DOI QR코드

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COMBINATORIAL WEBS OF QUANTUM LIE SUPERALGEBRA sl(1|1)

  • 투고 : 2009.03.11
  • 심사 : 2009.10.23
  • 발행 : 2009.12.31

초록

Temperley-Lieb algebras had been generalized to web spaces for rank 2 simple Lie algebras which led us to link invariants for these Lie algebras as a generalization of Jones polynomial. Recently, Geer found a new generalization of Jones polynomial for some Lie superalgebras. In this paper, we study the quantum sl(1|1) representation theory using the web space and find a finite presentation of the representation category (for generic q) of the quantum sl(1|1).

키워드

참고문헌

  1. N. Chbili, Quantum invariants and finite group actions on three-manifolds, preprint.
  2. Q. Chen and T. Le, Quantum invariants and periodic links and periodic manifolds, preprint, arXiv:math.QA/0408358.
  3. I. Frenkel and M. Khovanov, Canonical bases in tensor products and graphical calculus for $U_q(sl_2)$, Duke Math. J., 87(3), (1997) 409-480. https://doi.org/10.1215/S0012-7094-97-08715-9
  4. N. Geer and B. Patureau-Mirand, Colored HOMFLY-PT and Multivariable Link Invariants, preprint.
  5. N. Geer and B. Patureau-Mirand, An invariant supertrace for the category of representations of Lie superalgebras, preprint.
  6. N. Geer and B. Patureau-Mirand, Multivariable link invariants arising from sl(2|1) and the Alexander polynomial, preprint.
  7. V. F. R. Jones, Index of subfactors, Invent. Math., 72 (1983), 1-25. https://doi.org/10.1007/BF01389127
  8. V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math., 126 (1987), 335-388. https://doi.org/10.2307/1971403
  9. C. Kassel, M. Rosso and V. Turaev, Quantum groups and knot invariants, Panoramas et Syntheses, 5, Societe Mathematique de France, 1997.
  10. M. Khovanov, Categorifications of the colored Jones polynomial, J. Knot Theory Ramifications, 2005, 14(1), 111-130. https://doi.org/10.1142/S0218216505003750
  11. D. Kim, Graphical Calculus on Representations of Quantum Lie Algebras, Thesis, UCDavis, 2003, arXiv:math.QA/0310143.
  12. D. Kim and J. Lee, The quantum sl(3) invariants of cubic bipartite planar graphs, preprint.
  13. G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys., 180(1), (1996) 109-151, arXiv:q-alg/9712003. https://doi.org/10.1007/BF02101184
  14. Scott Morrison, A Diagrammatic Category for the Representation Theory of $U_q(sl_n)$, UC Berkeley Ph.D. thesis, arXiv:0704.1503.
  15. H. Murakami and T. Ohtsuki and S. Yamada, HOMFLY polynomial via an invariant of colored plane graphs, L'Enseignement Mathematique, t., 44 (1998), 325-360.
  16. T. Ohtsuki and S. Yamada: Quantum su(3) invariants via linear skein theory, J. Knot Theory Ramifications, 6(3) (1997), 373-404. https://doi.org/10.1142/S021821659700025X
  17. N. Yu. Reshetikhin and V. G. Turaev, Ribbob graphs and their invariants derived from quantum groups, Comm. Math. Phys., 127 (1990), 1-26. https://doi.org/10.1007/BF02096491
  18. N. Yu. Reshetikhin and V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math., 103 (1991), 547-597. https://doi.org/10.1007/BF01239527
  19. A. Sikora and B. Westbury, Confluence theory for graphs, preprint, arXiv:math.QA/0609832.
  20. T. Van Zandt. PSTricks: PostScript macros for generic TEX. Available at ftp://ftp.princeton.edu/ pub/tvz/.
  21. O. Viro, Quantum relatives of Alexander polynomials, preprint, arXiv:math.GT/0204290.
  22. E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys., 121 (1989), 300-379.