DUAL PRESENTATION AND LINEAR BASIS OF THE TEMPERLEY-LIEB ALGEBRAS
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
The braid group Bn maps homomorphically into the Temperley-Lieb algebra TLn. It was shown by Zinno that the homomorphic images of simple elements arising from the dual presentation of the braid group Bn form a basis for the vector space underlying the Temperley-Lieb algebra TLn. In this paper, we establish that there is a dual presentation of Temperley-Lieb algebras that corresponds to the dual presentation of braid groups, and then give a simple geometric proof for Zinno’s theorem,using the interpretation of simple elements as non-crossing partitions.
Abstract
The braid group Bn maps homomorphically into the Temperley-Lieb algebra TLn. It was shown by Zinno that the homomorphic images of simple elements arising from the dual presentation of the braid group Bn form a basis for the vector space underlying the Temperley-Lieb algebra TLn. In this paper, we establish that there is a dual presentation of Temperley-Lieb algebras that corresponds to the dual presentation of braid groups, and then give a simple geometric proof for Zinno’s theorem,using the interpretation of simple elements as non-crossing partitions.
참고문헌 (13)
- D. Bessis, "The dual braid monoid", Ann. Sci. Ecole Norm. Sup., 36(5), pp.647~683 (2003)
- D. Bessis, "Non-crossing partitions of type (e, e, r)", Adv. Math., 202(1), pp.1~49 (2006)
- D. Bessis, "Springer theory in braid groups and the Birman-Ko-Lee monoid", Pacific J. Math., 205(2), pp.287~309 (2002)
- J. S. Birman, "A new approach to the word and conjugacy problems in the braid groups", Adv. Math., 139(2), pp.322~353 (1998)
- T. Brady, "A partial order on the symmetric group and new K(¼, 1)’s for the braid groups", Adv. Math., 161(1), pp.20~40 (2001)
- P. Dehornoy, "Gaussian groups and Garside groups, two generalisations of Artin groups", Proc. London Math. Soc., 79(3), pp.569~604 (1999)
- V. F. R. Jones, "Index for subfactors", Invent. Math., 72(1), pp.1~25 (1983)
- V. F. R. Jones, "Hecke algebra representations of braid groups and link polynomials", Ann. of Math., 126(2), pp.335~388 (1987)
- L. Kauffman, "Knots and Physics" (1993), World Scientific
- W. B. R. Lickorish, "Three-manifolds and the Temperley-Lieb algebra", Math. Ann., 290(4), pp.657~670 (1991)
- J. McCammond, "Noncrossing partitions in surprising locations", Amer. Math. Monthly, 113(7), pp.598~610 (2006)
- V. G. Turaev, "Quantum Invariants of Knots and 3-manifolds, de Gruyter Studies in Mathematics, 18" (1994), Walter de Gruyter & Co.
- M. G. Zinno, "A Temperley-Lieb basis coming from the braid group", J. Knot Theory Ramifications, 11(4), pp.575~599 (2002)