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Invariant prime ideals in quantizations of nilpotent Lie algebras. (English) Zbl 1229.17020

Let \(\mathfrak{g}\) be a split finite dimensional semisimple Lie algebra over a field of chracteristic zero. Let \(q\) an element in the ground field of \(\mathfrak{g}\) that is transcendental and let \(\mathcal{U}_q(\mathfrak{g})\) denote the quantized universal enveloping algebra of \(\mathfrak{g}\) at \(q\) with standard generators \(X_1^\pm,\dots,X_r^\pm\) and \(K_1^{\pm 1},\dots,K_r^{\pm 1}\). C. De Concini,V. G. Kac and C. Procesi [Some quantum analogues of solvable Lie groups. in: Geometry and analysis. Papers presented at the Bombay colloquium, India, January 6–14, 1992. Oxford: Oxford University Press. Stud.Math., Tata Inst.Fundam.Res.13, 41–65 (1995; Zbl 0878.17014)] defined for a given element \(w\) of the Weyl group of \(\mathfrak{g}\) certain subalgebras \(\mathcal{U}_\pm^w\) of \(\mathcal{U}_q(\mathfrak{g})\) generated by the Lusztig root vectors obtained from reduced expressions of \(w\) and showed that these subalgebras are independent of the choice of the reduced expression.
The goal of the paper under review is to study the prime ideals of \(\mathcal{U}_-^w\) invariant under the conjugation action of the group-like elements \(H:=\langle K_1^{\pm 1}, \dots,K_r^{\pm 1}\rangle\) of \(\mathcal{U}_q(\mathfrak{g})\). The main results are as follows: 1) an explicit description of these invariant prime ideals using Demazure modules, 2) a construction of a small generating set for each invariant prime ideal, and 3) an identification of the poset structure on invariant prime ideals with a Bruhat interval. Even in the special case of quantum matrices 1) and 2) are new. The proof uses M. Gorelik’s investigation of the spectra of quantum Bruhat cell translates [J.Algebra 227, No.1, 211–253 (2000; Zbl 1038.17006)], A. Joseph’s results on generating sets for ideals of the quantized coordinate ring of the simply connected semisimple algebraic group \(G\) with Lie algebra \(\mathfrak{g}\) [C.R.Acad.Sci.Paris ,Sér.I Math.321, No.2, 135–140 (1995; Zbl 0837.17007)], and an interpretation of \(\mathcal{U}_-^w\) as a quantized function algebra on the Schubert cell \(B_+w\cdot B_+\). Similar results are also obtained for vanishing ideals of torus orbit closures of symplectic leaves of related Poisson structures on Schubert cells in flag varieties. The results of the paper under review play a crucial role in the author’s recent classification of the \(H\)-invariant prime ideals of arbitrary quantum partial flag varieties of \(G\) [Proc.Am.Math.Soc.138, No.4, 1249–1261 (2010; Zbl 1245.16030)].

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16T20 Ring-theoretic aspects of quantum groups
53D17 Poisson manifolds; Poisson groupoids and algebroids
14M15 Grassmannians, Schubert varieties, flag manifolds
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