# Polyhedral Realizations of Crystal Bases for Quantized Kac-Moody Algebras

@article{Nakashima1997PolyhedralRO,
title={Polyhedral Realizations of Crystal Bases for Quantized Kac-Moody Algebras},
author={Toshiki Nakashima and Andrei Zelevinsky},
year={1997},
volume={131},
pages={253-278}
}
• Published 28 March 1997
• Mathematics
Let B(\infty) be the crystal corresponding to the nilpotent part of a quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as the set of integer solutions of a system of linear inequalities. As an application, we treat in a unified manner all Kac-Moody algebras of rank 2 (sharpening the result by Kashiwara), as well as the algebras of types A_n and A_{n-1}^{(1)}.
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## References

SHOWING 1-10 OF 17 REFERENCES
Crystalizing theq-analogue of universal enveloping algebras
For an irreducible representation of theq-analogue of a universal enveloping algebra, one can find a canonical base atq=0, named crystal base (conjectured in a general case and proven forAn, Bn, Cn
Canonical bases for the quantum group of type $A_r$ and piecewise-linear combinatorics
• Mathematics
• 1996
This work was motivated by the following two problems from the classical representation theory. (Both problems make sense for an arbitrary complex semisimple Lie algebra but since we shall deal only
A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras
In the representation theory of the group GLn(C), an important tool are the Young tableaux. The irreducible representations are in one-to-one correspondence with the shapes of these tableaux. Let T
String bases for quantum groups of type ᵣ
• Mathematics
• 1993
This is the quantum deformation (or q−deformation) of the algebra of polynomial functions on the group Nr+1 of upper unitriangular (r + 1) × (r + 1) matrices. In this paper we introduce and study a
Introduction to Quantum Groups
THE DRINFELD JIMBO ALGERBRA U.- The Algebra f.- Weyl Group, Root Datum.- The Algebra U.- The Quasi--Matrix.- The Symmetries of an Integrable U-Module.- Complete Reducibility Theorems.- Higher Order
Crystal Graphs for Representations of the q-Analogue of Classical Lie Algebras
• Mathematics
• 1994
The explicit form of the crystal graphs for the finite-dimensional representations of the q-analogue of the universal enveloping algebras of type A, B, C, and D is given in terms of semi-standard
Combinatorics of representations of $$U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))$$ atq=0
• Mathematics
• 1991
AbstractTheq=0 combinatorics for $$U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))$$ is studied in connection with solvable lattice models. Crystal bases of highest weight representations of U_q