What I want to ask is there any other way to define quantized flag variety? In the classical case, it is well known that flag variety can be defined as $G/B$, say $G$ is general linear group and B is Borel subgroup. Is there any analogue for quantum case? Is there a definition like $G_q$ as "quantum linear group" and $B_q$ as quantum analogue of Borel subgroup?

The Borel subgroup of the quantized function algebra $G_q$ is of course certain quotient Hopf algebra $p:\mathcal{O}(G_q)\to \mathcal{O}(B_q)$ which then canonically coacts on $G_q$ from both sides (from the left by $(p\otimes id)\circ\Delta_{G_q}:\mathcal{O}(G_q)\to \mathcal{O}(B_q)\otimes \mathcal{O}(G_q)$. In the case of $SL_q(n)$ you get the lower Borel by quotienting by the Hopf ideal generate by the entries of the matrxi $T$ of the generators standing above the diagonal.

Of course, if you define $Qcoh(G_q)$ as the category ${}_{\mathcal{O}(G_q)}\mathcal{M}$ of left modules over $\mathcal{O}(G_q)$ then the category of relative left-right Hopf modules ${}_{\mathcal{O}(G_q)}\mathcal{M}^{\mathcal{O}(B_q)}$ is precisely the category of quasicoherent sheaves over the quantum flag variety $G_q/B_q$. If you look at several of my earlier articles they together show that in the case of $SL_q(n)$ at least, there is a collection of localizations of the above category of relative Hopf modules which are affine, which gives it a structure of a noncommutative scheme with a canonical atlas whose cardinality is the cardinality of the Weyl group (the generalization to other parabolics is easy). Moreover the forgetful functor from the category of Hopf modules to ${}_{\mathcal{O}(G_q)}\mathcal{M}$ is the inverse image part of the geometric morphism which is a Zariski locally trivial $B_q$-fibration; the local triviality, which is obtained with the help of the quantum Gauss decomposition, boils down to the Schneider's equivalence for the extension of the algebras of localized coinvariants into the corresponding localization of $G_q$. The sketch of the whole program is in

• Localizations for construction of quantum coset spaces, in "Noncommutative geometry and Quantum groups", W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265--298, Warszawa 2003, math.QA/0301090.

For the Peter-Weyl look at the original works of S. Woronowicz or

• A. Klimyk, K. Schmüdgen, Quantum groups and their representations, Springer 1997.

What I want to ask is there any other way to define quantized flag variety? In the classical case, it is well known that flag variety can be defined as $G/B$, say $G$ is general linear group and B is Borel subgroup. Is there any analogue for quantum case? Is there a definition like $G_q$ as "quantum linear group" and $B_q$ as quantum analogue of Borel subgroup?

The Borel subgroup of the quantized function algebra $G_q$ is of course certain quotient Hopf algebra $p:\mathcal{O}(G_q)\to \mathcal{O}(B_q)$ which then canonically coacts on $G_q$ from both sides (from the left by $(p\otimes id)\circ\Delta_{G_q}:\mathcal{O}(G_q)\to \mathcal{O}(B_q)\otimes \mathcal{O}(G_q)$. In the case of $SL_q(n)$ you get the lower Borel by quotienting by the Hopf ideal generate by the entries of the matrxi $T$ of the generators standing above the diagonal.

Of course, if you define $Qcoh(G_q)$ as the category ${}_{\mathcal{O}(G_q)}\mathcal{M}$ of left modules over $\mathcal{O}(G_q)$ then the category of relative left-right Hopf modules ${}_{\mathcal{O}(G_q)}\mathcal{M}^{\mathcal{O}(B_q)}$ is precisely the category of quasicoherent sheaves over the quantum flag variety $G_q/B_q$. If you look at several of my earlier articles they together show that in the case of $SL_q(n)$ at least, there is a collection of localizations of the above category of relative Hopf modules which are affine, which gives it a structure of a noncommutative scheme with a canonical atlas whose cardinality is the cardinality of the Weyl group (the generalization to other parabolics is easy). Moreover the forgetful functor from the category of Hopf modules to ${}_{\mathcal{O}(G_q)}\mathcal{M}$ is the inverse image part of the geometric morphism which is a Zariski locally trivial $B_q$-fibration; the local triviality, which is obtained with the help of the quantum Gauss decomposition, boils down to the Schneider's equivalence for the extension of the algebras of localized coinvariants into the corresponding localization of $G_q$. The sketch of the whole program is in

• Localizations for construction of quantum coset spaces, in "Noncommutative geometry and Quantum groups", W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265--298, Warszawa 2003, math.QA/0301090.

For the Peter-Weyl look at the original works of S. Woronowicz or

• A. Klimyk, K. Schmüdgen, Quantum groups and their representations, Springer 1997.