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

Abstract 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 $$\mathcal {O}_q(\textrm{F}_n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mtext>F</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Moreover, the associated differential calculus $$\Omega ^{(0,\bullet )}_q(\textrm{F}_n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>Ω</mml:mi> <mml:mi>q</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>∙</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mtext>F</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> was shown to have classical dimension, giving a direct q -deformation of the classical anti-holomorphic Dolbeault complex of $$\textrm{F}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>F</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> </mml:math> . Here, we examine in detail the rank two case, namely the full quantum flag manifold of $$\mathcal {O}_q(\textrm{SU}_3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mtext>SU</mml:mtext> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . In particular, we examine the $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> -differential calculus associated with $$\Omega ^{(0,\bullet )}_q(\textrm{F}_3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>Ω</mml:mi> <mml:mi>q</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>∙</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mtext>F</mml:mtext> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$\mathfrak {sl}_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sl</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> ) to 4 (that is 2 to the power of the number of simple roots of $$\mathfrak {sl}_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>sl</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:math> ). 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 $$\mathcal {O}_q(\textrm{SU}_3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mtext>SU</mml:mtext> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -covariant noncommutative Kähler structure.