Perfect crystals of \(U_q(G_2^{(1)})\)

DOI10.1006/jabr.1998.7597zbMath0929.17013arXivq-alg/9712012OpenAlexW1964742344MaRDI QIDQ1279920

Shigenori Yamane

Publication date: 7 April 1999

Published in: Journal of Algebra (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/q-alg/9712012

zbMATH Keywords

perfect crystalquantum affine algebraintegrable highest weight representationsnon-classical quantum affine algebrapath realizations

Mathematics Subject Classification ID

Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) (17B10) Quantum groups (quantized enveloping algebras) and related deformations (17B37) Exactly solvable models; Bethe ansatz (82B23)

Related Items (29)

Scattering rule in soliton cellular automaton associated with crystal base of Uq(D4(3)) ⋮ Level 0 Monomial Crystals ⋮ Connecting marginally large tableaux and rigged configurations via crystals ⋮ Virtualization map for the Littelmann path model ⋮ Kirillov-Reshetikhin crystals for nonexceptional types ⋮ Existence of Kirillov-Reshetikhin crystals of type \(G_2^{(1)}\) and \(D_4^{(3)}\) ⋮ Kirillov–Reshetikhin crystals B^7,s for type E₇⁽¹⁾ ⋮ Enumeration of non-positive planar trivalent graphs ⋮ Categorical relations between Langlands dual quantum affine algebras: exceptional cases ⋮ Uniform description of the rigged configuration bijection ⋮ Crystal energy functions via the charge in types \(A\) and \(C\) ⋮ Cluster algebra structures on module categories over quantum affine algebras ⋮ Monoidal categorification and quantum affine algebras ⋮ Kirillov-Reshetikhin crystals \(B^{1, s}\) for \(\widehat{\mathfrak{s}\mathfrak{l}}_n\) using Nakajima monomials ⋮ A crystal to rigged configuration bijection and the filling map for type \(D_4^{(3)}\) ⋮ Combinatorial structure of Kirillov-Reshetikhin crystals of type \(D_n^{(1)}\), \(B_n^{(1)}\), \(A^{(2)}_{2n-1}\) ⋮ A uniform approach to soliton cellular automata using rigged configurations ⋮ Perfect crystals for \(U_q(D_4^{(3)})\) ⋮ Ultra-Discretization of the affine G_2 Geometric Crystals to Perfect Crystals ⋮ A generalization of adjoint crystals for the quantized affine algebras of type \(A^{(1)}_n\), \(C^{(1)}_n\) and \(D^{(2)}_{n+1}\) ⋮ A uniform realization of the combinatorial \(R\)-matrix for column shape Kirillov-Reshetikhin crystals ⋮ On Demazure crystals for $U_{q}(G_{2}^{(1)})$ ⋮ Existence of Kirillov–Reshetikhin crystals for nonexceptional types ⋮ Virtual crystals and fermionic formulas of type 𝐷_{𝑛+1}⁽²⁾, 𝐴_{2𝑛}⁽²⁾, and 𝐶_{𝑛}⁽¹⁾ ⋮ $A^{(1)}_{n}$ -Geometric Crystal Corresponding to Dynkin Index i=2 and Its Ultra-Discretization ⋮ On higher level Kirillov-Reshetikhin crystals, Demazure crystals, and related uniform models ⋮ Soliton cellular automaton associated with $G_2^{(1)}$G2(1) crystal base ⋮ Level one perfect crystals for \(B_n^{(1)}\), \(C_n^{(1)}\), and \(D_n^{(1)}\) ⋮ A generalization of the alcove model and its applications

Cites Work

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• A q-difference analogue of \(U({\mathfrak g})\) and the Yang-Baxter equation
• On crystal bases of the \(q\)-analogue of universal enveloping algebras
• Global crystal bases of quantum groups
• Perfect crystals of quantum affine Lie algebras
• A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras
• Crystal bases and tensor product decompositions of \(U_ q(G_ 2)\)- modules
• Crystal graphs for representations of the \(q\)-analogue of classical Lie algebras
• Crystalizing the q-analogue of universal enveloping algebras
• Quantum R matrix for G₂and a solvable 175-vertex model
• AFFINE CRYSTALS AND VERTEX MODELS

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