A class of non symmetric solutions for the integrability condition of the knizhinik-zamolodchikov equation: a hopf algebra approach

Abstract

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 R ¹² R ¹³ R ¹³ R ¹².and R ¹² R ²³ = R ²³ R ¹² 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: [R ¹² R ¹³+R ²³]. 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 = R _σ, where M has a structure of a right comodule over a Long bialgebra (L(R),σ), and R _σ is the special map .