Bitableaux bases of the quantum coordinate algebra of a semisimple group. (English) Zbl 1147.17013
Summary: We extend the Standard Basis Theorem of Rota et al. [J. Désarménien, J. P. S. Kung and G.-C. Rota, Adv. Math. 27, 63–92 (1978; Zbl 0373.05010)] to the setting of quantum symmetrizable Kac-Moody algebras. In particular, we obtain a procedure to give a presentation of the quantum coordinate algebra of any semisimple group, for generic \(q\). More precisely, given any integrable module \(V\) of a quantum symmetrizable Kac-Moody algebra \(U_{q}(\mathfrak g)\), we obtain a generating set of the ideal of relations among the matrix coefficients of \(V\), and we give an upper bound for the degrees of these polynomials. Our approach is based on the theory of crystal bases and Littelmann’s generalization of the plactic algebra [cf. P. Littelmann, Adv. Math. 124, 312–331 (1996; Zbl 0892.17009)].
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
Keywords:
Bitableaux; quantum function algebra; plactic monoid; crystal bases; Robinson-Schensted correspondence; Kac-Moody algebraReferences:
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