# Drinfeld't map, centre of quantum group, representation category of quantum group

My question is about the Drinfeld't map between $Rep(U_q(\mathfrak{g}))$ and $Z(U_q(\mathfrak{g}))$. I have heard the reference 1989 paper by Drinfeld't "Almost cocommutative Hopf algebras" - but this paper I cannot find in English.

Define the Drinfeld't map as follows. Given a representation $\pi_V: U_q(\mathfrak{g}) \rightarrow End(V)$, $(1 \otimes \pi_V)(R^{21}R) \in U_q(\mathfrak{g}) \otimes End(V)$, so let $c_V = tr((1 \otimes \pi_V)(R^{21}R)) \in U_q(\mathfrak{g})$. It is easy to see that the map $c$ is compatible with the additive structure on $Rep(U_q(\mathfrak{g}))$; is it also compatible with the multiplicative (tensor) structure on that category (my calculations say otherwise, but I may have messed up somewhere)? Then the claim is that $c_V \in Z(U_q(\mathfrak{g}))$ and these elements generate the center under suitable hypothesis. Why is this?

If there is an ambiguity: by $U_q(\mathfrak{g})$, I mean the quantum group defined in Jantzen's "Lectures on Quantum Groups", Chapter 4.

Edit: Cf. the comment below, feel free to make any assumptions that make the problem easier, e.g. characteristic $0$ and $q$ not a root of unity.

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This kind of question is natural but usually difficult, since even in some classical cases (such as prime characteristic) the centers of relevant associative algebras are elusive. That reflects in part complications in the related representation theory related to finding "blocks". There is an English translation of Drinfeld's paper if you can find it through a library, and a MathSciNet review: MR1025154 (91b:16046), Drinfel′d, V. G., Almost cocommutative Hopf algebras. (Russian), Algebra i Analiz 1 (1989), no. 2, 30–46; translation in Leningrad Math. J. 1 (1990), no. 2, 321–342. –  Jim Humphreys May 2 '11 at 22:08
I am having trouble locating it at the library unfortunately. What happens in the simplest possible case - characteristic 0, and q not a root unity? Someone told me a short outline of how the map is defined, but I don't remember anymore. –  Vinoth May 2 '11 at 23:05
Is that a(nother) correct transliteration for Drinfel’d's name? –  Mariano Suárez-Alvarez May 3 '11 at 0:56
@Mariano: Transliteration of names is too complicated for most users of English like myself to cope with. There is a "soft" sound in Russian which has no letter equivalent in English, so one often sees Drinfel'd or Gelfan'd. But after a while it's easier to omit that extra mark. East Asian languages are really impossible to deal with, as people at MathSciNet sorting out authors Wong/Wang and such can tell you. In mainland China and Taiwan it's even a political issue of sorts. –  Jim Humphreys May 5 '11 at 22:33

Without directly answering the question about Drinfeld's paper, I'd emphasize the variations in the definition of "quantum group" in the literature and the resulting variation in discussions about centers, representations, etc. The early work by Drinfeld and Jimbo was partly motivated by mathematical physics, so the $q$ in the definition might be a complex number (say related to the Planck constant). But in later work by Lusztig and others the $q$ is at first an indeterminate and can then be specialized, so that $q=1$ recovers something close to the universal enveloping algebra of a complex semisimple Lie algebra whereas $q \neq 1$ a root of unity leads elsewhere. Jantzen's book mostly follows Lusztig's lead in the treatment of quantized universal enveloping algebras, where a version of Kostant's $\mathbb{Z}$-form leads to finite dimensional versions, etc. (Other generalizations involve affine Lie algebras, while very different "quantum groups" arise from function algebras of algebraic groups.)

In all of these settings it is a problem to describe the center of the quantum group and relate it to the (initially finite dimensional) representation theory. As far as I know, only partial information about centers has been developed so far, while at least for Lusztig's quantum groups the finite dimensional representations have mostly been studied without full knowledge of the center. That has certainly been true in modular representation theory as well.

