Dual presentation and linear basis of the Temperley-Lieb algebras - PDF document

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/2111720 Dual presentation and linear basis of the Temperley-Lieb algebras Article in Journal of the Korean Mathematical Society · March 2004 DOI: 10.4134/JKMS.2010.47.3.445 · Source: arXiv CITATIONS READS 5 36 2 authors , including: Sang-Jin Lee Konkuk University 16 PUBLICATIONS 82 CITATIONS SEE PROFILE All content following this page was uploaded by Sang-Jin Lee on 27 November 2014. The user has requested enhancement of the downloaded file.

DUAL PRESENTATION AND LINEAR BASIS arXiv:math/0403429v3 [math.GR] 23 Apr 2006 OF THE TEMPERLEY-LIEB ALGEBRAS EON-KYUNG LEE AND SANG JIN LEE Abstract. 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 al- gebra 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. 1. Introduction Since Jones [7, 8] discovered the Jones polynomial for links by investi- gating representations of braid groups into Hecke algebras and Temperley- Lieb algebras, Temperley-Lieb algebras have played important roles in the quantum invariants of links and 3-manifolds. The Temperley-Lieb algebra TL n is defined on non-invertible generators e 1 , . . . , e n − 1 with the relations: e i e j = e j e i for | i − j | ≥ 2; e 2 i = e i ; e i e i ± 1 e i = τe i along with a complex number τ . It is well-known that the dimension of TL n is the n th Catalan . Setting t such that τ − 1 = 2 + t + t − 1 , and then � 2 n 1 � number C n = n +1 n setting h i = ( t + 1) e i − 1, we get an alternative presentation of TL n with invertible generators h 1 , . . . , h n − 1 satisfying the relations: (1) h i h j = h j h i if | i − j | ≥ 2; (2) h i h i +1 h i = h i +1 h i h i +1 ; h 2 (3) i = ( t − 1) h i + t ; (4) h i h i +1 h i + h i h i +1 + h i +1 h i + h i + h i +1 + 1 = 0 . The braid group B n is defined by the Artin presentation, where the gen- erators are σ 1 , . . . , σ n − 1 and the defining relations are σ i σ j = σ j σ i if | i − j | ≥ 2; σ i σ i +1 σ i = σ i +1 σ i σ i +1 for i = 1 , . . . , n − 2 . 2000 Mathematics Subject Classification. Primary 20F36; Secondary 57M27. Key words and phrases. Temperley-Lieb algebra, braid group, dual presentation, non- crossing partition. 1

2 E.-K. LEE AND S. J. LEE The braid group B n maps homomorphically into the Temperley-Lieb algebra TL n under π : σ i �→ h i . There is another presentation [4] with generators a ji (1 ≤ i < j ≤ n ) and defining relations a lk a ji = a ji a lk if ( l − j )( l − i )( k − j )( k − i ) > 0; a kj a ji = a ji a ki = a ki a kj for i < j < k. The generators a ji ’s are related to the σ i ’s by a ji = σ j − 1 σ j − 2 · · · σ i +1 σ i σ − 1 i +1 · · · σ − 1 j − 2 σ − 1 j − 1 . Bessis [1] showed that there is a similar presentation, called the dual pre- sentation , for Artin groups of finite Coxeter type. Both the Artin and dual presentations of the braid group B n determine a Garside monoid, as defined by Dehornoy and Paris [6], where the sim- ple elements play important roles. Nowadays, it becomes more and more popular to describe simple elements arising from the dual presentation via non-crossing partitions. Non-crossing partitions are useful in diverse ar- eas [1, 5, 2, 3, 9], because they have beautiful combinatorial structures. Let P 1 , . . . , P n be the points in the complex plain given by P k = exp( − 2 kπ n i ). See Figure 1. Recall that a partition of a set is a collection of pairwise disjoint subsets whose union is the entire set. Those subsets (in the collection) are called blocks. A partition of { P 1 , . . . , P n } is called a non-crossing partition if the convex hulls of the blocks are pairwise disjoint. A positive word of the form a i 1 i 2 a i 2 i 3 · · · a i k − 1 i k , i 1 > i 2 > · · · > i k , is called a descending cycle and denoted [ i 1 , i 2 , . . . , i k ]. Two descending cycles [ i 1 , . . . , i k ] and [ j 1 , . . . , j l ] are said to be parallel if the convex hulls of { P i 1 , . . . , P i k } and of { P j 1 , . . . , P j l } are disjoint. The simple elements are the products of parallel descending cycles. We remark that the definition of simple elements depends on the presenta- tions. For example, the simple elements arising from the Artin presentation are in one-to-one correspondence with permutations. Throughout this note, we consider only the simple elements arising from the dual presentation of braid groups as above. Note that simple elements are in one-to-one correspondence with non- crossing partitions. Our convention is that if a block in a non-crossing partition consists of a single point, then the corresponding descending cycle is the identity (i.e. the descending cycle of length 0). In particular, the number of the simple elements is the n th Catalan number C n , which is the dimension of TL n . Zinno [10] established the following result. Theorem 1 (Zinno’s theorem) . The homomorphic images of the simple elements arising from the dual presentation of B n form a linear basis for the Temperley-Lieb algebra TL n . We explain briefly Zinno’s proof. It is known that the ordered reduced words ( h j 1 h j 1 − 1 · · · h k 1 )( h j 2 h j 2 − 1 · · · h k 2 ) · · · ( h j p h j p − 1 · · · h k p ) ,

DUAL PRESENTATION AND LINEAR BASIS OF THE TEMPERLEY-LIEB ALGEBRAS 3 Figure 1. The shaded regions show the blocks in the non-crossing partition corresponding to the simple element [12 , 11 , 2] [10 , 4 , 3] [9 , 8 , 6 , 5] in B 12 . where j i ≥ k i , j i +1 > j i and k i +1 > k i , form a linear basis of TL n , and Zinno showed that the matrix for writing the images of simple elements as the linear combination of the ordered reduced words is invertible. Because the number of the simple elements is equal to the dimension of TL n , this proves the theorem. In this note, we first establish that there is a dual presentation of TL n . We are grateful to David Bessis for pointing out that the relation (4) in the Temperley-Lieb algebra presentation is equivalent to the forth relation in the dual presentation in the following theorem. Theorem 2 (dual presentation of TL n ) . The Temperley-Lieb algebra TL n has a presentation with invertible generators g ji (1 ≤ i < j ≤ n ) satisfying the relations: g lk g ji = g ji g lk if ( l − j )( l − i )( k − j )( k − i ) > 0; g kj g ji = g ji g ki = g ki g kj for i < j < k ; g 2 ji = ( t − 1) g ji + t for i < j ; g ji g kj + tg kj g ji + g kj + g ji + tg ki + 1 = 0 for i < j < k. The new generators are related to the old ones by g ji = h j − 1 h j − 2 · · · h i +1 h i h − 1 i +1 · · · h − 1 j − 2 h − 1 j − 1 . Using the above presentation, we give a new proof of Zinno’s theorem in § 3. We exploit non-crossing partitions so as to make the proof easy and intuitive. For the proof, we show that any monomial in the h ± 1 i ’s can be written as a linear combination of the images of simple elements. Therefore the images of simple elements span TL n . As a result, they form a linear basis of TL n because the number of simple elements is equal to the dimension of TL n . We remark that it seems possible to prove the linear independence of the images of the simple elements directly from the relations in the dual