Bridges with pillars A graphical calculus of knot algebra

Tammo tom Dieck

1. Introduction

This paper comprises a graphical calculus which is designed to deal with the Coxeter-Dynkin series of type E and some generalizations.

A Coxeter matrix (S, m) consists of a finite set S and a symmetric mapping m: S × S → IN ∪ {∞} with m(s, s) = 1 and m(s, t) ≥ 2 for s ≠ t. A Coxeter matrix (S, m) is often specified by its weighted Coxeter graph Γ(S, m). It has S as its set of vertices and an edge with weight m(s, t) connecting s and t whenever m(s, t) ≥ 3. Usually, the weight m(s, t) = 3 is omitted in the notation. If m(S×S) ⊂ {1, 2, 3}, we define the associated Temperley-Lieb algebra T_{d}(S, m) as follows. Let 𝔎 be a commutative ring and d ∈ 𝔎^{∗} an invertible parameter. Then T_{d}(S, m) is the associative algebra with 1 over 𝔎 with generators (e_{s} | s ∈ S) and relations

(1.1) \begin{matrix}e_{s}^{2} & = & de_{s} \\ e_{s}e_{t} & = & e_{t}e_{s} & m(s, t) = 2 \\ e_{s}e_{t}e_{s} & = & e_{s} & m(s, t) = 3. \end{matrix}

Recall that the Hecke algebra H_{q}(S, m) associated to the Coxeter matrix (S, m) and an invertible parameter q ∈ 𝔎 is the associative algebra with 1 over 𝔎 with generators (x_{s} | s ∈ S) and relations x_{s}^{2} = (q − 1)x_{s} + q and braid relations x_{s}x_{t} … = x_{t}x_{s} … (m(s, t) factors x_{s}, x_{t} alternating on each side). Suppose p ∈ 𝔎^{∗}, q = p^{2}, d = p + p^{−1}. Then T_{d}(S, m) = T (S, m). Then the assignment x_{s} ↦ pe_{s} − 1 yields a surjection H_{q}(S, m) → T_{d}(S, m) (see [3]). The classical Coxeter matrices (Coxeter graphs) of simple Lie groups yield finite dimensional Hecke algebras. It turns out that there are additional graphs which yield finite dimensional Temperley-Lieb algebras, e. g. those associated to the graphs E_{n}(k) below.

In general, the algebras T_{d}(S, m) are difficult to analyze, because of their definition by generators and relations. The purpose of this paper is to describe a graphical calculus which is adapted to Coxeter graphs of type E. We denote by E_{n}(k) the graph with n vertices of the following shape.

[figure: e_{1}; e_{2}; e_{k}; e_{n−1}; e_{0}; …; …]

We also use the standard notation of Lie theory

E_{n}(1) = A_{n}, E_{n}(2) = D_{n}, E_{n}(3) = E_{n}.

Note that E_{n}(k) contains the linear subgraph A_{n−1} with n − 1 vertices (e_{0} omitted). Although the graphical calculus is designed for the finite dimensional algebras T_{d}E_{n}(k), it has other uses as well.

The starting point for our calculus is the graphical notation of Kauffman [5, p. 100] for the standard Temperley-Lieb algebra T_{d}A_{n} associated to the Coxeter graph A_{n}. A basis element of T_{d}A_{n−1} over 𝔎 consists of n disjoint arcs in IR×[0, 1] with endpoint set {1, …, n} × {0, 1}. We call such figures (n, n)-bridges. We use the decomposition of IR × [0, 1] into planar domains produced by the arcs of a bridge. This decomposition is used to incorporate additional information into the figure: We single out certain regions by placing pillars. A 𝔎-basis of the algebra T_{d}E_{n}(k) will then, hopefully, consist of certain such bridges with pillars. For typographical reasons we use a bracket notation for bridges with pillars. The reader is advised to decode this into ordinary planar figures. The advantage of the graphical calculus is its semi-global nature, compared with generators and relations. (This is similar to the difference between the global definition of the symmetric group by permutations and its definition by generators and relations from the Coxeter graph.) As an example of the use of the calculus we mention: A geometrically defined filtration on the set of figures allows a splitting of the resulting algebras into matrix components (in the generic semi-simple case), and also allows a geometric construction of modules.

This paper presents the calculus and applies it to the basic example E_{6} = E_{6}(3). A typical result is:

Theorem A. The algebra T_{d}E_{6}(3) has rank 662. For generic parameters d in a field the algebra is semi-simple and has simple modules M(0), M(1), M(2), and M(3) of rank 1, 6, 20, and 15, respectively.

