Double product integrals and Enriquez quantization of Lie bialgebras I: The quasitriangular identities

Abstract

Let 𝒯(ℒ) be the space of all tensors over a Lie algebra ℒ in which the Lie bracket is obtained by taking commutators in an associative algebra. We show that 𝒯(ℒ) becomes a Hopf algebra when equipped with a noncommutative modification of the shuffle product together with the standard coproduct. A definition is given of directed double product integrals as iterated single product integrals driven by formal power series with coefficients in the tensor product of ℒ with an appropriate associative algebra. For the Hopf algebra T(L)[[h]] of formal power series we show that elements R[h] of (T(L)⊗T(L))[[h]] satisfying (Δ⊗id)R[h]=R[h]13R[h]23, (id⊗Δ)R[h]=R[h]13R[h]12,and which are unitalized by the counit in either copy of 𝒯(ℒ), can be characterized as such directed double product integrals ∏∏(1+d⃗⊗d↙r[h]) where r[h] is a formal power series with coefficients in L⊗L and vanishing constant term.