Noncommutative complex structures for the full quantum flag manifold of \(\mathcal {O}_q(\textrm{SU}_3)\)

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 \(\mathcal {O}_q(\textrm{F}_n)\). Moreover, the associated differential calculus \(\Omega ^{(0,\bullet )}_q(\textrm{F}_n)\) was shown to have classical dimension, giving a direct q-deformation of the classical anti-holomorphic Dolbeault complex of \(\textrm{F}_n\). Here, we examine in detail the rank two case, namely the full quantum flag manifold of \(\mathcal {O}_q(\textrm{SU}_3)\). In particular, we examine the \(*\)-differential calculus associated with \(\Omega ^{(0,\bullet )}_q(\textrm{F}_3)\) and its noncommutative 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 \(\mathfrak {sl}_3\)) to 4 (that is 2 to the power of the number of simple roots of \(\mathfrak {sl}_3\)). 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 \(\mathcal {O}_q(\textrm{SU}_3)\)-covariant noncommutative Kähler structure.