A while ago I was doing a reading course on Link Invariants and I came across the notion of a Drinfeld Double: given a Hopf algebra, H, the Drinfeld Double, D(H), was a quasi-triangular Hopf algebra.
Recently I was reading some physics papers and I again came across the notion of a Drinfeld Double, but I'm not sure about the relation between the two. In this paper, the Drinfeld Double was defined as a Lie group whose Lie algebra could be decomposed into a pair of maximally isotropic subalgebras with respect to a non-degenerate invariant bilinear form.
Are these two notions related, and if so, how?
Yes, they are related. A Lie Algebra $D(L)$ is called a Drinfeld double, if it can be endowed with a Manin triple structure $(L,S^+,S^-)$. This is the so-called "semiclassical analogue" of Drinfeld's quantum double of a Hopf algebra $H$, yielding a quasi-triangular Hopf algebra $D(H)$. In fact, $D(L)$ is a quasi-triangular Lie bialgebra, and hence a semiclassical Hopf algebra. References are given in the paper you have linked (in particular, Drinfeld's Quantum Groups).