MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 completed definition of deformed coordinate ring of SL(2)

The group $G$ being simply connected is not a notion that one can detect from the usual universal enveloping algebra $U(\mathfrak g)$, so one shouldn't expect there to be a notion for quantum groups $U_q(\mathfrak g)$ corresponding to $G$ being simply connected. You should think of $U_q(\mathfrak g)$ as just giving the local structure around the identity, and so global notions like being simply connected don't make sense (for exactly the same reason that $G$ and $G/N$ have the same Lie algebra for any finite normal subgroup $N\lhd G$).

An "intrinsic" perspective is as follows. Let $\hat{\mathcal O}_{G,e}$ be the complete local ring of $G$ at the identity $e\in G$. Then $U(\mathfrak g)$ is a certain dual of $\hat{\mathcal O}_{G,e}$ (it's the direct limit of $(\hat{\mathcal O}_{G,e}/\mathfrak m_{G,e}^n)^\ast$). Thus $U(\mathfrak g)$ only sees the local group structure in a formal neighborhood of $e\in G$.

There may indeed, however, be a natural notion of being simply connected if you look at the models of quantum groups like $\mathbb Z[q,q^{-1}]\langle a,b,c,d\rangle/(ab-qba,bc-cb,cd-qdc,\text{etc})$ a,b,c,d\rangle$modulo the relations$ab=q^{-1}ba$,$ac=q^{-1}ca$,$cd=q^{-1}dc$,$bd=q^{-1}db$,$bc=cb$,$ad-da=(q^{-1}-q)bc$, and$ad-q^{-1}bc=1$(corresponding to the classical group$\operatorname{GL}(2)$).\operatorname{SL}(2)$).

1

The group $G$ being simply connected is not a notion that one can detect from the usual universal enveloping algebra $U(\mathfrak g)$, so one shouldn't expect there to be a notion for quantum groups $U_q(\mathfrak g)$ corresponding to $G$ being simply connected. You should think of $U_q(\mathfrak g)$ as just giving the local structure around the identity, and so global notions like being simply connected don't make sense (for exactly the same reason that $G$ and $G/N$ have the same Lie algebra for any finite normal subgroup $N\lhd G$).

An "intrinsic" perspective is as follows. Let $\hat{\mathcal O}_{G,e}$ be the complete local ring of $G$ at the identity $e\in G$. Then $U(\mathfrak g)$ is a certain dual of $\hat{\mathcal O}_{G,e}$ (it's the direct limit of $(\hat{\mathcal O}_{G,e}/\mathfrak m_{G,e}^n)^\ast$). Thus $U(\mathfrak g)$ only sees the local group structure in a formal neighborhood of $e\in G$.

There may indeed, however, be a natural notion of being simply connected if you look at the models of quantum groups like $\mathbb Z[q,q^{-1}]\langle a,b,c,d\rangle/(ab-qba,bc-cb,cd-qdc,\text{etc})$ (corresponding to the classical group $\operatorname{GL}(2)$).