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.
I don't think that you really need to learn much more algebra before you start on Hopf algebras. As long as you know about groups, rings, etc, you should be fine. An abstract perspective on these things is useful; e.g. think about multiplication in an algebra $A$ as being a linear map $m : A \otimes A \to A$, and then associativity of multiplication as being a certain commutative diagram involving some $m$'s. This naturally leads to dualization, i.e. coalgebras, comultiplication, coassociativity, etc, and then Hopf algebras come right out of there by putting the algebra and coalgebra structures together and asking for some compatibility (and an antipode).
For the Drinfeld-Jimbo type quantum groups, it is helpful to know some Lie theory, especially the theory of finite-dimensional semisimple Lie algebras over the complex numbers. If you don't know that stuff, the definitions will probably not be that enlightening for you.
There are a lot of books on quantum groups by now. They have a lot of overlap, but each one has some stuff that the others don't. Here are some that I have looked at:
- Quantum Groups and Their Representations, by Anatoli Klimyk and Konrad Schmudgen. They have a penchant for doing things in excruciating, unenlightening formulas, but this book is the first one that I learned quantum groups from, so it remains the most familiar to me. This one has a lot more about Hopf $*$-algebras than any of the others.
- A Guide to Quantum Groups, by Vijayanthi Chari and Andrew Pressley. Has an approach based more on Poisson geometry and deformation quantization.
- Foundations of Quantum Group Theory, by Shahn Majid. Goes into more detail on braided monoidal categories, braided Hopf algebras, reconstruction theorems (i.e. reconstructing a Hopf algebra from its category of representations) than most other books, although some of this is covered in Chari-Pressley.
- Quantum Groups, by Christian Kassel. Focuses mainly on $U_q(\mathfrak{sl}_2)$ and $\mathcal{O}_q(SL_2)$, and does a lot of stuff with knot invariants coming from quantum groups.
- Hopf Algebras and Their Actions on Rings, by Susan Montgomery. This one is more about the theory of Hopf algebras than about Drinfeld-Jimbo quantum groups.
There are some other ones which I know are out there, but I haven't read. These include Lectures on Algebraic Quantum Groups, by Ken Brown and Ken Goodearl, Lectures on Quantum Groups, by Jens Jantzen, Introduction to Quantum Groups, by George Lusztig, and Quantum Groups and Their Primitive Ideals, by Anthony Joseph. Having glanced a little bit at the last two in this list, I found both of them more difficult to read than the ones in my bulleted list above.
So, as you can see, there is a lot of choice available. I would advise you to check a few of them out of the library and just see which one you like the best.
Best Answer
I will try to use notations compatible with those in Theorem 3.2.12 from the mentioned Timmerman's book "An invitation to quantum groups and duality".
So for each $\alpha$ i will denote by $V_\alpha$ the space $\mathfrak C(u^\alpha)$, by $\delta_\alpha$ the corepresentation of $A$ on $V_\alpha$ defined by the restriction of the (right regular corepresentation) $\Delta$, by $V$ the whole space $A$ (seen as a vector space after applying the forgetful functor).
Assume there is some $v\ne 0$ in the intersection $$ V_\alpha\cap\underbrace{\left(\sum_{\beta\ :\ \beta\ne \alpha}V_\beta\right)}_{=:W^\alpha} \subseteq V\ . $$ In the lattice of invariant subspaces of $V$ (with $\min$ given by intersection) let us consider the smallest invariant subspace $U\ne 0$ containing $v$. (The family of such spaces is not empty, contains at least $V$.) Then $U\subseteq V_\alpha$, (else consider $U\cap V_\alpha$, and contradict the assumed minimality,) and $U\subseteq W^\alpha$.
For the convenience of the reader, i am roughly citing from loc. cit.
Applied for $\delta_\alpha$ instead of $\delta$, this gives: $ \displaystyle \Delta\Big|_{V_\alpha} = \Delta\Big|_{\mathfrak C(\delta_\alpha)} \cong \delta_\alpha^{\boxplus n_\alpha} $ for some $n_\alpha\in\Bbb N$.
