Crystal bases for the quantum superalgebra $U_q(\mathfrak {gl}(m,n))$
HTML articles powered by AMS MathViewer
- 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
- PDF | Request permission
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
Bibliographic Information
- Georgia Benkart
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706–1388
- MR Author ID: 34650
- Email: benkart@math.wisc.edu
- Seok-Jin Kang
- Affiliation: Department of Mathematics, Seoul National University, Seoul 151-742, Korea
- MR Author ID: 307910
- Email: sjkang@math.snu.ac.kr
- Masaki Kashiwara
- Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606–8502, Japan
- MR Author ID: 98845
- Email: masaki@kurims.kyoto-u.ac.jp
- Received by editor(s): November 2, 1998
- Received by editor(s) in revised form: June 21, 1999
- Published electronically: January 31, 2000
- Additional Notes: The first author was supported in part by National Science Foundation Grant #DMS-9622447.
The second author was supported in part by the Basic Science Research Institute Program, Ministry of Education of Korea, BSRI-98-1414, and GARC-KOSEF at Seoul National University. - © Copyright 2000 American Mathematical Society
- Journal: J. Amer. Math. Soc. 13 (2000), 295-331
- MSC (1991): Primary 17B65, 17B37, 81R50, 05E10
- DOI: https://doi.org/10.1090/S0894-0347-00-00321-0
- MathSciNet review: 1694051