2024/11/12 by Carotenuto, Alessandro, Buachalla, Réamonn Ó, Razzaq, Junaid · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2411.07767
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 Oq(Fn). Moreover, the associated differential calculus Ω(0,\bullet)q(Fn) was shown to have classical dimension, giving a direct q-deformation of the classical anti-holomorphic Dolbeault complex of Fn. Here we examine in detail the rank two case, namely the full quantum flag manifold of Oq(SU3). In particular, we examine the *-differential calculus associated to Ω(0,\bullet)q(F3) 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 \fraksl3) to 4 (that is 2 to the power of the number of simple roots of \fraksl3). 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 Oq(SU3)-covariant noncommutative Kähler structure.