# Can one define quantized universal enveloping algebras in a basis-free way?

(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.

• A related question? mathoverflow.net/questions/105221/… – LSpice Jul 19 '17 at 21:36
• Probably Vogel's approach could be used - it builds only on invariant quantities like Casimir eigenvalues. Quantization has been worked on by Westbury – მამუკა ჯიბლაძე Jul 19 '17 at 21:37
• @LSpice Good catch, I wouldn't object to my question being marked as a duplicate, although André Henriques's question is perhaps more restrictive, but Theo Johnson-Freyd's answer to it is definitely enlightening. I'll let other people decide whether to close this. – Gro-Tsen Jul 19 '17 at 22:13
• Quantum geometric Satake can at least recover the category of representations of the quantum group, not sure if it can recover the fiber functor as well: front.math.ucdavis.edu/0705.4571 – Qiaochu Yuan Jul 20 '17 at 0:11
• @Gro-Tsen, sorry; I didn't mean to say that it was a duplicate (the answers seem meaningfully different), literally just that it appeared to be an interesting related question. – LSpice Jul 20 '17 at 14:00

For complex simple $$\mathfrak g$$, Drinfeld (1986, p. 807) already characterized his $$\mathrm U_h\mathfrak g$$ as the unique (up to equivalence and change of parameter) deformation of $$\mathrm U\mathfrak g$$ admitting a “quantum Cartan involution”, i.e. an algebra automorphism and coalgebra antiautomorphism extending the Cartan involution and for which comultiplication is cocommutative on the Cartan subalgebra. (This exact statement is from Shnider (1991, Cor. 3), more details and a proof in Shnider-Sternberg (1993, p. 285).)

• I just went and found a copy of Shnider–Sternberg cough somewhere cough, and, irrespective of the mathematics, am pleased to see on the referenced page that probably-dying artefact of e-mail mangling + TeX, the string "¿From" as which TeX renders ">From" in Computer Modern. – LSpice Jul 19 '17 at 21:43
• the first link is dead now. It would be nice if it were replaced. – Konstantinos Kanakoglou Nov 5 '18 at 23:03
• @KonstantinosKanakoglou Done. – Francois Ziegler Nov 6 '18 at 0:04
• @Francois Ziegler, great! – Konstantinos Kanakoglou Nov 6 '18 at 0:07

I do not know the answer in general. But towards the end of the OP you say:

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

Motivated from this remark, I was wondering whether this description might be of interest for your purposes. If yes, you can find more details at C. Kassel's book on Quantum groups, ch. $VI$, sect. $2$, p. 125-126 (Prop. 2.1, 2.2).

• Of course, p. 125–126 are excerpted from the otherwise-generous Google Books preview of Chapter VI …. – LSpice Jul 19 '17 at 21:32