Nakajima Monomials Research Articles

Zigzag strip bundles and crystals II

Zigzag strip bundles are combinatorial models realizing the crystals B(∞) and the highest weight crystals B(λ) over quantum affine algebras Uq(g), and they are closely related with Nakajima monomials and Kashiwara embeddings of specific type. In this paper, we introduce new zigzag strip bundles for Uq(g), which we call reverse zigzag strip bundles in order to distinguish from the previous ones, and we give explicit 1-1 correspondences with the set of Nakajima monomials and the image of Kashiwara embedding which are different from those appeared in the study of the zigzag strip bundles. Another main result is to give different points of view about zigzag strip bundles and reverse zigzag strip bundles. Indeed, the previous 1-1 correspondences between the sets of zigzag strip bundles or reverse zigzag strip bundles, and the sets of Nakajima monomials or images of Kashiwara embeddings were determined based on the number of blocks in the columns of these strip bundles. In our work, we use the number of blocks in the rows of zigzag strip bundles and reverse zigzag strip bundles to provide characterizations of Nakajima monomials and images of Kashiwara embeddings that differ from the previous ones. That is, we give characterizations of Nakajima monomials and Kashiwara embeddings of 4 types. Further, we discuss the connection between zigzag strip bundles and reverse zigzag strip bundles.

Kirillov–Reshetikhin Crystals B1, s for $\widehat {\mathfrak {s}\mathfrak {l}}_{n}$ Using Nakajima Monomials

We give a realization of the Kirillov--Reshetikhin crystal $B^{1,s}$ using Nakajima monomials for $\widehat{\mathfrak{sl}}_n$ using the crystal structure given by Kashiwara. We describe the tensor product $\bigotimes_{i=1}^N B^{1,s_i}$ in terms of a shift of indices, allowing us to recover the Kyoto path model. Additionally, we give a model for the KR crystals $B^{r,1}$ using Nakajima monomials.

Strip bundle realization of the crystals over Uq(G2(1))

Motivated by the zigzag strip bundles which are combinatorial models realizing the crystals B(∞) for the quantum affine algebras Uq(g), where g=Bn(1),Dn(1),Dn+1(2),Cn(1), A2n−1(2),A2n(2), we introduce a new combinatorial model called strip bundles for the quantum affine algebra Uq(G2(1)). We give new realizations S(∞) and S(λ) of the crystal B(∞) and the highest weight crystals B(λ) over Uq(G2(1)) using strip bundles, and as subsets of S(∞) and S(λ), we also give realizations of the crystal B(∞) and the highest weight crystals B(λ) over the quantum finite algebra Uq(G2). Moreover, we give characterizations of the image of the crystal embedding Ψi and the connected component C1 in the set M of all Nakajima monomials which are isomorphic to the crystal B(∞) over Uq(G2(1)).

Virtual Crystals and Nakajima Monomials

An explicit description of the virtualization map for the (modified) Nakajima monomial model for crystals is given. We give an explicit description of the Lusztig data for modified Nakajima monomials in type $A_n$.

Strip Bundles and the Highest Weight Crystals Over Quantum Infinite Rank Affine Algebras

We give new realizations of the highest weight crystals B(λ) over the quantum infinite rank affine algebras U q (A ∞), U q (B ∞), U q (C ∞), and U q (D ∞) using strip bundles, which are related with Nakajima monomials.

Monomial Realization of the Tensor Product of Crystals for Quantum Finite Algebras

In this paper, we give a combinatorial description of the crystal bases of the tensor products of the irreducible highest weight modules over quantum finite algebras in terms of Nakajima monomials. Also, we explain how to decompose the tensor product of irreducible highest weight crystals into irreducibles by giving a bijection defined by multiplication of monomials.

CRYSTAL B(λ) IN B(∞) FOR G2TYPE LIE ALGEBRA

A previous work gave a combinatorial description of the crystal B(<TEX>${\infty}$</TEX>), in terms of certain simple Young tableaux referred to as the marginally large tableaux, for finite dimensional simple Lie algebras. Using this result, we present an explicit description of the crystal B(<TEX>${\lambda}$</TEX>), in terms of the marginally large tableaux, for the <TEX>$G_2$</TEX> Lie algebra type. We also provide a new description of B(<TEX>${\lambda}$</TEX>), in terms of Nakajima monomials, that is in natural correspondence with our tableau description.

Nakajima Monomials, Strip Bundles, and the Crystals B(∞) Over U q (B ∞), and U q (C ∞), and U q (D ∞)

In this article, we give two new realizations of the crystal base B(∞) of the negative part of the quantum generalized Kac–Moody algebra U q (𝔤) (𝔤 =B ∞, C ∞ or D ∞) using Nakajima monomials and strip bundles, respectively. Furthermore, we discuss the relation between monomial realization and strip bundle realization.

