type theorem: ifM is a finite dimensional vector space and R ∈ End k(M⊗M) is a solution for the Hopf equation, then there exists a bialgebra B(R) such that (M, ·, ρ) ∈ B(R)M B(R) and R = R(M,·,ρ). By applying this result, we construct new examples of noncommutative and noncocommutative bialgebras which are different from the ones arising from quantum group theory. In particular, over a field of characteristic two, an example of five dimensional noncommutative and noncocommutative bialgebra is given. 0 Introduction Let H be a bialgebra over a field k. There are two fundamental categories in the theory of Hopf algebras and quantum groups: HM H , the category of H-Hopf modules and HYD H , the category of quantum Yetter-Drinfel’d modules. The objects in these categories are k-vector 1 spaces M which are left H-modules (M, ·), right H-comodules (M, ρ), such that the following quite distinct compatibility relations hold: ρ(h ·m) = ∑ h(1) ·m<0> ⊗ h(2)m<1> (1) in the case HM , and respectively ∑ h(1) ·m<0> ⊗ h(2)m<1> = ∑ (h(2) ·m)<0> ⊗ (h(2) ·m)<1>h(1) (2) for the Yetter-Drinfel’d categories. Traditionally, these two categories have been studied for completely different reasons: the classical category HM H (or immediate generalisations of it: AM H , AM(H) ) is involved in the theory of integrals for a Hopf algebra (see [1], [27] or the more recent [19]), Clifford theory of representations ([17], [23], [25], [26]) and Hopf-Galois theory ([19], [24], etc.). The cateory HYD H , introduced in [31], plays an important role in the quantum Yang-Baxter equation, quantum groups, low dimensional topology and knot theory (see [13], [14], [20], [21], or [29]). However, there are two connections between these categories. The first one was given by P. Schauenburg in [22]: it was proven that the category HYD H is equivalent to the category H HM H H of two-sided, two-cosided Hopf modules. The second was given recentely in [4]. For A an H-comodule algebra and C an H-module coalgebra, Doi (see [7]) and independently Koppinen (see [12]) defined AM(H) , the category of Doi-Koppinen Hopf modules, whose objects are left A-modules and right C-comodules and satisfy a compatibility relation which generalises (1). In [4] it was proven that HYD H is isomorphic to HM(H op ⊗H) , where H can be viewed as an H ⊗ H-module (comodule) coalgebra (algebra). The isomorhism is the identity functor M → M . We hereby obtain a strong link between the categories HM H and HYD H : both are particular cases of the same general category AM(H) . This led us in [5], [6] to study the implications of the category HYD H in the classic, non-quantic part of Hopf algebra theory. In [5] we start with the following classic theorem (see [19]): any finite dimensional Hopf algebra is Frobenius. In the language of categories, this result is interpreted as follows: the forgetful functor HM H → HM is Frobenius (i.e., cf. [5], by definition has the same left and right adjoint) if and only if H is finite dimensional. The next step is easy to take: we must generalize this result for the forgetful functor AM(H) C → AM and then apply it in the case of Yetter-Drinfel’d modules for the forgetful functor HYD H → HM. We thus obtain the fact that the forgetful functor HYD H → HM is Frobenius if and only if H is finite dimensional and unimodular (see theorem 4.2 of [5]). The same treatment was applied in [6] for the classic Maschke theorem. One of the major obstacles was to correctly define the notion of integral for the Doi-Hopf datum (H,A,C), such as to be connected to the classic integral on a Hopf algebra (corresponding to the case C=A=H), as well as to the notion of total integral (corresponding to the case C=A) defined by Doi in [8]. This technique can be looked upon as a ”quantisation” of the theorems from the classic theory of Hopf algebras. There are two steps to it: first, we seek to generalize a result for the category AM(H) , then to apply it to the particular HYD H case. There is also another approach to this ”quantisation” technique, recently evidenced in [9] for the same Frobenius type theorem. It 2 was first proven, by generalizing the classic result, that any finite dimensional Hopf algebras extensions is a β-Frobenius extension (or a Frobenius extension of second kind). Then, this theorem was ”quantised” to the case of Hopf algebras extensions in HYD H . The result includes the case of enveloping algebras of Lie coloralgebras. Beginning