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DUAL PRESENTATION AND LINEAR BASIS OF THE TEMPERLEY-LIEB ALGEBRAS
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 Title & Authors
DUAL PRESENTATION AND LINEAR BASIS OF THE TEMPERLEY-LIEB ALGEBRAS
Lee, Eon-Kyung; Lee, Sang-Jin;
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 Abstract
The braid group maps homomorphically into the Temperley-Lieb algebra . It was shown by Zinno that the homomorphic images of simple elements arising from the dual presentation of the braid group form a basis for the vector space underlying the Temperley-Lieb algebra . 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.
 Keywords
Temperley-Lieb algebra;braid group;dual presentation;noncrossing partition;
 Language
English
 Cited by
1.
Noncrossing partitions, fully commutative elements and bases of the Temperley–Lieb algebra, Journal of Knot Theory and Its Ramifications, 2016, 25, 06, 1650035  crossref(new windwow)
2.
Dual braid monoids, Mikado braids and positivity in Hecke algebras, Mathematische Zeitschrift, 2016  crossref(new windwow)
 References
1.
D. Bessis, The dual braid monoid, Ann. Sci. Ecole Norm. Sup. (4) 36 (2003), no. 5, 647-683.

2.
D. Bessis and R. Corran, Non-crossing partitions of type (e, e, r), Adv. Math. 202 (2006), no. 1, 1-49.

3.
D. Bessis, F. Digne, and J. Michel, Springer theory in braid groups and the Birman-Ko-Lee monoid, Pacific J. Math. 205 (2002), no. 2, 287-309.

4.
J. S. Birman, K. H. Ko, and S. J. Lee, A new approach to the word and conjugacy problems in the braid groups, Adv. Math. 139 (1998), no. 2, 322-353.

5.
T. Brady, A partial order on the symmetric group and new K(${\pi}$, 1)'s for the braid groups, Adv. Math. 161 (2001), no. 1, 20-40.

6.
P. Dehornoy and L. Paris, Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. London Math. Soc. (3) 79 (1999), no. 3, 569-604.

7.
V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1-25.

8.
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335-388. crossref(new window)

9.
L. Kauffman, Knots and Physics, World Scientific, Singapore, 1993.

10.
W. B. R. Lickorish, Three-manifolds and the Temperley-Lieb algebra, Math. Ann. 290 (1991), no. 4, 657-670. crossref(new window)

11.
J. McCammond, Noncrossing partitions in surprising locations, Amer. Math. Monthly 113 (2006), no. 7, 598-610. crossref(new window)

12.
V. G. Turaev, Quantum Invariants of Knots and 3-manifolds, de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., Berlin, 1994.

13.
M. G. Zinno, A Temperley-Lieb basis coming from the braid group, J. Knot Theory Ramifications 11 (2002), no. 4, 575-599. crossref(new window)