Double Product Integrals and Enriquez Quantisation of Lie Bialgebras II: The Quantum Yang–Baxter Equation

Abstract

For a Lie algebra with Lie bracket got by taking commutators in a nonunital associative algebra \({\cal L}\), let \({\cal T}({\cal L})\) be the vector space of tensors over \({\cal L}\) equipped with the Itô Hopf algebra structure derived from the associative multiplication in \({\cal L}\). It is shown that a necessary and sufficient condition that the double product integral \({\buildrel \rightarrow \leftarrow \over \prod} (1+ \hbox{d}r[h])\) satisfy the quantum Yang–Baxter equation over \({\cal T}({\cal L})\) is that \(1 + r[h]\) satisfy the same equation over the unital associative algebra \({\cal L}^{\prime}\) got by adjoining a unit element to \({\cal L}\). In particular, the first-order coefficient r1 of r[h] satisfies the classical Yang–Baxter equation. Using the fact that the multiplicative inverse of \({\buildrel \rightarrow \leftarrow \over \prod} (1+ \hbox{d}r[h])\) is \({\buildrel \rightarrow \leftarrow \over \prod} (1+ \hbox{d}r^{\prime}[h])\) where \(1+r^{\prime} [h]\) is the inverse of \(1 + r[h]\) in \({\cal L}^{\prime}h\) we construct a quantisation of an arbitrary quasitriangular Lie bialgebra structure on \({\cal L}\) in the unital associative subalgebra of \({\cal T}({\cal L})[[h]]\) consisting of formal power series whose zero order coefficient lies in the space \({\cal S}({\cal L})\) of symmetric tensors. The deformation coproduct acts on \({\cal T}({\cal L})\) by conjugating the undeformed coproduct by \({\buildrel \rightarrow \leftarrow \over \prod} (1+ \hbox{d}r[h])\) and the coboundary structure r of \({\cal L}\) is given by \(r = r_{1} - \tau_{(2,1)}r_{1}\) where \( \tau_{(2,1)}\) is the flip.