with this paper, we shall tackle the reverse problem: we shall try to involve the category HM H in fields dominated until now by HYD H , i.e. try a ”dequantisation”. For the beginning, it is enough to remind that the category HYD H is deeply involved in the quantum Yang-Baxter equation: RRR = RRR (3) where R ∈ End k(M ⊗ M), M being a k-vector space. The starting point of this paper is the following question: ”Can the category HM H be studied in connection with a certain non-linear equation?” The answer is affirmative and, surprisingly, the equation in which the category HM H is involved (which we shall call Hopf equation) is very close to the quantum Yang-Baxter equation. More precisely, it is RR = RRR (4) The simple way of obtaining it from the quantum Yang-Baxter equation by just deleting the term R on the left hand side has nevertheless unpleasant effects: first of all, if the Yang-Baxter equation is reduced to the solution of a homogenous system, in the case of the Hopf equation the system is not homogenous any more; secondly, if R is a solution of the Hopf equation, W := τRτ (or W := R, if R is bijective) is not a solution for the Hopf equation, but for the pentagonal equation: W W W 23 = W W . An interesting connection between the pentagonal and the quantum Yang-Baxter equations is given in [30]. The pentagonal equation plays a fundamental role in the duality theory for operator algebras (see [2] and the references indicated here). If H is a Hopf algebra, then R : H ⊗H → H ⊗H, R(g ⊗ h) = ∑ h(1)g ⊗ h(2) is a bijective solution of the Hopf equation. Moreover, the comultiplication ∆ can be rebuilt from R by means of ∆(h) = R((1⊗ h)z), where z ∈ H ⊗ H such that R(z) = 1 ⊗ 1. This operator was defined first by Takesaki in [28] for a Hopf-von Neumann algebra. The operator W := τRτ is called in [15] the evolution operator for a Hopf algebra and plays an important role in the description of the Markov transition operator for the quantum random walks (see [15] or [16]). The Hopf equation can be viewed as a natural generalisation of the idempotent endomorphisms of a vector space: more precisely, if f ∈ End k(M), then f ⊗ I (or I⊗ f) is a solution 3 of the Hopf equation if and only if f 2 = f . We shall prove that if (M, ·, ρ) ∈ HM H then the natural map R(M,·,ρ)(m⊗ n) = ∑ n<1> ·m⊗ n<0> is a solution of the Hopf equation. Conversely, the main result of this paper is a FRT type theorem which shows that in the finite dimensional case, any solution R of the Hopf equation has this form, i.e. there exists a bialgebra B(R) such that (M, ·, ρ) ∈ B(R)M B(R) and R = R(M,·,ρ). Similarly to [2], a solution R of the Hopf equation is called commutative if RR = RR. In the finite dimensional case, any commutative solution of the Hopf equation has the form R = R(M,·,ρ), where (M, ·, ρ) is a Hopf module over a commutative bialgebra B(R). This result can be viewed as the algebraic version of the theorem 2.2 of [2], which classifies all multiplicative, unitary and commutative operators which can be defined on a Hilbert space. In the last part we shall apply our theorem to constructing new examples of noncommutative, noncocommutative bialgebras which differ from those arising from the FRT theorem for the quantum Yang-Baxter equation. These bialgebras arise from the elementary maps of plane euclidian geometry: the projections of k on the Ox and Oy coordinate axis. Surprisingly, over a field of characteristic 2, our FRT type construction supplies us an example of noncommutative and noncocommutative bialgebra of dimension 5. Obviously, substituting the key map R(M,·,ρ) with τR(M,·,ρ)τ , all results of this paper remain valid if we replace the Hopf equation with the pentagonal equation. We have preferred however to work with the Hopf equation, for historical reasons: this is how the issue has been raised for the first time in lemma 4.9 of [28]. This study was continued in [18], where new classes of bialgebras arising from the Hopf equation are introduced and analyzed. 