Polyhedral Realizations of Crystal Bases for Quantum Algebras of Finite Types

Abstract

Polyhedral realization of crystal bases is one of the methods for describing the crystal base $B(\infty )$ explicitly. This method can be applied to symmetrizable Kac-Moody types. We can also apply this method to the crystal bases $B(\lambda )$ of integrable highest weight modules and of modified quantum algebras. But, the explicit forms of the polyhedral realizations of crystal bases $B(\infty )$ and $B(\lambda )$ are only given in the case of arbitrary rank 2, of $A_n$ and of $A_n^{(1)}$. So, we will give the polyhedral realizations of crystal bases $B(\infty )$ and $B(\lambda )$ for all simple Lie algebras in this paper.

27 pages, LaTeX

References in corpus (1)

• Polyhedral Realizations of Crystal Bases for Modified Quantum Algebras of Arbitrary Rank 2 Cases

Cited by in corpus (4)

• Young tableaux and crystal $B(\infty )$ for finite simple Lie algebras
• Polyhedral realizations for crystal bases of integrable highest weight modules and combinatorial objects of type ${\mathrm{A}}_{n-1}^{(1)}$, ${\mathrm{C}}_{n-1}^{(1)}$, ${\mathrm{A}}_{2n-2}^{(2)}$, ${\mathrm{D}}_n^{(2)}$
• Alcove path model for $B(\infty )$
• A new description of the bicrystal $B(\infty )$ and the extended crystal