(For the background, I am learning about quantum groups — essentially in order to understand crystal/global/canonical bases in the context of this question — from the books by Jantzen and by Hong&Kang and the 1995 paper "On Crystal Bases" by Kashiwara, along with a few others.)

The definition given of a quantized universal enveloping algebra (at least in the sources mentioned in the above parenthesis, or here on Wikipedia) is an explicit construction by generators and relations. What I would like to understand is whether this is merely convenient (simply construct the objects we care about and then work with them) or if there is some deeper reason:

Is there an alternative definition of the quantized universal enveloping algebra of a semisimple Lie algebra $\mathfrak{g}$ that does not involve giving an explicit construction with generators and relations?

(I am willing to restrict myself to finite dimensional $\mathfrak{g}$ if that helps.)

This could mean, for example, a combination of one of the following ideas that come to my mind:

• Abstractly defining a quantum deformation of the universal enveloping algebra of a semisimple Lie algebra.

• Defining a particular condition on Hopf algebras (along the lines of "the category of <mumble mumble mumble> modules is semisimple") and then classifying the algebras satisfying this condition using root systems just like one does for finite-dimensional semisimple Lie algebras.

• Describing the quantized universal enveloping algebra as solution to a universal problem (or representing some functor).

• Perhaps only in the classical ($A_n$, $B_n$, $C_n$, $D_n$) cases, constructing the algebra starting from a "standard representation" that itself can be obtained from basis-free data (such as a vector space perhaps with a quadratic form attached to it, or something).

• Using the "canonical basis" to define the algebra in the first place.

(Maybe some of these ideas are completely stupid. I merely list them in order to explain the sort of thing I'd be happy to see.)

Even a construction that still involves generators and relations but avoids choosing a basis of the root system would be interesting to see.

As things stand, I don't even understand to what extent the $e,f,k$ generators of the algebra can be recovered from the algebra itself, or what choices have to be made for that (this is admittedly a different question, but I imagine it is strongly related), so answers along that line are also welcome.