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)xT(L))[[h]] satisfying (Deltaxid)R[h]=R[h]R-13[h](23), (idxDelta)R[h]=R[h]R-13[h](12),and which are unitalized by the counit in either copy of T(L), can be characterized as such directed double product integrals PiPi(1+(d) over right arrowx (d) over left arrowr[h]) where r[h] is a formal power series with coefficients in LxL and vanishing constant term. (C) 2004 American Institute of Physics.
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机译：设T（L）为李代数L上所有张量的空间，其中李括号是通过将换向子带入相联代数获得的。我们显示，当T（L）随同洗牌产品和标准副产品一起进行非交换性修改时，就变成了Hopf代数。给出了有向双乘积积分的定义，即由正规幂级数驱动的迭代单乘积积分，其中L的张量积中的系数具有适当的关联代数。对于形式幂级数的霍夫夫代数T（L）[[h]]，我们证明（T（L）xT（L））[[h]]的元素R [h]满足（Deltaxid）R [h] = R [h] R-13 [h]（23），（idxDelta）R [h] = R [h] R-13 [h]（12），并且在T（L）的两个副本中均被同位单元统一）可以表征为这样的有向双积积分PiPi（1+（d）超过右箭头x（d）超过左箭头r [h]），其中r [h]是系数为LxL且不为常数项的形式幂级数。 （C）2004年美国物理研究所。
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