Lusztig's positive root vectors and a Dolbeault complex for the A-series full quantum flag manifolds

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F25%3A10510276" target="_blank" >RIV/00216208:11320/25:10510276 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=wO8WLYzXQM" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=wO8WLYzXQM</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.jalgebra.2025.03.035" target="_blank" >10.1016/j.jalgebra.2025.03.035</a>

Result language
angličtina
Original language name
Lusztig's positive root vectors and a Dolbeault complex for the A-series full quantum flag manifolds
Original language description
For the Drinfeld-Jimbo quantum enveloping algebra Uq(sln+1), we show that the span of Lusztig's positive root vectors, with respect to Littlemann's nice reduced decompositions of the longest element of the Weyl group Sn+1, form quantum tangent spaces for the full quantum flag manifold Oq(Fn+1). The associated differential calculi are direct q-deformations of the anti-holomorphic Dolbeault complex of the classical full flag manifold Fn+1. As an application we establish a quantum Borel-Weil theorem for Oq(Fn+1), giving a noncommutative differential geometric realisation of all the finite-dimensional type-1 irreducible representations of Uq(sln+1). Restricting this differential calculus to the quantum Grassmannians is shown to reproduce the celebrated Heckenberger-Kolb anti-holomorphic Dolbeault complex. Lusztig's positive root vectors for non-nice decompositions of the longest element of Sn+1 are examined for low orders, and are exhibited to either not give tangents spaces, or to produce differential calculi of non-classical dimension. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GA22-00091S" target="_blank" >GA22-00091S: Geometric structures, invariance and differential equations related to mathematical physics</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2025
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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