Polyhedral Realizations of Crystal Bases for Quantized Kac–Moody Algebras

Polyhedral Realizations of Crystal Bases for Modified Quantum Algebras of Type A

“…By the similar argument in [8] Lemma4.2, we obtain the explicit form of ϕ (l) −k (x) up to constant term as follows:…”

“…ι − ) a crystal structure on Z +∞ (resp. Z −∞ ) (see [8]) and denote it by Z +∞ ι + (resp. Z −∞ ι − ).…”

“…We can define tensor product structure on crystals in a similar manner to crystal bases. Indeed, some crystals (and their tensor products) are used to realize crystal base B(∞)( [8]) and B(λ)( [7]) by the polyhedral realization method. Polyhedral realization of crystal bases is the method embedding crystal bases in some infinite-dimensional vector space and describing its image as a set of lattice points in certain convex polyhedron( [7], [8]).…”

Polytopes for Crystallized Demazure Modules and Extremal Vectors

“…with several axioms (see [3], [9], [10]). In fact, B(∞) and B(λ) are the typical examples of crystals.…”

“…In [9], [10], we developed the polyhedral realization of crystal bases. We shall explain the relations between crystal bases of Demazure modules and the polyhedral realizations briefly.…”