On the other side, let us build $U^\perp$ inside $V_\alpha$ (not inside $V$), it is also an invariant subspace, since $U$ is, and we split $\Delta$ restricted on $V_\alpha$ into the pieces. We obtain: $$ \delta_\alpha^{\boxplus n_\alpha} \cong \Delta\Big|_{V_\alpha} = \Delta\Big|_U \boxplus \Delta\Big|_{U^\perp} \ . $$ The projectors on the $\boxplus$-summands are intertwiners. Further splitting the two summands on the R.H.S. above, and applying Schur' Lemma for corepresentations, we see that only $\delta_\alpha$ components may occur in $\Delta\Big|_U$.
With the same argument applied for the restriction of $\Delta$ to $W^\alpha$, (and with an orthocomplement of $U$, built inside $W^\alpha$,) we see that only $\delta_\beta$ components for one or more values of $\beta\ne \alpha$ may occur in $\Delta\Big|_U$.
Contradiction. (The identity of $U$ is inside a sum of homomorphisms from the $n_\alpha$ pieces $ \delta_\alpha$ to corresponding copies of pieces $\delta_\beta$, but $\operatorname{Hom}(\delta_\alpha,\delta_\beta)=0$ for $\beta\ne\alpha$.)
$\square$
Later edit.
I'm very thankful for the comments, and try here to say more on the way to use the "same argument" as presented for $\Delta|_{V_\alpha}$ also for the restriction $\Delta|_{W_\beta}$. There is a constraint in doing so. The question addresses the situation from the implication (iii) $\Rightarrow$ (iv) from Theorem 3.2.12 in the cited book of Timmerman. So the arguments have to be given in this in-between situation. (For this reason a Haar functional $h$ will not appear.)
Some references for these details / discussion:
[T] An invitation to Quantum Groups and Duality, Thomas Timmermann, EMS/AMS, 2008
[K] General Compact Quantum Groups, a Tutorial, Tom H. Koornwinder, 1994
[DK] CQG Algebras: A Direct Algebraic Approach to Compact Quantum Groups, Mathijs S. Dijkhuizen, Tom H. Koornwinder, Lett. Math. Phys. 32 (1994), no. 4, 315–330.
[MvD] Notes on Compact Quantum Groups, Ann Maes, Alfons Van Daele, 1998
([T] is target oriented, algebraic CQG's are in focus, although it collects many results not involving a Haar state $h$. Koornwinder and Dijkhuizen are extract the linear algebra for the first time, [K] §2 is a survey. The $C^*$-algebraic framework of CGQ as presented for instance in [MvD] is isolated first a the Hopf-algebraic level. So [K] is well suited for this answer. We use only this reference below.)
Relevant results:
[K] Proposition 1.28, showing that matrix elements of mutually inequivalent corepresentations are linearly independent, i think this is the missing piece in the puzzle, claim and proof are added below.
[K] Lemma 1.25, that will be cited here with slightly changed notations to match those in [T]:
The proof of Lemma 1.25 uses adapted Schur Lemmas ([K], Lemma 1.24 (b), (c)), or [T], Proposition 3.2.2 (ii)) for corepresentations, and the following facts.
[K] Lemma 1.24 (a) from [K], Proposition 3.2.2 (i) from [T] - image and kernel of an intertwiner $T:\delta_V\to\delta_W$ are invariant subspaces in $W$, respectively $V$.
