# Non-Drinfeld--Jimbo Deformations and Finite Quantum Groups

As is well-known, for the compact semi-simple Lie groups $G$, there exist non-commutative Hopf algebra deformations ${\cal O}_q[G]$ of their coordinate algebras ${\cal O}[G]$, the so-called Drinfeld--Jimbo quantized coordinate algebras. Do there exist other examples of noncommutative Hopf algebra deformations of ${\cal O}[G]$? Have such deformations been classified?

Moreover, does there exist analogous constructions for "quantizing" the group algebra of a finite group.

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In addition to the Hopf algebra $O_q(G)$ which you mentioned, there is an important twist-equivalent Hopf algebra $A_q$ introduced by Majid, as the "equivariantized coordinate algebra". In modern terminology, O_q(G) is an algebra in a rather unnatural category: $C^{op}\boxtimes C$, where $C$ is the braided tensor category of locally finite $U_q(g)$-modules. Majid's algebra is the image of $O_q$ under the composition $C^{op}\boxtimes C\to C\boxtimes C \to C$, first applying the braiding on the $C^{op}$ factor, and then applying the functor of tensor product. By construction $A_q$ is equivariant for the adjoint action (hence the name) of $U_q$ on itself. Generally speaking, if you are wanting to quantize something related to the diagonal or adjoint action of $G$, $A_q$ is the one you want. If you are considering rather the one-sided action of some subgroup of $G$, then you want $O_q(G)$.

For $GL_n$, the algebra $A_q$ is sometimes called the reflection equation algebra, because it's defining relations related to affine reflection groups.

The algebra $A_q$ also has a nice interpretation as the CoEnd of the tensor functor $C\boxtimes C\to C$; equivalently, it is a direct limit of $V^*\otimes V$, over all finite dimensional representations $V$ of $U_q(g)$, subject to certain natural relations involving duals.

Finally work of Caldero and Joseph-Letzter exhibits $A_q$ as a certain canonical sub-algebra of $U_q$, so that one can view $U_q$ as degenerating both to $U(g)$ and $O(G)$ at the same time; this can be regarded as a non-commutative Fourier transform.

Regarding the question of quantizing the group algebra of a finite group: one issue is that even infinitessimal deformations are necessarily trivial, since (at least over a field of char zero), the group ring of a finite group is semi-simple and so admits no non-trivial deformations. That said, there are the Hecke algebras which "deform" the group algebras of reflection groups; although these deformations are trivial for generic parameters (are actually isomorphic to the group algebra itself), they are still interesting for many reasons.

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I do not know of a general method for quantizing the group algebra of a finite group. However, there is a way to do it for Coxeter groups (finite or not): the result is called an Iwahori-Hecke algebra. These are closely related to Drinfeld-Jimbo quantum groups, at least when the Coxeter group is the Weyl group of a finite-dimensional semisimple Lie algebra (and probably this extends to Kac-Moody algebras, but I don't know enough about that to make a definitive statement).

One thing to think about is that if $G$ is a finite group, the group algebra $kG$ is semisimple as long as the characteristic of the field doesn't divide the order of $G$. And semisimple algebras are somewhat resistant to deformation. See this question for some more details on that story.

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Sorry, link to Iwahori-Hecke algebra isn't working. The wiki page has a long dash in the title which seems to make copy/paste not work properly. But if you click the link I put there it will take you to the disambiguation page for "Hecke algebra", which has a link to the page for Iwahori-Hecke algebras. – MTS Mar 15 '13 at 3:13
I may have forgotten/mixed things, but the problem of semisimplicity was already there for D-J quantum groups; actually, the 'trick' is that one doesn't deform the algebra (it's essentially impossible), but rather the coproduct. – Amin Mar 15 '13 at 7:47

Much depending on what you want to do with it.... ;-)

There is a duality between coordinate algebras to the Drinfel'd-Jimbo $U_q(\mathfrak{g})$ (you're title suggests you're interested rather in the finite-dimensional truncations??). Generalizations of the latter (so-called finite-dimensional pointed Hopf algebras) have indeed be classified in certain cases (Andruskiewitsch-Schneider, arXiv). You give a Dynkin diagram of Cartan type, q-decorations and so-called linkings (dotted lines) fulfilling certain diagramactic rules. There are exotic examples with several different "Borel-algebras".

Maybe your problem can be reformulated using the duality?

Can you give further information, then I'll be glad to try to help?

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In the compact case: twisted quantized function algebras (which shouldn't be the same as what David Jordan is discussing, since they are true Hopf algebras over $\mathbb C$) where introduced by Levendorskij and Soibelman:

1. Levendorskij Twisted function algebras on a compact quantum group and their representations St. Petersburg Math. J. 3, (1992)

2. Levendorskij and Soibelman Algebras of functions on compact quantum groups, Schubert cells and quantum tori, Comm. Math. Phys. 139, (1991).

Such Hopf algebras are a family parametrized by a real number $a$ and an element $u\in\wedge^2\mathfrak t$, where $\mathfrak t$ is the Lie algebra of the maximal torus $T$ of $G$. When $a=1$ and $u=0$ one gets back the more familiar quantization. In the $SU(2)$ case, since $\wedge^2\mathfrak t=0$ and the real parameter $a$ can be "rescaled" than there is nothing differing from Drinfel'd-Jimbo. It is often the case that the term "twisted" is reserved only for the $a=0$ case, so my terminology here could be not completely standard.

Are they classified? Well in a sense yes. The point is that we know that Hopf algebra quantizations of compact groups are (from the work of Etingof-Kazhdan) functorial from Poisson--Lie group structures. So we have one, up to iso, for any non isomorphic Poisson-Lie group structure on $G$. Now this twisted quantized function algebra are quantizing all compact Poisson-Lie group structure, the latter being classified. So in this setting this list is complete.

A good place to study such things is the book "Algebras of functions on quantum groups PartI" Math. Surv. and Monographs 56, by Korogodski and Soibelman. However references there are quite limited in number.

The "complexified" part of the story is much more complicated. A good starting point is

Hodges-Levasseur-Toro "Algebraic structure of multiparametric quantum groups" Advances in Math. 126 (1997).

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