Hyeonmi Lee
We study the crystal base of the negative part of a quantum group. Two explicit descriptions of the crystal $B(\infty)$ for types $G_2$ are given. The first is given in terms of extended Nakajima monomials and the second realization follows a similar result given for other finite types by Cliff.
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We study the crystal base of the negative part of a quantum group. An explicit realization of the crystal is given in terms of Young tableaux for types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$. Connection between our realization and a previous realization of Cliff is also given.
Using Lakshmibai-Seshadri paths, we give a combinatorial realization of the crystal basis of an extremal weight module of integral extremal weight over the quantized universal enveloping algebra associated to the infinite rank affine Lie algebra of type $B_{\infty}$, $C_{\infty}$, or $D_{\infty}$. Moreover, via this realization, we obtain an explicit description (in terms of Littlewood-Richardson coefficients) of how tensor products of these crystal bases decompose into connected components when their extremal weights are of nonnegative levels. These results, in types $B_{\infty}$, $C_{\infty}…
A new description of the bicrystal $B(\infty)$ and the extended crystal
本文利用多面体实现为$B(\infty)$定义了新晶体映射，提出滑动菱形规则组合描述其双晶体结构，并据此定义扩展晶体，证明其与Kashiwara和Park引入的扩展晶体同构。
In the preceding paper, we formulated a conjecture on the relations between certain classes of irreducible representations of affine Hecke algebras of type B and symmetric crystals for $\gl_\infty$. In the present paper, we prove the existence of the symmetric crystal and the global basis for $\gl_\infty$.
We consider a category of $\gl_\infty$-crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight $\gl_\infty$-modules with respect to derivations shifting the characters of fundamental modules. A Littlewood-Richardson rule of extremal weight crystals with non-negative level is described ex…
A previous work gave a combinatorial description of the crystal $B(\infty)$, 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 $B(λ)$, in terms of the marginally large tableaux, for the $G_2$ Lie algebra type. We also provide a new description of $B(\la)$, in terms of Nakajima monomials, that is in natural correspondence with our tableau description.
A new categorical crystal structure for the quantum affine algebras is presented. We introduce the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$ for an arbitrary quantum group, which is the product of infinite copies of the crystal $B(\infty)$. For a complete duality datum in the Hernandez-Leclerc category $\mathcal{C}^0_{\mathfrak{g}}$ of a quantum affine algebra $U_q'(\mathfrak{g})$, we prove that the set of the isomorphism classes of simple modules in $\mathcal{C}^0_{\mathfrak{g}}$ has an extended crystal structure isomorphic to the extended crystal $\widehat{B}_{\mathfrak{g}}(\inft…
Tristan Bozec gave a definition of generalized quantum groups that extends the usual definition of quantum groups to finite quivers with loops at vertices, and he introduced a theory of generalized crystals for this new family of Hopf algebras. We explicitly characterize the generalized crystal $\mathcal{B}(\infty)$ associated to a certain family of quivers with multiple loops by providing a complete set of relations among the Kashiwara operators themselves.
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 $B(λ)$ of integrable highest weight modules and of modified quantum algebras. But, the explicit forms of the polyhedral realizations of crystal bases $B(\infty)$ and $B(λ)$ are only given in the case of arbitrary rank 2, of $A_n$ and of $A^{(1)}_n$. So, we will give the polyhedral realizations of crystal bases $B(\infty)$ and $B(λ)$ for all simple Lie algebras in this…
We present explicit descriptions of the crystals $\mathcal{B}(\infty)$ and $\mathcal{B}(λ)$ over special linear Lie algebras in the language of \emph{extended Nakajima monomials}. There is a natural correspondence between the monomial description and Young tableau realization, which is another realization of crystals $\mathcal{B}(\infty)$ and $\mathcal{B}(λ)$.
It is shown that the direct limit of the semistandard decomposition tableau model for polynomial representations of the queer Lie superalgebra exists, which is believed to be the crystal for the upper half of the corresponding quantum group. An extension of this model to describe the direct limit combinatorially is given. Furthermore, it is shown that the polynomial representations may be recovered from the limit in most cases.
OCHA is the homotopy algebra of open-closed strings. It can be defined as a sequence of multilinear operations on a pair of DG spaces satisfying certain relations which include the $L_\infty$ relations in one space and the $A_\infty$ relations in the other. In this paper we show that the OCHA structure is intrinsic to the tensor product of the symmetric and tensor coalgebras. We also show how an OCHA can be obtained from $A_\infty$-extesions and define the {\it universal enveloping} $A_\infty$-algebra of an OCHA as an $A_\infty$-extension of the universal enveloping of its $L_\infty$ part by i…
Let B(\infty) be the crystal corresponding to the nilpotent part of a quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as the set of integer solutions of a system of linear inequalities. As an application, we treat in a unified manner all Kac-Moody algebras of rank 2 (sharpening the result by Kashiwara), as well as the algebras of types A_n and A_{n-1}^{(1)}.
