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Quantum flag and Schubert schemes. (English) Zbl 0792.17012

Deformation theory and quantum groups with applications to mathematical physics, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Amherst/MA (USA) 1990, Contemp. Math. 134, 145-181 (1992).
The results of this paper are a generalization of those in the authors’ paper [Special functions, Proc. Hayashibara Forum, Okayama/Jap. 1990, ICM-90 Satell. Conf. Proc., 149-168 (1991)]. Let \(k_ q[G]\) be the quantum algebra of functions on a semisimple algebraic group \(G\) of rank \(\ell\) (in [loc. cit.], the considered \(G= \text{SL}(N)\)). Let \(B\) be a Borel subgroup of \(G\) and let \(P\supseteq B\) be a maximal parabolic subgroup of \(G\). Let \(k_ q[B]\) be the quantum Hopf algebra of functions on \(B\). Let \(w\) be an element of the Weyl group and let \(X(w) \subset G/B\) be the corresponding Schubert variety. The authors define the quantum algebras \(k_ q[G/P]\), \(k_ q[G/B]\), \(k_ q[X(w)]\); the first two are subcomodules of \(k_ q[G]\), the last is a quotient of \(k_ q[G/B]\). Each of these algebras has, in the classical case, a basis consisting of standard monomials—compatible with canonical \(\mathbb{Z}\) or \(\mathbb{Z}^ \ell\)-gradations. The authors prove the existence of such basis and gradations in the quantum case and give a presentation for \(k_ q[G/B]\).
For the entire collection see [Zbl 0755.00012].

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
14M15 Grassmannians, Schubert varieties, flag manifolds
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