1 Preliminaries Throughout this paper, k will be a field. All vector spaces, algebras, coalgebras and bialgebras that we consider are over k. ⊗ and Hom will mean ⊗k and Hom k. For a coalgebra C, we will use Sweedler’s Σ-notation, that is, ∆(c) = ∑ c(1)⊗c(2), (I⊗∆)∆(c) = ∑ c(1)⊗c(2)⊗c(3), etc. We will also use Sweedler’s notation for right C-comodules: ρM(m) = ∑ m<0> ⊗m<1>, for any m ∈ M if (M, ρM) is a right C-comodule. M C will be the category of right Ccomodules and C-colinear maps and AM will be the category of left A-modules and A-linear maps, if A is a k-algebra. Recall the following well known lemmas: Lemma 1.1 Let M be a finite dimensional vector space with {m1, · · · , mn} a basis for M and let C be a coalgebra. We define the k-linear map ρ : M → M⊗C, ρ(ml) = ∑n v=1 mv⊗cvl, for all l = 1, · · · , n, where (cvl)v,l is a family of elements of C. The following statements are equivalent: 4 1. (M, ρ) is a right C-comodule. 2. The matrix (cvl)v,l is comultiplicative, i.e. ∆(cjk) = n ∑ u=1 cju ⊗ cuk, ε(cjk) = δjk (5) for all j, k = 1, · · · , n If we denote B = (cvl)v,l, then, as usual, the relations (5) can formally be written: ∆(B) = B ⊗ B, ε(B) = In. Lemma 1.2 Let (C,∆, ε) be a coalgebra. Then, on the tensor algebra (T (C),M, u), there exists a unique bialgebra structure (T (C),M, u,∆, ε) such that ∆(c) = ∆(c) and ε(c) = ε(c) for all c ∈ C. In addition, the inclusion map i : C → T (C) is a coalgebra map. Furthermore, if M is a vector space and μ : C ⊗M → M , μ(c⊗m) = c ·m is a linear map, then there exists a unique left T (C)-module structure on M , μ : T (C)⊗M → M , such that μ(c⊗m) = c ·m, for all c ∈ C, m ∈ M . Let H be a bialgebra. Recall that an (left-right) H-Hopf module is a left H-module (M, ·) which is also a right H-comodule (M, ρ) such that ρ(h ·m) = ∑ h(1) ·m<0> ⊗ h(2)m<1> (6) for all h ∈ H , m ∈ M . HM H will be the category of H-Hopf modules and H-linear Hcolinear homomorphisms. Lemma 1.3 Let H be a bialgebra, (M, ·) a left H-module and (M, ρ) a right H-comodule. Then the set {h ∈ H | ρ(h ·m) = ∑ h(1) ·m<0> ⊗ h(2)m<1>, ∀m ∈ M} is a subalgebra of H. Proof Straightforward. ⊓⊔ We obtain from this lemma that if a left H-module and right H-comodule M satisfies the condition of compatibility (6) for a set of generators as an algebra of H and for a basis of M , then M is an H-Hopf module. If (M, ·) is a left H-module and (M, ρ) is a right H-comodule, the special map R(M,·,ρ) : M ⊗M → M ⊗M, R(M,·,ρ)(m⊗ n) = ∑ n<1> ·m⊗ n<0> (7) will play an important role in the present paper. It is useful to point out the following lemma. The proof is left to the reader. 5 Lemma 1.4 Let H be a bialgebra, (M, ·) a left H-module and (M, ρ) a right H-comodule. If I is a biideal of H such that I ·M = 0, then, with the natural structures, (M, ·) is a left H/I-module, (M, ρ) a right H/I-comodule and R(M,·′,ρ′) = R(M,·,ρ) For a vector space V , τ : V ⊗V → V ⊗V will denote the switch map, that is, τ(v⊗w) = w⊗v for all v, w ∈ V . If R : V ⊗V → V ⊗V is a linear map we denote by R, R, R the maps of End k(V ⊗ V ⊗ V ) given by R = R⊗ I, R = I ⊗ R, R = (I ⊗ τ)(R⊗ I)(I ⊗ τ). Using the notation R(u⊗ v) = ∑ u1 ⊗ v1 then R(u⊗ v ⊗ w) = ∑ u1 ⊗ v1 ⊗ w0 where the subscript (0) means that w is not affected by the application of R. Let H be a bialgebra and (M, ·) a left H-module which is also a right H-comodule (M, ρ). Recall that (M, ·, ρ) is a Yetter-Drinfel’d module if the following compatibility relation holds: ∑ h(1) ·m<0> ⊗ h(2)m<1> = ∑ (h(2) ·m)<0> ⊗ (h(2) ·m)<1>h(1) for all h ∈ H , m ∈ M . HYD H will be the category of Yetter-Drinfel’d modules and H-linear H-colinear homomorphism. If (M, ·, ρ) is a Yetter-Drinfel’d module then the special map R = R(M,·,ρ) given by the equation (7) is a solution of the quantum Yang-Baxter equation RRR = RRR. If M is a finite dimensional vector space and R is a solution of the quantum Yang-Baxter equation, then there exists a bialgebra A(R) such that (M, ·, ρ) ∈ A(R)M A(R) and R = R(M,·,ρ) (see [20]). For a further study of the Yetter-Drinfel’d category we refer to [14], [20], [21], [31], or to the more recent [4], [5], [6], [9]. 2 The Hopf equation We will start with the following Definition 2.1 Let V be a vector space and R ∈ End k(V ⊗ V ). 1. We shall say that R is a solution for the Hopf equation if RRR = RR (8) 2. We shall say that R is a solution for the pentagonal equation if RRR = RR (9) 6 Remarks 2.2 1. The Hopf equation is obtained from the quantum Yang-Baxter equation RRR = RRR by deleting the midle term from the right hand side. 2. Let {mi}i∈I be a basis of the vector space V . Then an endomorphism R of V ⊗V is given by a family of scalars (x ij )i,j,k,l∈I of k such that R(mv ⊗mu) = ∑ i,j x uvmi ⊗mj (10) for all v, u ∈ I. A direct computation shows us that R is a solution of the Hopf equation if and only if (x ij )i,j,k,l∈I is a solution of the nonlinear equation