[K] Proposition 1.23 from [K], or 3.2.1 (ii) [T]:
Proof: (a) Consider $w\in W$. Let $W^\perp\subseteq V$ be the orthogonal complement. We fix $v\in W^\perp$, show $\delta v\in W^\perp\otimes A$. We write: $$ \begin{aligned} \delta v &= \sum v_0\otimes a_1\ ,\\ \delta w &= \sum w_0\otimes b_1\ ,\\ \end{aligned} $$ with corresponding components $(v_0)$ in $V$, $(w_0)$ in $W\subseteq V$, and $(a_1)$, $(b_1)$ chosen respectively linearly independent in $A$. The compatibility of the scalar product on $A$ with $\delta$ (making $\delta$ unital), $$ (v,w)\;1_A = \underbrace{\sum(v_0,w_0)\; b_1^*a_1}_{=:(\delta v,\delta w)} $$ is rewritten as ([K] (1.42)) : $$ \sum (v_0,w)\; Sa_1 =\sum \underbrace{(v,w_0)}_{=0}\; b_1^*=0\ .$$ So the components $(v_0,w)$ are each equal to zero. Since $w\in W$ is arbitrary, each component $v_0$ is in $W^\perp$, so $\delta v\in W^\perp\otimes A$.
(b) Apply the previous result inductively w.r.t. the dimension of $V$.
$\square$
[K] Proposition 1.28:
(See also the comments on this Proposition in [DK] after Theorem 2.1 in §2, CQG Algebras. There is no scalar product required.)
Proof:
(i)
Let us work first with only one corepresentation $\delta_\alpha$, so we rename it to $\delta:V\to V\otimes A$. We fix a basis of $V$ and obtain the matrix coefficients $(a_{ij})$ of $\delta$, spanning a subspace $\mathfrak C(\delta)$.
Recall that associated to $A$ we have two regular corepresentations induced by $\Delta$: $$ \begin{aligned} \delta^A_r &:A\to A\otimes A\ , & \delta_r &= \Delta\ , & \delta_r a_{ij} &= \sum_k a_{ik}\otimes a_{kj}\ ,\\ \delta^A_l &:A\to A\otimes A\ , & \delta_l &= \tau(S\otimes\operatorname{id})\Delta\ , & \delta_r a_{ij} &= \tau(S\otimes\operatorname{id})\sum_k a_{ik}\otimes a_{kj}\\ &&&&& = \tau\sum_k S a_{ik}\otimes a_{kj}\\ &&&&& = \sum_k a_{kj} \otimes S a_{ik}\\ &&&&& = \sum_k a_{kj} \otimes a'_{ki}\ . \end{aligned} $$ $\tau=\tau_{12}$ is the switch of corresponding $\otimes$-factors (on the positions $1$ and $2$). Here we use the notation $$ a'_{ij} := Sa_{ji}\ . $$ The matrix (coefficients) $(a'_{ij})_{ij}$ also correspond to a corepresentation, the contragredient corepresentation: $$ \sum_k a'_{ik}\otimes a'_{kj} = \sum_k Sa_{ki}\otimes Sa_{jk} = \sum_k (S\otimes S)\tau a_{jk}\otimes a_{ki} = (S\otimes S)\tau\sum_k a_{jk}\otimes a_{ki} = (S\otimes S)\tau\Delta a_{ji} = \Delta S a_{ji} = \Delta a'_{ij}\ . $$ Now consider the corepresentation $\tilde \Delta$ of the Hopf algebra $B:=A\otimes A$ on the space $A$. It is defined as a composition $$ \begin{aligned} A&\overset{\delta_r}\longrightarrow A\otimes A\overset{\delta_l \otimes\operatorname{id}}\longrightarrow (A\otimes A)\otimes A\cong A\otimes (A\otimes A)=A\otimes B\ ,\\ \\ &\qquad\text{ and explicitly on elements $a\in A$:} \\ a&\overset{\delta_r}\longrightarrow \Delta a = \sum a_1\otimes a_2\\ &\qquad \overset{\delta_l \otimes\operatorname{id}}\longrightarrow \sum \delta_l a_1\otimes a_2 \\ &\qquad\qquad = \sum \tau(S\otimes \operatorname{id})\Delta a_1\otimes a_2\\ &\qquad\qquad = \sum (\tau(S\otimes \operatorname{id}) a_{11}\otimes a_{12})\otimes a_2\\ &\qquad\qquad = \sum a_{12}\otimes S a_{11}\otimes a_2= \sum a_2\otimes S a_1\otimes a_3\ ,\\ &\qquad\text{ so its action on the matrix element $a_{ij}$ is:}\\ \Delta^{(2)}\underbrace{a_{ij}}_a &=(\Delta\otimes\operatorname{id})\Delta a_{ij}=\sum_{k,j} \underbrace{a_{ik}}_{a_1}\otimes\underbrace{a_{kl}}_{a_2}\otimes\underbrace{a_{lj}}_{a_3} \\ a_{ij} &\overset{\tilde\Delta}\longrightarrow \sum_{k,l}a_{kl}\otimes S a_{ik}\otimes a_{lj} \\ &\qquad = \sum_{k,l}\underbrace{a_{kl}}_{\in\mathfrak C(\delta)}\otimes \underbrace{a'_{ki}\otimes a_{lj}}_{\in B=A\otimes A} \in\mathfrak C(\delta)\otimes B\ . \end{aligned} $$ So we can consider the restriction $\tilde\Delta|=\tilde\Delta|_{\mathfrak C(\delta)}:\mathfrak C(\delta)\to\mathfrak C(\delta)\otimes B$.
