Crystal Bases for Kac–Moody Superalgebras

Jeong, Ki‐Hun

doi:10.1006/jabr.2000.8590

Cited by 11 publications

(11 citation statements)

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“…2011-0006735). [7,11]. For example, it is twisted compared to the usual crystal base over U q (gl(m)⊕ gl(n)) in the sense that it is a lower crystal base as a U q (gl(m|0))-module but is a upper crystal base as a U q (gl(0|n))-module.…”

Section: Algebrasmentioning

confidence: 99%

Crystal Bases of q-deformed Kac Modules Over the Quantum Superalgebra Uq(𝔤𝔩(m|n))

Kwon

2012

International Mathematics Research Notices

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We introduce the notion of a crystal base of a finite dimensional qdeformed Kac module over the quantum superalgebra Uq(gl(m|n)), and prove its existence and uniqueness. In particular, we obtain the crystal base of a finite dimensional irreducible Uq(gl(m|n))-module with typical highest weight. We also show that the crystal base of a q-deformed Kac module is compatible with that of its irreducible quotient V (λ) given by Benkart, Kang and Kashiwara when V (λ) is an irreducible polynomial representation.where P(Φ − 1 ) is the power set of the set of negative odd roots of gl(m|n) and B m,n (λ) is the crystal of V m,n (λ). Also, the crystal structure on (1.1) can be described easily (Section 5.1).We next show that the crystal base of K(λ) is compatible with that of its irreducible quotient V (λ) when V (λ) is an irreducible polynomial representation (Theorem 4.11), that is, the canonical projection from K(λ) to V (λ) sends the crystal base of K(λ) onto that of V (λ). Hence we may regard the crystal of V (λ) as a subgraph of the crystal of K(λ). We give a combinatorial description of its embedding (Section 5.3) using the (m|n)-hook tableaux crystal model for V (λ) and the skew dual RSK algorithm introduced by Sagan and Stanley [16].The paper is organized as follows. In Section 2, we give necessary background on the quantum superalgebra U q (gl(m|n)). In Section 3, we review the crystal base theory developed in [1]. In Section 4, we define the notion of a crystal base of a Kac module and state the main results, whose proofs are given in the following two sections.Acknowledgement Part of this work was done while the author was visiting the Institute of Mathematics in Academia Sinica, Taiwan on January 2012. He would like to thank S.-J. Cheng for the invitation and helpful discussion, and the staffs for their warm hospitality.2. Quantum superalgebra U q (gl(m|n)) 2.1. Lie superalgebra gl(m|n). For non-negative integers m, n, let [m|n] be a Z 2graded set with [m|n] 0 = { m, . . . , 1 }, [m|n] 1 = { 1, . . . , n } and a linear ordering m < . . . < 1 < 1 < . . . < n. We denote by |a| the degree of a ∈ [m|n]. Let C [m|n] be the complex superspace with a basis { v a | a ∈ [m|n] }, where the parity of v a is |a|.Let gl(m|n) denote the Lie superalgebra of [m|n] × [m|n] complex matrices, which is spanned by E ab (a, b ∈ [m|n]) with 1 in the ath row and the bth column, and 0 elsewhere [8].Let P ∨ = a∈[m|n] ZE aa be the dual weight lattice and h = C ⊗ Z P ∨ the Cartan subalgebra of gl(m|n). Define ǫ a ∈ h * by E bb , ǫ a = δ ab for a, b ∈ [m|n], where ·, · denotes the natural pairing on h × h * . Let P = a∈[m|n] Zǫ a be the weight lattice of gl(m|n). For λ = a∈[m|n] λ a ǫ a ∈ P , the parity of λ is defined to be λ 1 + · · · + λ n mod 2 and denoted by |λ|. Let ( · | · ) denote a symmetric bilinear form on h * = C ⊗ Z P given by (ǫ a | ǫ b ) = (−1) |a| δ ab for a, b ∈ [m|n].Let I = I m|n = { m − 1, . . . , 1, 0, 1 . . . , n − 1 }, where we assume that I m|0 = { m − 1, . . . , 1 } and I 0|n = { 1 . . . , n−1 }. Then with respect to ...

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Section: Algebrasmentioning

confidence: 99%

Crystal Bases of q-deformed Kac Modules Over the Quantum Superalgebra Uq(𝔤𝔩(m|n))

Kwon

2012

International Mathematics Research Notices

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“…There is another important class in classical Lie superalgebras called ortho-symplectic Lie superalgebras osp k|2l . However, there is little known about the existence of its crystal bases except for osp 1|2l , which is of type B(0, l), and where we can apply the crystal base theory for a Kac-Moody superalgebra without odd isotropic simple root [17].…”

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confidence: 99%

Super Duality and Crystal Bases for Quantum Ortho-Symplectic Superalgebras

Kwon¹

2015

Int Math Res Notices

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We introduce a semisimple tensor category O int q (m|n) of modules over an quantum ortho-symplectic superalgebra. It is a natural counterpart of the category of finitely dominated integrable modules over the quantum classical (super) algebra of type Bm+n, Cm+n, Dm+n or B(0, m + n) from a viewpoint of super duality. We classify the irreducible modules in O int q (m|n) and show that an irreducible module in O int q (m|n) has a unique crystal base in case of type B and C. An explicit description of the crystal graph is given in terms of a new combinatorial object called ortho-symplectic tableaux.1 T m+n (a) = T g J m+n (a), T m+n (λ, ℓ) = T g J m+n (λ, ℓ), T sp m+n = T sp J m+n ,

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“…Note that V V = 0 unless = . Moreover, the symmetric bilinear form on V defined by (3.3) is nondegenerate (see [8] or [9]). The weights of the irreducible highest weight U q -module V satisfy the following properties on the imaginary simple coroots.…”

Section: Remark 33mentioning

confidence: 99%