Since A_{5} ⊂ E_{6}, we have an inclusion T_{d}A_{5} ⊂ T_{d}E_{6}. We determine the decomposition of the simple T_{d}E_{6}-modules when restricted to T_{d}A_{5}. Recall that the algebra T_{d}A_{5} has simple modules M_{0}, M_{1}, M_{2}, and M_{3} of rank 1, 5, 9, and 5 (see e. g. [4, 2.8] for the module theory of T_{d}A_{n} in general).

T. tom Dieck 2. Bridges with pillars 3

Theorem B. The following isomorphisms hold for the restricted modules:

resM(0) ≅ M_{0}
resM(1) ≅ M_{1} ⊕ M_{0}
resM(2) ≅ M_{2} ⊕ 2M_{1} ⊕ M_{0}
resM(3) ≅ M_{3} ⊕ M_{2} ⊕ M_{0}.

We mention (but do not prove in this paper) the module structure of T_{d}E_{7}(3):

Theorem C. The algebra T_{d}E_{7}(3) has in the generic case simple modules N(4), N(3), N′(3), N(2), N′(2), N(1), and N(0) of rank 15, 35, 35, 27, 18, 7, and 1, respectively. □

In the statement of the next result we use the fact that the algebra T_{d}A_{6} has simple modules N_{0}, N_{1}, N_{2}, N_{3} of rank 1, 6, 14, 14.

Theorem D. The restriction properties of these modules to A = T_{d}A_{6} and E = T_{d}E_{6} are:

res_{E}N(4) = M(3)
res_{A}N(4) = N_{0} ⊕ N_{3}
res_{E}N(3) = M(2) ⊕ M(3)
res_{A}N(3) = N_{0} ⊕ N_{1} ⊕ N_{2} ⊕ N_{3}
res_{E}N(2) = M(2) ⊕ M(1) ⊕ M(0)
res_{E}N′(2) = 3M(1)
res_{A}N(2) = N_{2} ⊕ 2N_{1} ⊕ N_{0}
res_{A}N′(2) = 3N_{1}
res_{E}N(1) = M(1) ⊕ M(0)
res_{A}N(1) = N_{1} ⊕ N_{0}.

The module N′(3) has the same restriction properties as N(3). □

In the next section we define the calculus and various related algebras. Section 3 presents some elementary examples. In section 4 and 5 we give a detailed description of the algebra related to E_{6}(3). We conclude by mentioning other uses of the calculus.

2. Bridges with pillars

We remind the reader that an (n, n)-bridge consists of a system of n arcs in IR × [0, 1] without crossings such that the set of its boundary points is P_{n} := {1, 2, …, n} × {0, 1}. We think of the arcs meeting IR × {0, 1} transversely. Two bridges are equal if they connect the same points. An upper (lower) arc has both of its boundary points in IR × 1 (IR × 0). The upper and lower arcs are called horizontal, the others vertical. If a bridge b has k upper arcs, we call k its horizontal edge number and write HE(b) = k. The configuration which consists of the upper arcs of an (n, n)-bridge b is called the upper n-bridge of b. An (n, n)- bridge is determined by its upper and lower n-bridge. An upper n-bridge per se of horizontal edge number k can be specified by a system of k admissible bracket pairs with additional marks for the end points of the vertical arcs. Thus, the 9 upper 6-bridges b with HE(b) = 2 are in this notation:

(())||, ()()||, |(())|, |()()|, ||(()), ||()(), ()|()|, ()||(), |()|().

An (n, n)-bridge will, in a similar manner, be denoted by a pair of bracket systems with a fraction stroke (compare the figure below for the notation of a pillar bridge).

We need bridges with additional structure. The definition needs some preparation. The arcs of a bridge subdivide the strip IR×[0, 1] into domains. There are one or two unbounded domains. An (n, n)-bridge yields n + 1 domains. The distance between two bridge domains is the minimal numbers of arcs which a path from one domain to the other has to cross. The distance modulo two is a topological intersection number. The boundary curve of a domain consists of certain intervals in IR × {0, 1} and some arcs of the bridge. The intervals are called the feet of the domain. A domain is determined by its feet. An upper (lower) domain has all its feet in IR × 1 (IR × 0). The remaining domains are called vertical.

Each domain D of a bridge has 4 separation numbers: SNL(D), SNR(D), SNT (D), SNB(D). Here L, R, T, B stands for left, right, top, bottom. The left separation number is the minimal number of arcs separating the domain from the left unbounded one. The bottom separation number is the minimal number of arcs separating the domain from IR × 0. Thus a vertical domain is characterized by SNB = SNT = 0.