We construct by fusion product new irreducible representations of the quantum affinization $U_q(\hat{sl}_\infty)$. The action is defined via the Drinfeld coproduct and is related to the crystal structure of semi-standard tableaux of type $A_\infty$. We call these representations extremal loop weight modules. The main motivations are applications to quantum toroidal algebras $U_q(sl_{n+1}^{tor})$: we prove the conjectural link between $U_q(\hat{sl}_\infty)$ and $U_q(sl_{n+1}^{tor})$ stated in [14] for this family of representations. We recover in this way the extremal loop weight modules obtain…
Infinite-level Fock spaces, crystal bases, and tensor product of extremal weight modules of type $A_{+\infty}$
研究由U_q(gl_{>0})的极值重量模生成的范畴C，证明其为张量范畴，并描述任意两个极值重量模张量积的socle滤膜，基于无限级福克空间的研究。
In order to see the behavior of $\imath$canonical bases at $q = \infty$, we introduce the notion of $\imath$crystals associated to an $\imath$quantum group of certain quasi-split type. The theory of $\imath$crystals clarifies why $\imath$canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of $\imath$crystals whose projective limit can be thought of as the $\imath$canonical basis of the modified $\imath$quantum group at $q = \infty$.
A new Young wall realization of $B(λ)$ and $B(\infty)$
本文利用 Young 墙的新组合学，为量子仿射代数构造了任意水平最高权晶体 $B(λ)$ 和 $B(\infty)$ 的新实现，并证明了约化 Young 墙晶体与 $B(λ)$ 同构。
The crystal bases are quite useful combinatorial tools to study the representations of quantized universal enveloping algebras $U_q(\mathfrak{g})$. The polyhedral realization for $B(\infty)$ is a combinatorial description of the crystal base, which is defined as an image of embedding $Ψ_ι:B(\infty)\hookrightarrow \mathbb{Z}^{\infty}_ι$, where $ι$ is an infinite sequence of indices and $\mathbb{Z}^{\infty}_ι$ is an infinite $\mathbb{Z}$-lattice with a crystal structure associated with $ι$. It is a natural problem to find an explicit form of the polyhedral realization ${\rm Im}(Ψ_ι)$. In this ar…
The crystals for finite dimensional representations of sl(n+1) can be realized using Young tableaux. The infinity crystal on the other hand is naturally realized using multisegments, and there is a simple description of the embedding of each finite crystal into the infinity crystal in terms of these realizations. The infinity crystal is also parameterized by Lusztig's PBW basis with respect to any reduced expression for the longest word in the Weyl group. We give an explicit description of the unique crystal isomorphism from PBW bases to multisegments for one standard choice of reduced express…
In this paper, 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 $B(\infty)$ and $B(\la)$.
We provide a geometric realization of the crystal $B(\infty)$ for quantum generalized Kac-Moody algebras in terms of the irreducible components of certain Lagrangian subvarieties in the representation spaces of a quiver.
In this paper, we give polyhedral realization of the crystal $B(\infty)$ of $U_q^-(\mathfrak g)$ for the generalized Kac-Moody algebras. As applications, we give explicit descriptions of crystals for the generalized Kac-Moody algebras of rank 2, 3 and {\it Monster Lie algebras}.
Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras. The first realization we consider is a geometric construction in terms of irreducible components of certain quiver varieties established by Kashiwara and Saito. The second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke using Ringel's Hall algebra approach to quantum groups. We show that these two constructions ar…
Let g = n^- + h + n^+ be a symmetrizable Kac-Moody algebra. Let B(\infty) be the Kashiwara crystal of U_q(n^-), let λbe a dominant integral weight, let T_λ= {t_λ} be the crystal with one element of weight λ, and let B(λ) \subset B(\infty) \otimes T_λbe the crystal of the integrable representation of highest weight λ. We compute the descending string parameters of an element b \otimes t_λin B(λ) in terms of the Lusztig parameters of b.
The polyhedral realization of crystal base has been introduced by A.Zelevinsky and the second author([T.Nakashima, A.Zelevinsky, Adv. Math. 131, no. 1 (1997)]), which describe the crystal base $B(\infty)$ as a polyhedral convex cone in the infinite $\mathbb{Z}$-lattice $\mathbb{Z}^{\infty}$. To construct the polyhedral realization, we need to fix an infinite sequence $ι$ from the indices of the simple roots. According to this $ι$, one has certain set of linear functions defining a polyhedral convex cone and under the `positivity condition' on $ι$, it has been shown that the polyhedral convex c…
Let B be the crystal basis of the minus part of the quantized enveloping algebra of a semi-simple Lie algebra. Kashiwara has shown that B has a combinatorial description in terms of an embedding of B into the tensor product of B and k abstract crystals B_{i_j}, j=1,2,...,k, where the longest word in the Weyl group is s_{i_1}...s_{i_k}. We give an explicit description of the image of this embedding for classical Lie algebras of types A, B, C, D. This description is in terms of semi-standard Young tableaux of types A, B, C, D defined by Kashiwara and Nakashima.
