Geometric and Combinatorial Realizations of Crystals of Enveloping Algebras

@article{Savage2006GeometricAC,
  title={Geometric and Combinatorial Realizations of Crystals of Enveloping Algebras},
  author={Alistair Savage},
  journal={arXiv: Quantum Algebra},
  year={2006}
}
  • Alistair Savage
  • Published 2006
  • Mathematics
  • arXiv: Quantum Algebra
  • Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig's quiver varieties. This gives a geometric realization of the crystal graph of the lower half of the quantum group asso- ciated to a simply-laced Kac-Moody algebra. Using an enumeration of the irreducible components of Lusztig's quiver varieties in finite and affine type A by combinatorial data, we compute the geometrically defined crystal structure in terms of this combinatorics. We conclude by… CONTINUE READING
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    References

    SHOWING 1-10 OF 11 REFERENCES
    Geometric and Combinatorial Realizations of Crystal Graphs
    • 26
    • PDF
    Geometric construction of crystal bases
    • 216
    • Highly Influential
    • PDF
    Crystal bases and quiver varieties
    • 47
    Quivers, perverse sheaves, and quantized enveloping algebras
    • 414
    • Highly Influential
    • PDF
    Introduction to Quantum Groups and Crystal Bases
    • 450
    • PDF
    Realizations of Crystals
    • 53
    • PDF