The above $\tilde\Delta|$ was built on $A$ using the Hopf algebra structure, mainly $\Delta$.
On the other hand, using the matrix coefficients $(a_{ij})$ and $(a'_{ij})$ and the corresponding corepresentations $\delta$ and $\delta'$ we obtain a corepresentation of $B=A\otimes A$. $$\tilde\delta:=\delta'\otimes\delta\ .$$ It is irreducible, since $\delta,\delta'$ are irreducible, [K] Lemma 1.26. It is given by: $$ \tilde\delta e_{ij} = \sum_{k,l} e_{kl}\otimes \underbrace{a'_{ki}\otimes a_{lj}}_{\in B=(A\otimes A)}\ . $$ From the match of the $B$-coefficients in the formulas for $\tilde \Delta a_{ij}|$ and $\tilde\delta e_{ij}$ [K] constructs the intertwiner $L:\tilde \delta\to\tilde\Delta|$ given by $L(e_{ij})=a_{ij}$.
The kernel of $L$ is an invariant space. If it is not $0$, then it is also proper since $\epsilon(a_{ii})=1\ne 0$, contradicting $\tilde\delta$ irreducible. So the kernel of $L$ is trivial, making it a bijection. Since $(e_{ij})$ is a linear independent system, the same holds for $(a_{ij})$.
(ii)
Let us consider now a list of inequivalent, irreducible corepresentations $\delta_\alpha$. Set $V_\alpha=\mathfrak C(\delta_\alpha)$ and the corresponding matrix elements $(a^\alpha_{ij})$. For each $\alpha$ we use a different index set $I(\alpha)$, so that the (disjoint) union of all $I(\alpha)$ is a set $I$.
Let $V$ be the span of all $V_\alpha$. Then $\Delta$ maps $\mathfrak C(\delta_\alpha)$ into $\mathfrak C(\delta_\alpha)\otimes \mathfrak C(\delta_\alpha)$. So $V$ is an invariant space of $\tilde\Delta$ over $B=A\otimes A$.
On the other side, consider elements $(e^\alpha_{ij})$ generating spaces $W_\alpha$ and set $W=\bigoplus W_\alpha$. $W$ comes naturally with a corepresentation $$ \boxplus_\alpha (\delta_\alpha' \boxtimes\delta_\alpha) $$ The pieces $(\delta_\alpha' \boxtimes\delta_\alpha)$ are inequivalent and irreducible (over $B=A\otimes A$), [K] Lemma 1.26.
Define again an intertwiner $L$ by setting $e^\alpha_{ij}\to a^\alpha_{ij}$. Its kernel is invariant. Suppose it is not trivial, so it has an invariant subspace $U\ne 0$. By [K] Lemma 1.25 $U$ is $W_\alpha$ for some suitable $\alpha$. For this index $\alpha$ we then have $a^\alpha_{ii}=0$ contradicting $\epsilon(a^\alpha_{ii})=1$.
$\square$
(All arguments are reproduced from [K].)