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Seok-Jin Kang

Publications and source records attributed to Seok-Jin Kang.

At least 55 records · Page 3Linked to original sources
arXiv2013

Quiver Hecke superalgebras

We introduce a new family of superalgebras which should be considered as a super version of the Khovanov-Lauda-Rouquier algebras. Let $I$ be the set of vertices of a Dynkin diagram with parity. To this data, we associate a family of graded superalgebras, the quiver Hecke superalgebras. When there are no odd vertices, these algebras are nothing but the usual Khovanov-Lauda-Rouquier algebras. We then define another family of graded superalgebras, the quiver Hecke-Clifford superalgebras, and show that they are weakly Morita superequivalent to the quiver Hecke superalgebras. Moreover, we prove that the affine Hecke-Clifford superalgebras, as well as their degenerate version, the affine Sergeev superalgebras, are isomorphic to the quiver Hecke-Clifford superalgebras after a completion.

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arXiv2012

Geometric realization of Khovanov-Lauda-Rouquier algebras associated with Borcherds-Cartan data

We construct a geometric realization of the Khovanov-Lauda-Rouquier algebra $R$ associated with a symmetric Borcherds-Cartan matrix $A=(a_{ij})_{i,j\in I}$ via quiver varieties. As an application, if $a_{ii} \ne 0$ for any $i\in I$, we prove that there exists a 1-1 correspondence between Kashiwara's lower global basis (or Lusztig's canonical basis) of $U_\A^-(\g)$ (resp.\ $V_\A(λ)$) and the set of isomorphism classes of indecomposable projective graded modules over $R$ (resp.\ $R^λ$).

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arXiv2012

Construction of Irreducible Representations over Khovanov-Lauda-Rouquier Algebras of Finite Classical Type

We give an explicit construction of irreducible modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^λ$ for finite classical types using a crystal basis theoretic approach. More precisely, for each element $v$ of the crystal $B(\infty)$ (resp. $B(λ)$), we first construct certain modules $\nabla(\mathbf{a};k)$ labeled by the adapted string $\mathbf{a}$ of $v$. We then prove that the head of the induced module $\ind \big(\nabla(\mathbf{a};1) \boxtimes...\boxtimes \nabla(\mathbf{a};n)\big)$ is irreducible and that every irreducible $R$-module (resp. $R^λ$-module) can be realized as the irreducible head of one of the induced modules $\ind (\nabla(\mathbf{a};1) \boxtimes...\boxtimes \nabla(\mathbf{a};n))$. Moreover, we show that our construction is compatible with the crystal structure on $B(\infty)$ (resp. $B(λ)$).

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arXiv2012

Mixed Schur-Weyl-Sergeev duality for queer Lie superalgebras

We introduce a new family of superalgebras $\overrightarrow{B}_{r,s}$ for $r, s \ge 0$ such that $r+s>0$, which we call the walled Brauer superalgebras, and prove the mixed Scur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let $\mathfrak{q}(n)$ be the queer Lie superalgebra, ${\mathbf V} =\mathbb{C}^{n|n}$ the natural representation of $\mathfrak{q}(n)$ and ${\mathbf W}$ the dual of ${\mathbf V}$. We prove that, if $n \ge r+s$, the superalgebra $\overrightarrow{B}_{r,s}$ is isomorphic to the supercentralizer algebra $_{\mathfrak{q}(n)}({\mathbf V}^{\otimes r} \otimes {\mathbf W}^{\otimes s})^{\op}$ of the $\mathfrak{q}(n)$-action on the mixed tensor space ${\mathbf V}^{\otimes r} \otimes {\mathbf W}^{\otimes s}$. As an ingredient for the proof of our main result, we construct a new diagrammatic realization $\overrightarrow{D}_{k}$ of the Sergeev superalgebra $Ser_{k}$. Finally, we give a presentation of $\overrightarrow{B}_{r,s}$ in terms of generators and relations.

