Noncommutative complex structures for the full quantum flag manifold of $$\mathcal {O}_q(\textrm{SU}_3)$$
Year: 2025
Type: article
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 $$\mathcal {O}_q(\textrm{F}_n)$$ O q ( F n ) . Moreover, the associated differential calculus $$\Omega ^{(0,\bullet )}_q(\textrm{F}_n)$$ Ω q ( 0 , ∙ ) ( F n ) was shown to have classical dimension, giving a direct q-deformation of the classical anti-holomorphic Dolbeault complex of $$\textrm{F}_n$$ F n . Here, we examine in detail the rank two case, namely the full quantum flag manifold of $$\mathcal {O}_q(\textrm{SU}_3)$$ O q ( SU 3 ) . In particular, we examine the $$*$$ ∗ -differential calculus associated with $$\Omega ^{(0,\bullet )}_q(\textrm{F}_3)$$ Ω q ( 0 , ∙ ) ( 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 $$\mathfrak {sl}_3$$ sl 3 ) to 4 (that is 2 to the power of the number of simple roots of $$\mathfrak {sl}_3$$ 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 $$\mathcal {O}_q(\textrm{SU}_3)$$ O q ( SU 3 ) -covariant noncommutative Kähler structure.
Source: Letters in Mathematical Physics
Institutions: University of Parma, Charles University, University of Bologna
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4411151971