Crystal bases for the quantum superalgebra 𝑈_{𝑞}(𝔤𝔩(𝔪,𝔫))

Benkart, Georgia; Kang, Seok-Jin; Kashiwara, Masaki

doi:10.1090/S0894-0347-00-00321-0

Crystal bases for the quantum superalgebra $U_q(\mathfrak {gl}(m,n))$
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by Georgia Benkart, Seok-Jin Kang and Masaki Kashiwara

J. Amer. Math. Soc. 13 (2000), 295-331

DOI: https://doi.org/10.1090/S0894-0347-00-00321-0

Published electronically: January 31, 2000

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Abstract:

A crystal base theory is introduced for the quantized enveloping algebra of the general linear Lie superalgebra $\mathfrak {gl}(m,n)$, and an explicit realization of the crystal base is given in terms of semistandard tableaux.

References

Georgia Benkart, Chanyoung Lee Shader, and Arun Ram, Tensor product representations for orthosymplectic Lie superalgebras, J. Pure Appl. Algebra 130 (1998), no. 1, 1–48. MR 1632811, DOI 10.1016/S0022-4049(97)00084-4
A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math. 64 (1987), no. 2, 118–175. MR 884183, DOI 10.1016/0001-8708(87)90007-7
Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1359532, DOI 10.1090/gsm/006
V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, DOI 10.1016/0001-8708(77)90017-2
V. Kac, Representations of classical Lie superalgebras, Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977) Lecture Notes in Math., vol. 676, Springer, Berlin, 1978, pp. 597–626. MR 519631
Seok-Jin Kang and Kailash C. Misra, Crystal bases and tensor product decompositions of $U_q(G_2)$-modules, J. Algebra 163 (1994), no. 3, 675–691. MR 1265857, DOI 10.1006/jabr.1994.1037
Masaki Kashiwara, Crystalizing the $q$-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), no. 2, 249–260. MR 1090425
M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
Masaki Kashiwara and Toshiki Nakashima, Crystal graphs for representations of the $q$-analogue of classical Lie algebras, J. Algebra 165 (1994), no. 2, 295–345. MR 1273277, DOI 10.1006/jabr.1994.1114
S. M. Khoroshkin and V. N. Tolstoy, Universal $R$-matrix for quantized (super)algebras, Comm. Math. Phys. 141 (1991), no. 3, 599–617. MR 1134942
C. Lee Shader, Representations for Lie superalgebra $\mathfrak {spo}(2m,1)$, to appear.
Peter Littelmann, Crystal graphs and Young tableaux, J. Algebra 175 (1995), no. 1, 65–87. MR 1338967, DOI 10.1006/jabr.1995.1175
Kailash Misra and Tetsuji Miwa, Crystal base for the basic representation of $U_q(\mathfrak {s}\mathfrak {l}(n))$, Comm. Math. Phys. 134 (1990), no. 1, 79–88. MR 1079801
Ian M. Musson and Yi Ming Zou, Crystal bases for $U_q(\textrm {osp}(1,2r))$, J. Algebra 210 (1998), no. 2, 514–534. MR 1662280, DOI 10.1006/jabr.1998.7591
Toshiki Nakashima, Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras, Comm. Math. Phys. 154 (1993), no. 2, 215–243. MR 1224078
V. Rittenberg and M. Scheunert, A remarkable connection between the representations of the Lie superalgebras $\textrm {osp}(1,\,2n)$ and the Lie algebras $\textrm {o}(2n+1)$, Comm. Math. Phys. 83 (1982), no. 1, 1–9. MR 648354
Sheila Sundaram, Orthogonal tableaux and an insertion algorithm for $\textrm {SO}(2n+1)$, J. Combin. Theory Ser. A 53 (1990), no. 2, 239–256. MR 1041447, DOI 10.1016/0097-3165(90)90059-6
Hiroyuki Yamane, Quantized enveloping algebras associated with simple Lie superalgebras and their universal $R$-matrices, Publ. Res. Inst. Math. Sci. 30 (1994), no. 1, 15–87. MR 1266383, DOI 10.2977/prims/1195166275
Yi Ming Zou, Crystal bases for $U_q(\textrm {sl}(2,1))$, Proc. Amer. Math. Soc. 127 (1999), no. 8, 2213–2223. MR 1486756, DOI 10.1090/S0002-9939-99-04821-2