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Using Nakajima’s monomials, we construct a new realization of crystal bases for finite dimensional irreducible modules over quantum classical algebras. We also give an explicit bijection between the monomial realization and the Young tableau realization of crystal bases.

In this article, we introduce the notion of abstract crystals for quantum generalized Kac-Moody algebras and study their fundamental properties. We then prove the crystal embedding theorem and give a characterization of the crystals B(∞) and B(λ).

In this paper, we give a polyhedral realization of the highest weight crystals $B(\lambda )$ associated with the highest weight modules $V(\lambda )$ for the generalized Kac-Moody algebras. As applications, we give explicit descriptions of crystals for the generalized Kac-Moody algebras of ranks 2, 3, and Monster algebras.

ABSTRACT In this article, we give a new realization of crystal bases for irreducible highest weight modules over U q (G 2) in terms of monomials. We also discuss the natural connection between the monomial realization and tableau realization. Communicated by K. Misra Key Words: Crystal baseMonomialMathematics Subject Classification: Primary 81R50Secondary 17B25 ACKNOWLEDGMENTS The author would like to express his sincere gratitude to Professor S.-J. Kang and Doctor J.-A. Kim for their interest i

We give a new realization of the crystal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">B(\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formu

In this paper, we give a new realization of crystal bases for finite-dimensional irreducible modules over classical Lie algebras. The basis vectors are parameterized by certain Young walls lying between highest weight and lowest weight vectors.

We give a new realization of the crystal B(∞) of (, , or ) using the combinatorial objects zigzag strip bundles. Moreover, we study the relation between zigzag strip bundle realization, Nakajima monomial realization, and polyhedral realization of B(∞).

We give a 1-1 correspondence with the Young wall realization and the Young tableau realization of the crystal bases for the classical Lie algebras.