The good classical prototype occurs in the older work of Chevalley and Harish-Chandra on the usual universal enveloping algebra of a complex semisimple Lie algebra (say of rank $\ell$). Here the center turns out to be just a polynomial algebra in $\ell$ indeterminates, closely related to the enveloping algebra of a Cartan subalgebra and characterized as a suitable algebra of Weyl group invariants. In turn, the "central characters" are easily described and help to sort out representations (as in the earlier use of isolated Casimir elements) even in the more complicated infinite dimensional setting of the BGG category. The notion of "infinitesimal character" even plays a role in the study of Lie group representations, but is only one of many ingredients there. Note that trace functions arising from representations are (as in Drinfeld's construction) a natural way to relate the center to the representation category.

In prime characteristic things get more complicated, as in Lusztig's work on quantum groups at a root of unity: here the finite Weyl group tends to give way to an infinite Coxeter group such as the affine Weyl group (of Langlands dual type) and much is still not understood even though the quantum group representations have been fairly well sorted out. Along the way the role of the center gets diminished, though is still potentially quite interesting.

If the question here is limited just to the Lusztig version of quantized enveloping algebras in characteristic 0 (for an indeterminate $q$), I'm not sure the center has yet been understood well enough from Drinfeld's viewpoint to contribute much to the study of finite dimensional representations. The latter closely resemble the familiar highest weight representations of a semisimple Lie algebra and can be viewed as "quantizations".

By now there is of course a lot of literature to consult, beyond the original papers by Drinfeld and Jimbo. Jantzen gives a good introduction to Lusztig's theory, while Lusztig's many papers (and one book) go farther. But the centers of the various Hopf algebras remain mysterious, to me at least.

ADDED: Probably the most useful book to consult is the 1994 Cambridge treatise A Guide to Quantum Groups by Chari and Pressley (corrected paperback reprint in 1995), which also has an extensive bibliography. They follow some of Drinfeld's 1989/1990 paper for their general discussion in 4.2A of "almost cocommutative" Hopf algebras. Chapter 10 deals with Lusztig's formulation of the quantum enveloping algebra, with $q$ an indeterminate. Here the work of Marc Rosso brings out the classical-looking role of the center in the study of finite dimensional representations. See especially Rosso's 1990 Ann. Scient. Ecole Norm. Sup. paper (and related Bourbaki seminar talk), both available online at www.numdam.org by doing a quick search for "Rosso".

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It reminds me at a remark by Yuri Manin, that so far no "natural" or "intrinsic" definition of quantum groups exists, but I don't remember how old that remark was. –  Thomas Riepe May 3 '11 at 13:34
@Thomas: The remark still seems to be correct. –  Jim Humphreys May 5 '11 at 22:29
@Jim Some surprising feature of quantum groups is that in certain cases it is more simple to deal with them rather than with classical Lie algebras. Description of the centers is such an example. Drinfeld construction works for generic q, but does not work for q=1. Because for q=1 R-matrix is identity because comultplication is co-cocommutative. So you get constants from R=1. Attempt to get classical center by limit q->1 is rather non-trivial. –  Alexander Chervov Oct 8 '11 at 18:00
@Jim For example. For q-analogues of loop algebras of semisimple Lie algebras Drinfeld's construction can be applied modula some technical modifications (there is paper by Etingof, Ding arxiv.org/abs/hep-th/9403064). However the problem of describing appropriate center for q=1 for B,C,D was solved only few months ago by A. Molev arxiv.org/abs/1105.2341. After breakthrough by D. Talalaev in \hat gl_n case several years ago hep-th/0404153 (see also hep-th/0604128, arXiv:0711.2236, arXiv:0808.1947). –  Alexander Chervov Oct 8 '11 at 18:11
The fact that these elements are in the center is quite direct, just using the basic property of $R$ and of the trace, and is true in any quasitriangular Hopf algebra. You really need to know what you want the quantum group for, because it is certainly possible to define the QG so that this is all the center, or so that it is not!