J. Korean Math. Soc. 2010; 47(3): 445-454
Printed May 1, 2010
https://doi.org/10.4134/JKMS.2010.47.3.445
Copyright © The Korean Mathematical Society.
Eon-Kyung Lee and Sang-Jin Lee
Sejong University and Konkuk University
The braid group $B_n$ maps homomorphically into the Temperley-Lieb algebra $TL_n$. It was shown by Zinno that the homomorphic images of simple elements arising from the dual presentation of the braid group $B_n$ form a basis for the vector space underlying the Temperley-Lieb algebra $TL_n$. 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, non-crossing partition
MSC numbers: Primary 20F36; Secondary 57M27
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