Proceedings of the Japan Academy, Series A, Mathematical Sciences

Quantum queer superalgebra and crystal bases

Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara, and Myungho Kim

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

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Proc. Japan Acad. Ser. A Math. Sci. Volume 86, Number 10 (2010), 177-182.

First available in Project Euclid: 6 December 2010

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Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]

Quantum queer superalgebra crystal bases odd Kashiwara operators


Grantcharov, Dimitar; Jung, Ji Hye; Kang, Seok-Jin; Kashiwara, Masaki; Kim, Myungho. Quantum queer superalgebra and crystal bases. Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 10, 177--182. doi:10.3792/pjaa.86.177.

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  • G. Benkart, S.-J. Kang and M. Kashiwara, Crystal bases for the quantum superalgebra $U_{q}(\mathfrak{gl}(m,n))$, J. Amer. Math. Soc. 13 (2000), no. 2, 295–331.
  • D. Grantcharov et al., Highest weight modules over quantum queer superalgebra $U_{q}(\mathfrak{q}(n))$, Commun. Math. Phys. 296 (2010), no. 3, 827–860.
  • M. Kashiwara, Crystalizing the $q$-analogue of universal enveloping algebras, Commun. Math. Phys. 133 (1990), no. 2, 249–260.
  • M. Kashiwara, On crystal bases of the $q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516.
  • M. Kashiwara, The crystal base and Littelmann's refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839–858.
  • M. Kashiwara and T. Nakashima, Crystal graphs for representations of the $q$-analogue of classical Lie algebras, J. Algebra 165 (1994), no. 2, 295–345.
  • I. Penkov and V. Serganova, Characters of finite-dimensional irreducible $\mathfrak{q}(n)$-modules, Lett. Math. Phys. 40 (1997), no. 2, 147–158.
  • A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras $\mathfrak{gl}(n,m)$ and $Q(n)$, Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422–430.