Compression of Nakajima monomials in type A and C

We describe an explicit crystal morphism between Nakajima monomials and monomials which give a realization of crystal bases for finite dimensional irreducible modules over the quantized enveloping algebra for Lie algebras of type A and C. This morphism provides a connection between arbitrary Nakajima monomials and Kashiwara---Nakashima tableaux. This yields a translation of Nakajima monomials to the Littelmann path model. Furthermore, as an application of our results we define an insertion scheme for Nakajima monomials compatible to the insertion scheme for tableaux.

Nakajima monomials, Young walls and Kashiwara embedding for [formula omitted

In this paper, we realize the crystal basis B ( λ ) of the irreducible highest weight module V ( λ ) of level 1 for U q ( A n ( 1 ) ) using Nakajima monomials satisfying some conditions. Also, from this monomial realization, we obtain the image of Kashiwara embedding Ψ ι λ : B ( λ ) ↪ Z ∞ ⊗ R λ , where ι is some infinite sequence from the index set of simple roots. Finally, we give a U q ( A n ( 1 ) ) -crystal isomorphism between Young wall realization and monomial realization, and so we can understand the image of Kashiwara embedding Ψ ι λ : B ( λ ) ↪ Z ∞ ⊗ R λ using the combinatorics of Young walls.

Monomial Realization of the Crystals Over U q (A ∞) and the Quantum Monster Algebra

We give new realizations of the crystal bases of the Verma modules and the irreducible highest weight modules over the quantum generalized Kac–Moody algebra U q (A ∞) and the quantum Monster algebra using Nakajima monomials. Moreover, another realization of the crystals B(∞) and B(λ) over U q (A ∞) using triangular matrices and tableaux are given.

The crystal ℬ(∞) and extended Nakajima monomials

We construct the set of extended Nakajima monomials containing monomials realizing the irreducible highest weight crystals, introduced by Nakajima. We present explicit descriptions of a crystal B ∞ \mathcal {B}_{\infty } and the crystal B ( ∞ ) \mathcal {B}(\infty ) in this language for the case A n ( 1 ) A_n^{(1)} .

Crystals and Nakajima monomials for quantum generalized Kac–Moody algebras

We introduce the notion of Nakajima monomials for quantum generalized Kac–Moody algebras and construct the crystals B ( ∞ ) and B ( λ ) in terms of Nakajima monomials. We also give an explicit description of the Nakajima monomials in the crystals B ( ∞ ) and B ( λ ) for the rank 2 quantum generalized Kac–Moody algebras and for the quantum Monster algebra.

Nakajima Monomials and Crystals for Special Linear Lie Algebras

AbstractNakajima introduced a certain set of monomials realizing the irreducible highest weight crystalsB(λ). The monomial set can be extended so that it contains crystalB(∞) in addition toB(λ). We present explicit descriptions of the crystalsB(∞) andB(λ) over special linear Lie algebras in the language of extended Nakajima monomials. There is a natural correspondence between the monomial description and Young tableau realization, which is another realization of crystalsB(∞) andB(λ).

Monomial Realization of Crystal Bases B(∞) for the Quantum Finite Algebras

In this paper, we give a new realization of the crystal basis B(∞) using modified Nakajima monomials for the quantum finite algebras. Moreover, as an application, we obtain the image of the Kashiwara embedding Ψ ι from this realization of B(∞).

Extended Nakajima monomials and crystal [formula omitted] for finite simple Lie algebras

Nakajima introduced a certain set of monomials realizing the irreducible highest weight crystals B ( λ ) . The monomial set can be extended so that it contains the crystal B ( ∞ ) in addition to B ( λ ) . We study the crystal base B ( ∞ ) of the negative part of a quantum group. For finite types B n , C n , D n , and G 2 , we present new descriptions of the crystal B ( ∞ ) in the language of extended Nakajima monomials. There is a natural correspondence between the monomial description and Young tableau description, which is another description of B ( ∞ ) .

Modified Nakajima monomials and the crystal [formula omitted

In this paper, we give a new realization of the crystal B ( ∞ ) using modified Nakajima monomials. For the quantum affine algebras of type A n ( 1 ) , we give an explicit characterization of the monomials in B ( ∞ ) , and construct a natural crystal isomorphism between monomial realization and path realization. As a corollary, we obtain a new realization of the crystal B ( ∞ ) for the quantum classical algebras of type A n .

Crystal Bases for Quantum Classical Algebras and Nakajima’s Monomials

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.