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.

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 ...

The best place to starting learning 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.