We describe a combinatorial realization of the crystals $B(\infty)$ and $B(λ)$ using rigged configurations in all symmetrizable Kac-Moody types up to certain conditions. This includes all simply-laced types and all non-simply-laced finite and affine types.
We give a new combinatorial model of the Kirillov-Reshetikhin crystals of type $A_n^{(1)}$ in terms of non-negative integral matrices based on the classical RSK algorithm, which has a simple description of the affine crystal structure without using a promotion operator. We also have a similar description of the Kirillov-Reshetikhin crystals associated to exceptional nodes in the Dynkin diagrams of classical affine or non-exceptional type.
Polyhedral realizations for crystal bases and Young walls of classical affine types
本文研究了经典仿射类型晶格基的多面体实现，通过Young墙和截断墙描述了晶格基的多面体和锥体的不等式，给出了ε_k*-函数的组合描述。
For any simple complex algebraic group, we define upper/lower half-decorated geometric crystals and show that their tropicalization will be upper/lower normal Kashiwara's crystals. In particular, we show that the tropicalization of the half-decorated geometric crystal on the big Bruhat cell(=$B^-_{w_0}:=B^-\cap U\bar w_0 U$) is isomorphic to the crystal $B(\infty)$ of the nilpotent subalgebra of quantum group $U_q^-(\mathfrak g)$. As an application, we shall show that any cellular crystal associated with a reduced word is connected in the sense of a crystal graph.
We study the description of the crystal structure on the set of Mirković-Vilonen polytopes. Anderson and Mirković defined an operator and conjectured that it coincides with the Kashiwara operator. Kamnitzer proved the conjecture for type A and gave an counterexample for type C_{3}. He also gave an explicit formula to calculate the Kashiwara operator for type A. In this paper we prove that a part of the AM conjecture still holds in general, answering an open question of Kamnitzer (2007). Moreover, we show that although the formula given by Kamnitzer does not hold in general, it is still valid i…
Combinatorial descriptions of the crystal structure on certain PBW bases
针对 PBW 基实现 B(∞) 晶体时算子算法复杂的问题，本文刻画了一类约化表达式，使晶体算子可由更简单的括号规则描述，并证明其在多数类型中存在。
In the previous paper "Symmetric Crystals and Affine Hecke Algebras of Type B", we formulated a conjecture on the relations between certain classes of irreducible representations of affine Hecke algebras of type B and symmetric crystals for $\gl_\infty$. In the first half of this paper (sections 2 and 3), we give a survey of the LLTA type theorem of the affine Hecke algebra of type $A$. In the latter half (sections 4, 5 and 6), we review the construction of the symmetric crystals and the LLTA type conjectures for the affine Hecke algebra of type $B$.
We present a derivation of a formula that gives dynamics of an integrable cellular automaton associated with crystal bases. This automaton is related to type D affine Lie algebra and contains usual box-ball systems as a special case. The dynamics is described by means of such objects as carriers, particles, and antiparticles. We derive it from an analysis of a recently obtained formula of the combinatorial R (an intertwiner between tensor products of crystals) that was found in a study of geometric crystals.
We give a new interpretation and proof of the dilogarithm identities, associated to the affine Kac-Moody algebra sl(2)^, using the path description of the corresponding crystal basis. We also discuss connections with algebraic K-theory.
Crystal base of the negative half of quantum orthosymplectic superalgebra
本研究构造了量子正交代数超代数负半部分的晶体基，给出其与q-振荡器表示晶体的嵌入的组合描述，借助正交代数型Burge对应新实现刻画了嵌入的复合结构。
The polyhedral realizations for crystal bases of the integrable highest weight modules of $U_q(\mathfrak{g})$ have been introduced in ([T.Nakashima, J. Algebra, vol.219, no. 2, (1999)]), which describe the crystal bases as sets of lattice points in the infinite $\mathbb{Z}$-lattice $\mathbb{Z}^{\infty}$ given by some system of linear inequalities, where $\mathfrak{g}$ is a symmetrizable Kac-Moody Lie algebra. To construct the polyhedral realization, we need to fix an infinite sequence $ι$ from the indices of the simple roots. If the pair ($ι$,$λ$) ($λ$: a dominant integral weight) satisfies th…
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