You might be interested in the notion of a braided Lie algebra due to Majid. Roughly speaking this is a coalgebra ${L}$ in a braided category (ie $L$ is an object in a braided category category, with morphisms $\Delta:L \otimes L \to L$, and $\epsilon:L \to C$ satisfying the natural generalization of the axioms of a coalgebra), and in addition a morphism $$ [ , ]:L \otimes L \to L, $$ satisfying a "braided version" of the axioms of a Lie algebra.
The notion of the universal enveloping algebra of a Lie algebra generalizes to this context, and, quoting from Majid's paper http://arxiv.org/pdf/hep-th/9303148v1.pdf,
... the standard quantum deformations $U_q({\frak g})$ are understood as the enveloping algebras of such underlying braided Lie algebras ...
A best place to learn about these structures is probably Majid's Quantum Groups Primer book
The paper arxiv.org/abs/q-alg/9510004 mentioned in Jake's answer contains some discussion of these structures.