Noncommutative Complex Structures for the Full Quantum Flag Manifold of Quantum SU(3)

In recent work, Lusztig's positive root vectors (with respect to a distinguished choice of reduced decomposition of the longest element of the Weyl group) were shown to give a quantum tangent space for every $A$-series Drinfeld--Jimbo full quantum flag manifold $\mathcal{O}_q(\mathrm{F}_n)$. Moreover, the associated differential calculus $\Omega^{(0,\bullet)}_q(\mathrm{F}_n)$ was shown to have classical dimension, giving a direct $q$-deformation of the classical anti-holomorphic Dolbeault complex of $\mathrm{F}_n$. Here we examine in detail the rank two case, namely the full quantum flag manifold of $\mathcal{O}_q(\mathrm{SU}_3)$. In particular, we examine the $*$-differential calculus associated to $\Omega^{(0,\bullet)}_q(\mathrm{F}_3)$ and its non-commutative complex geometry. We find that the number of almost-complex structures reduces from $8$ (that is $2$ to the power of the number of positive roots of $\frak{sl}_3$) to $4$ (that is $2$ to the power of the number of simple roots of $\frak{sl}_3$). Moreover, we show that each of these almost-complex structures is integrable, which is to say, each of them is a complex structure. Finally, we observe that, due to non-centrality of all the non-degenerate coinvariant $2$-forms, none of these complex structures admits a left $\mathcal{O}_q(\mathrm{SU}_3)$-covariant noncommutative K\"ahler structure.