RESEARCH ARTICLE

Path realization of crystal B()

  • Bin LI ,
  • Hechun ZHANG
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  • Department of Mathematical Science, Tsinghua University, Beijing 100084, China

Received date: 26 Feb 2010

Accepted date: 17 Jun 2010

Published date: 01 Aug 2011

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

A class of piecewise linear paths, as a generalization of Littelmann’s paths, are introduced, and some operators, acting on the above paths with fixed parametrization, are defined. These operators induce the ordinary Littelmann’s root operators’ action on the equivalence classes of paths. With these induced operators, an explicit realization of B() is given in terms of equivalence classes of paths, where B() is the crystal base of the negative part of a quantum group Uq(g). Furthermore, we conjecture that there is a complete set of representatives for the above model by fixing a parametrization, and we prove the case when g is of finite type.

Key words: Path; crystal; root operator

Cite this article

Bin LI , Hechun ZHANG . Path realization of crystal B()[J]. Frontiers of Mathematics in China, 2011 , 6(4) : 689 -706 . DOI: 10.1007/s11464-010-0073-x

1
Cliff G. Crystal bases and Young tableaux. J Algebra, 1998, 202(1): 10-35

DOI

2
Hong J, Kang S-J. Introduction to Quantum Groups and Crystal Bases. Graduate Studies in Mathematics, Vol 42. Providence: Amer Math Soc, 2002

3
Hong J, Lee H. Young tableaux and crystal B(∞) for finite simple Lie algebras. J Algebra, 2008, 320: 3680-3693

DOI

4
Kang S-J, Kashiwara M, Misra K, Miwa T, Nakashima T, Nakayashiki A. Affine Crystals and Vertex Models. Int J Mod Phys, A7, 1992, Suppl.1A: 449-484

5
Kang S-J, Kashiwara M, Misra K, Miwa T, Nakashima T, Nakayashiki A. Perfect crystals of quantum affine Lie algebras. Duke Math J, 1992, 68: 499-607

DOI

6
Kashiwara M. Crystalizing the q-analogue of universal enveloping algebras. Comm Math Phys, 1990, 133: 249-260

DOI

7
Kashiwara M. On crystal bases of the q-analogue of universal enveloping algebras. Duke Math J, 1991, 63: 465-516

DOI

8
Kashiwara M. The crystal bases and Littelmann’s refined Demazure character formula. Duke Math J, 1993, 71(3): 839-858

DOI

9
Kashiwara M. Realizations of crystals. In: Combinatorial and Geometric Representation Theory (Seoul, 2001). Contemp Math, Vol 325. Providence: Amer Math Soc, 2003, 133-139

10
Kashiwara M, Saito Y. Geometric construction of crystal bases. Duke Math J, 1997, 89(1): 9-36

DOI

11
Littelmann P. A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras. Invent Math, 1994, 116: 329-346

DOI

12
Littelmann P. Path and root operators in representation theory. Ann of Math, 1995, 142(3): 499-525

DOI

13
Nakashima T, Zelevinsky A. Polyhedral realizations of crystal bases for quantized Kac-Moody algebras. Adv Math, 1997, 131(1): 253-278

DOI

14
Savage A. Geometric and combinatorial realizations of crystal graphs. Alg Represent Theory, 2006, 9(2): 161-199

DOI

15
Savage A. Geometric and combinatorial realizations of crystals of enveloping algebras. In: Lie Algebras, Vertex Operator Algebras and Their Applications (Raleigh, NC, 2005). Contemp Math, 442. Providence: Amer Math Soc, 2007, 221-232

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