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arXiv2012

Categorification of Quantum Generalized Kac-Moody Algebras and Crystal Bases

We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^λ$ which give a categrification of quantum generalized Kac-Moody algebras. Let $U_\A(\g)$ be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \in I}$ and let $K_0(R)$ be the Grothedieck group of finitely generated projective graded $R$-modules. We prove that there exists an injective algebra homomorphism $Φ: U_\A^-(\g) \to K_0(R)$ and that $Φ$ is an isomorphism if $a_{ii}\ne 0$ for all $i\in I$. Let $B(\infty)$ and $B(λ)$ be the crystals of $U_q^-(\g)$ and $V(λ)$, respectively, where $V(λ)$ is the irreducible highest weight $U_q(\g)$-module. We denote by $\mathfrak{B}(\infty)$ and $\mathfrak{B}(λ)$ the isomorphism classes of irreducible graded modules over $R$ and $R^λ$, respectively. If $a_{ii}\ne 0$ for all $i\in I$, we define the $U_q(\g)$-crystal structures on $\mathfrak{B}(\infty)$ and $\mathfrak{B}(λ)$, and show that there exist crystal isomorphisms $\mathfrak{B}(\infty) \simeq B(\infty)$ and $\mathfrak{B}(λ) \simeq B(λ)$. One of the key ingredients of our approach is the perfect basis theory for generalized Kac-Moody algebras.

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arXiv2012

Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras

In this paper, we prove Khovanov-Lauda's cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras. Let $U_q(g)$ be the quantum group associated with a symmetrizable Cartan datum and let $V(Λ)$ be the irreducible highest weight $U_q(g)$-module with a dominant integral highest weight $Λ$. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra $R^Λ$ gives a categorification of $V(Λ)$.

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arXiv2011

Quantum Queer Superalgebra and Crystal Bases

In this paper, we develop the crystal basis theory for the quantum queer superalgebra $\Uq$. We define the notion of crystal bases, describe the tensor product rule, and present the existence and uniqueness of crystal bases for finite-dimensional $\Uq$-modules in the category $\mathcal{O}_{int}^{\ge 0}$.

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arXiv2011

Highest weight modules over quantum queer Lie superalgebra U_q(q(n))

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra $U_q(q(n))$. The key ingredients are the triangular decomposition of $U_q(q(n))$ and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for $U_q(q(n))$-modules in the category $O_q^{\geq 0}$.

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arXiv2010

Irreducible Modules over Khovanov-Lauda-Rouquier Algebras of type $A_n$ and Semistandard Tableaux

Using combinatorics of Young tableaux, we give an explicit construction of irreducible graded modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^λ$ of type $A_{n}$. Our construction is compatible with crystal structure. Let ${\mathbf B}(\infty)$ and ${\mathbf B}(λ)$ be the $U_q(\slm_{n+1})$-crystal consisting of marginally large tableaux and semistandard tableaux of shape $λ$, respectively. On the other hand, let ${\mathfrak B}(\infty)$ and ${\mathfrak B}(λ)$ be the $U_q(\slm_{n+1})$-crystals consisting of isomorphism classes of irreducible graded $R$-modules and $R^λ$-modules, respectively. We show that there exist explicit crystal isomorphisms $Φ_{\infty}: {\mathbf B}(\infty) \overset{\sim} \longrightarrow {\mathfrak B}(\infty)$ and $Φ_λ: {\mathbf B}(λ) \overset{\sim} \longrightarrow {\mathfrak B}(λ)$.

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arXiv2006

Abstract Crystals for Quantum Generalized Kac-Moody Algebras

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 $B(\la)$.

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arXiv2005

Level 1 Perfect Crystals and Path Realizations of Basic Representations at q=0

We present a uniform construction of level 1 perfect crystals $\mathcal B$ for all affine Lie algebras. We also introduce the notion of a crystal algebra and give an explicit description of its multiplication. This allows us to determine the energy function on $\mathcal B \otimes \mathcal B$ completely and thereby give a path realization of the basic representations at $q=0$ in the homogeneous picture.

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arXiv2004

Crystal bases of the Fock space representations and string functions

Let $U_q(\frak{g})$ a the quantum affine algebra of type $A_n^{(1)}$, $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_n^{(1)}$, $D_n^{(1)}$ and $D_{n+1}^{(2)}$, and let $\mathcal{F}(Λ)$ be the Fock space representation for a level 1 dominant integral weight $Λ$. Using the crystal basis of $\mathcal{F}(Λ)$ and its characterization in terms of abacus, we construct an explicit bijection between the set of weight vectors in $\mathcal{F}(Λ)_{λ-mδ}$ ($m\geq 0$) for a maximal weight $λ$ and the set of certain ordered sequences of partitions. As a corollary, we obtain the string function of the basic representation $V(Λ)$.

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