A class of non-symmetric solutions for the integrability condition of the Knizhnik-Zamolodchikov equation: a Hopf algebra approach

Let M be a vector space over a field k and R ∈ End k(M ⊗ M). This paper studies what shall be called the Long equation: that is, the system of nonlinear equations R12R13 = R13R12 and R12R23 = R23R12 in End k(M ⊗ M ⊗ M). Any symmetric solution of this system supplies us a solution of the integrability condition of the Knizhnik-Zamolodchikov equation: [R12, R13+R23] = 0 ([4] or [10]). We shall approach this equation by introducing a new class of bialgebras, which we call Long bialgebras: these are pairs (H,σ), where H is a bialgebra and σ : H ⊗H → k is a k-bilinear map satisfying certain properties. The main theorem of this paper is a FRT type theorem: if M is finite dimensional, any solution R of the Long equation has the form R = Rσ, where M has a structure of a right comodule over a Long bialgebra (L(R), σ), and Rσ is the special map Rσ(m⊗ n) = ∑ σ(m<1> ⊗ n<1>)m<0> ⊗ n<0>. 0 Introduction Let M be a finite dimensional vector space over a field k and R ∈ End k(M ⊗M). In this paper we shall describe all solutions of what we have called the Long equation, namely the system of nonlinear equations { RR = RR RR = RR (1) in End k(M⊗M⊗M). Our approach is similar to the one used by Faddeev, Reshetikhin and Takhtadjian in relating the solutions of the quantum Yang-Baxter equation to comodules over co-quasitriangular bialgebras (see [2]). 1 We shall introduce a new class of bialgebras which we shall call Long bialgebras. They are pairs (H, σ), where H is a bialgebra and σ : H ⊗ H → k is a linear map satisfying the conditions (L1) − (L5) from definition 3.3. The conditions (L2) − (L5) are identical with the conditions (B2)− (B5) from the definition of co-quasitriangular bialgebras. What differentiates Long bialgebras from the co-quasitriangular bialgebras is the condition (L1) versus (B1). This new class of bialgebras will play a fundamental role in solving the Long equation. More precisely, if (M, ρ) is a right comodule over a Long bialgebra (H, σ), then the special map Rσ : M ⊗M → M ⊗M, Rσ(m⊗ n) = ∑ σ(m<1> ⊗ n<1>)m<0> ⊗ n<0> is a solution of the Long equation. Conversely, the main theorem of this paper is a FRT type theorem: if M is a finite dimensional vector space and R is a solution of the Long equation, then there exists a Long bialgebra (L(R), σ) such that M has a structure of right L(R)-comodule and R = Rσ. Let us now look at the Long equation from a different angle, having in mind Radford’s version of the FRT theorem ([9]): in the finite dimensional case, any solution R of the quantum Yang-Baxter equation has the form R = R(M,·,ρ), where (M, ·, ρ) ∈ A(R)YD , the category of Yetter-Drinfel’d modules. This is obtained immediately, keeping in mind the fact that a comodule (M, ρ) over a co-quasitriangular bialgebra (like (A(R), σ)) has a structure of Yetter-Drinfel’d module via h ·m = ∑ σ(m<1> ⊗ h)m<0> for all h ∈ A(R), m ∈ M and R(M,·,ρ) = Rσ. A similar phenomenon happens to the comodules over a Long bialgebra (H, σ): they become objects in the category HL H of Hdimodules (see Proposition 3.6). This category has been introduced by Long ([5]) for the case of a commutative and cocommutative H and studied in connection with the Brauer group of an H-dimodule algebra. For this reason, we called system (1) the Long equation and the new class of bialgebras, Long bialgebras. Finally, we shall point out a connection between the symmetric solutions of the Long equation (e.g. the map R from proposition 2.4) and solutions of the integrability condition of the Knizhnik-Zamolodchikov equation. Suppose that R is a solution of the Long equation; then R satisfies the equation [R, R + R] = 0, which is called the integrability condition of the Knizhnik-Zamolodchikov equation (see [10]). Moreover, we assume that R is symmetric, that is R = R. W is a solution of the Knizhnik-Zamolodchikov equation if it is a solution of the following system of differential equations: ∂W ∂zi = h ∑ i6=j ( R zi − zj )W. (2) This is the equation for a function W (z) taking values in M, representing a covariant constant section of the trivial bundle Yn × M ⊗n → Yn with flat connection ∑ i6=j R ij dz zi−zj (here k is the complex field C, h is a complex parameter and Yn = C /multidiagonal). In other words, R is describing a family of flat connections on bundles with fiber M. For further details we refer to [4] and [10]. 2 1 Preliminaries Throughout this paper, k will be a field. All vector spaces, algebras, coalgebras and bialgebras considered 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 C-comodules and C-colinear maps and AM will be the category of left A-modules and A-linear maps, if A is a k-algebra. An important role in the present paper will be played by M(k), the comatrix coalgebra of order n, i.e. M(k) is the n-dimensional vector space with {cij | i, j = 1, · · · , n} a k-basis such that ∆(cjk) = n ∑ u=1 cju ⊗ cuk, ε(cjk) = δjk (3) for all j, k = 1, · · · , n. We view T (M(k)) with the unique bialgebra structure which can be defined on the tensor algebra T (M(k)), which extends the comultiplication ∆ and the counity ε of M(k). Let H be a bialgebra. An H-dimodule over H is a triple (M, ·, ρ), where (M, ·) is a left H-module, (M, ρ) is a right H-comodule such that the following compatibility condition holds: ρ(h ·m) = ∑ h ·m<0> ⊗m<1> (4) for all h ∈ H and m ∈ M . The category of H-dimodules over H and H-linear H-colinear maps will be denoted by HL H . This category was introduced for a commutative and cocommutative H by Long in [5]. For a vector spaceM , τ : M⊗M → M⊗M will denote the flip map, that is τ(m⊗n) = n⊗m for all m, n ∈ M . If R : M ⊗M → M ⊗M is a linear map, we denote by R, R, R the maps of End k(M ⊗M ⊗M) given by R = R⊗ I, R = I ⊗ R, R = (I ⊗ τ)(R⊗ I)(I ⊗ τ). 2 The Long equation We shall start with the following: Definition 2.1 Let M be a vector space and R ∈ End k(M ⊗M). We shall say that R is a solution for the Long equation if { RR = RR RR = RR (5) holds in End k(M ⊗M ⊗M). 3 Remarks 2.2 1. If R is a solution of the Long equation, then R satisfies the equation [R, R +R] = 0. Hence, any symmetric solution R (i.e. R = R) of the Long equation is a solution for the integrability condition of the Knizhnik-Zamolodchikov equation (see [4] or [10]). 2. Let M be a finite dimensional vector space and {m1, · · · , mn} a basis of M . Let R ∈ End k(M ⊗M) given by R(mv ⊗mu) = ∑ i,j x uvmi ⊗mj , for all u, v = 1, · · · , n, where (x uv)i,j,u,v is a family of scalars of k. Then R is a solution of the Long equation if and only if the following two equations hold: