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Double product integrals and Enriquez quantization of Lie bialgebras. I: The quasitriangular identities. (English) Zbl 1071.81073

Summary: Let \(\mathcal T(\mathcal L)\) be the space of all tensors over a Lie algebra \(\mathcal L\) in which the Lie bracket is obtained by taking commutators in an associative algebra. We show that \(\mathcal T(\mathcal L)\) 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 \(\mathcal L\) coefficients in the tensor product of with an appropriate associative algebra. For the Hopf algebra \(\mathcal T(\mathcal L)[[h]]\) of formal power series we show that elements \(R[h]\) of \((\mathcal T(\mathcal L) \otimes \mathcal T(\mathcal L))[[h]]\) satisfying \((\Delta \otimes \text{id})R[h] = R[h]^{13} R[h]^{23}\), \((\text{id} \otimes \Delta)R[h] = R[h]^{13}R[h]^{12}\), and which are unitalized by the counit in either copy of \(\mathcal T(\mathcal L)\), can be characterized as such directed double product integrals \(\prod \prod(1 +\overset \neg d \otimes \overleftarrow {d}r[h])\) where \(r[h]\) is a formal power series with coefficients in \(\mathcal L \otimes \mathcal L\) and vanishing constant term.

MSC:

81S25 Quantum stochastic calculus
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B62 Lie bialgebras; Lie coalgebras
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