For a compact quantum group $C_q[G]$, it was shown by Woronowicz that $C_q[G]$ contains a dense Hopf algebra generalising the algebra of representations of $G$. I am interested in the other way around, ie given a Hopf algebra $H$ (say a Drinfeld–Jimbo algebra if it makes things easier) can it always be completed to give a compact quantum group? If so, is this completion unique in anyway, or are there many ways to get a cqg from a Hopf algebra?
[Math] Compact Quantum Groups from Hopf Algebras
noncommutative-geometryoa.operator-algebrasqa.quantum-algebraquantum-groups
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These Hopf algebras are distinguished by the property that their representation categories admit a braided tensor structure. The space of deformations of the braided tensor structure on these particular categories is one dimensional (I believe this is a result of Drinfeld), so we get a universal family of braided deformations of U(g)-mod by varying q in U_q(g). The symmetric structure in U(g)-mod makes it a special point in this space, and one could argue that U(g)-mod is a "degeneration" of the generic braided behavior. There is unpublished work of Lurie on algebraic groups over the sphere spectrum that lends homotopy-theoretic support to this idea, since the symmetric structure doesn't manifest over the sphere.
Braided structures are important when studying topological (and conformal) field theories, since they describe the local behavior of embedded codimension 2 objects, such as points in a surface or links in a three-manifold. If you like homotopy theory, a braided tensor category is one that admits an action of the E[2] operad, whose spaces are (homotopy equivalent to) configuration spaces of points in the plane. Physically, these are the points where one inserts fields.
In principle, any statement about semisimple groups that can be phrased in braided-commutative (rather than fully commutative) language should be reconfigurable to a statement about these quantum groups. For example, there is a quantum local Langlands program (see the introduction of Gaitsgory's twisted Whittaker paper). Also, the representation theory of U_q(g) is interesting because of its connections to the representation theory of affine algebras and mod p representations (I think Kazhdan, Lusztig, and Bezrukavnikov are among the key names here).
I've been having trouble answering this question because I think your notion of "quantum group" is either too restrictive or too expansive. Hopf algebras suffer from annoying analytic issues as soon as they're infinite dimensional, so you should either be looking at finite dimensional quasitriangular Hopf algebras, or you should pick some particular world of analysis you want to work in (C* quantum groups, h-adic quantum groups, etc.). On the other hand, there's no real reason to restrict your attention to Hopf algebras, lots of things that go under the name "quantum group" (most notably the semisimplified categories at a root of unity which occur in Reshetikhin-Turaev's construction of 3-manifold invariants) are not the category of representations of a Hopf algebra, instead they're a braided tensor category.
Anyway some important constructions that you don't mention include:
- Drinfel'd twists of already known quasi-triangular Hopf algebras
- Triangular Hopf algebras (classified by Etingof-Gelaki)
- Woronowicz-style quantum groups in the C* setting
- Various flavors of quantum groups at roots of unity
- Bruguieres-Mueger quotients (often called "modularization" or "deequivariantization) of known braided tensor categories
Best Answer
I think this was solved in the paper:
MR1310296 (95m:16029)
Dijkhuizen, Mathijs S.(NL-MATH); Koornwinder, Tom H.(NL-AMST-CS)
CQG algebras: a direct algebraic approach to compact quantum groups. (English summary)
Lett. Math. Phys. 32 (1994), no. 4, 315–330.
They show that a Hopf $*$-algebra $A$ is the maximal Hopf $*$-algebra of a compact quantum group in the Woronowicz sense if (and only if) $A$ is spanned by the matrix coefficients of its finite-dimensional unitary (irreducible) corepresentations. The "only if" part was shown by Woronowicz, of course.
They also show that a Hopf $*$-algebra has this property if and only if it admits a positive definite Haar functional (in the sense Andreas talks about).
You also ask about uniqueness-- this occurs if and only if $A$ is "coamenable" (see various papers by Bedos, Tuset and coauthors). A classical example comes from a discrete group $G$ with Hopf $*$-algebra $\mathbb C[G]$ (under convolution product). Then there is the maximal $C^*$ completion $C^*(G)$ and the completion of $\mathbb C[G]$ acting on $\ell^2(G)$ by the left-regular representation, leading to $C^*_r(G)$. The quotient map $C^*(G) \rightarrow C^*_r(G)$ is injective if and only if $G$ is amenable.