If $f: [a,b] \to V$ is a (nice) function taking values in a vector space, one can define the definite integral $\int\_a^b f(t)\ dt \in V$ as the limit of Riemann sums $\sum\_{i=1}^n f(t\_i^\*) dt\_i$, or as the final value $F(b)$ of the solution $F: [a,b] \to V$ to the ODE problem $F'(t) = f(t); F(a) = 0$.


In a similar spirit, given a (nice) function $f: [a,b] \to {\mathfrak g}$ taking values in a Lie algebra $\mathfrak g$ of a Lie group $G$, one can define the multiplicative definite integral, which for sake of discussion I will denote $\Pi\_a^b \exp(f(t)\ dt) \in G$, either as the limit of Riemann products $\prod\_{i=1}^n \exp(f(t\_i^\*) dt\_i)$ (with the product read from left to right), or as the final value $F(b)$ of the solution $F: [a,b] \to G$ of the ODE $F'(t) = F(t) f(t); F(a) = 1$.


Thus, for instance, when the Lie algebra is abelian, the multiplicative integral is just the exponential of the ordinary integral,


$$\Pi\_a^b \exp(f(t)\ dt) = \exp( \int\_a^b f(t)\ dt)$$


but in general the two are a little bit different, though still quite analogous.


This notion arises implicitly in many places (solving ODE, integrating connections along curves, dynamics and random walks on Lie groups (e.g. in the work of Terry Lyons), the "noncommutative Fourier transform" from scattering theory, etc.), but I am sure that it must be studied explicitly in some body of literature (and even vaguely recall seeing such at some point in the past). But I am having difficulty locating this literature because I am not sure I have the correct terminology for this concept. So my questions are:


1. What is the accepted name and notation for this concept in the literature? (Perhaps there is more than one such notation, coming from separate bodies of literature.)
2. What are the references for the theory of this concept?



What is the standard notation for a multiplicative integral?
This type of construction also arises in topology and algebraic geometry as "iterated integrals" or "Chen's iterated integrals". There are many sources of which a famous one by Chen himself is: [Link](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-83/issue-5/Iterated-path-integrals/bams/1183539443.full) .


Path-ordered (or time-ordered) exponential, as suggested in the other answer, is the most common term, or at least would get the most hits in a search, but this is due to the usage in physics.


ADDED: this paper by Hain (Chen's student) calls the construction "iterated line integrals". <https://arxiv.org/abs/math.AG/0109204> . Another paper calls it "iterated integrals" in a more specific context matching that of the question: p.21 of <http://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf> .