We fix a base number k ∈ \{0, 1, 2, …, [\frac{n}{2}]\}. A bridge then has a basic domain with foot ]∞, 0] × 1 in case k = 0 or [k, k + 1] × 1 in case k > 0. A domain is called even (odd) if the distance to the basic domain is even (odd). This depends only on k mod 2; but later k itself will play a role.

Now the basic notion of our calculus! An (n, n)-bridge with pillars and base number k consists of an (n, n)-bridge together with a (possibly empty) subset of its even domains. We specify graphically the chosen domains by placing a point (a pillar) into the domain. For typographical reasons we use the bracket notation introduced above. We add a • in order to specify a pillar domain. The following figure explains the usage. It displays a (5, 5)-bridge with even base number and two pillar domains; also the bracket notation for its upper and lower bridge is given.

[figure: =; \frac{((•)) • |}{()|()} .]

Observe that one may have a choice where to put certain • in the bracket notation, in this case we could also use \frac{((•))|}{()•|()}. Let E^{∗}(n, k) denote the set of (n, n)- bridges with pillars and base number k. Later we have to use a certain subset: A pillar bridge is called reduced if the distance of any two pillar domains is at least 4. Let E(n, k) denote the set of reduced bridges in E^{∗}(n, k). We note that the distance between any two pillar domains is even.

We recall the graphical definition of the Temperley-Lieb algebra T_{n} = T_{d}A_{n−1} associated to the Coxeter graph A_{n−1} (see [5, p. 100]), since we have to use it in the definition of our algebras. Let 𝔎 be a commutative ring and d ∈ 𝔎 a parameter. Additively, T_{n} is the free 𝔎-module on the set of (n, n)-bridges. The multiplication is 𝔎-bilinear. Thus it remains to define the product of two bridges S and T. Let T ∘ S denote the figure which is obtained by placing the T -strip above the S-strip and squeezing the result affinely into IR × [0, 1]. In general, the figure T ∘ S is not a bridge, there may be circles in the interior of the strip. Suppose there exist k(T, S) circles. Let T ∧ S denote the bridge obtained by removing the circles. Then the product is defined by

(2.1) T · S = d^{k(T,S)}T ∧ S.

We now use pillar bridges to define other algebras in an analogous manner. The next section collects a few elementary examples which illustrate the following definitions.

The algebra T E^{∗}(n, k). It is additively the free 𝔎-module on E^{∗}(n, k). The multiplication is again 𝔎-bilinear. Its definition uses two further parameters c, f ∈ 𝔎. Thus T E^{∗}(n, k) depends on c, d, f. In order to define the product of two pillar bridges we consider the underlying ordinary bridges and form T ∘ S as before. Then we look at the position of the pillars. The deviation from a pillar bridge can have several reasons. We correct the deviation by the processes (2.2) and (2.3) below.

(2.2) Multiple pillars. Suppose a domain of T ∘ S contains a > 1 pillars. We remove a − 1 pillars from the domain and multiply the result by c^{a−1}. Let T ∘_{1}S denote the resulting figure with pillars. Suppose a(T, S) pillars are removed altogether by this process. ♡

(2.3) Pillar circles. Consider the circles of T ∘_{1} S. If such a circle can be connected with a pillar in its interior without crossing of other circles (pillar circle), we remove circle and pillar and multiply the result by f. The other circles are treated with the parameter d as above. Suppose altogether there exist b(T, S) pillar circles and k(T, S) ordinary ones. ♡

The above two processes yield a bridge T · S with pillars and base number k. The underlying ordinary bridge is T ∧ S. The product of T and S is defined by

c^{a(T,S)}f^{b(T,S)}d^{k(T,S)}T · S.

The reader should draw a figure and verify the following product

\frac{() • |(())()}{|(•())(•())} · \frac{()()|(•())}{((•)) • ()|()} = c^{3}fd^{2} \frac{() • |(())()}{((•))()|()}.

In this paper, the algebras T E^{∗}(n, k) are technical tools. But they have some independent interest. For instance, T E^{∗}(n, 1) is related to the Temperley-Lieb algebra associated to the root system B_{n}, see [1].

The algebra T E(n, k). It is additively the free 𝔎-module on E(n, k). In order to define the multiplication, we begin as for E^{∗}(n, k). But there is now another reason why the result may be wrong: If two pillar domains have distance 2. We correct by the next process.

(2.4) Reduction. If two pillar domains have distance 2, we modify as in the following figure (• = pillar).

[figure: ⇒]

Thus, the two pillar domains are connected by a corridor, and there results a single new pillar domain. One has to show that reduction is well defined. Before doing this, we have to restrict the possible parameters, namely we set in T E(n, k)

c = d, f = 1.

The following example demonstrates why this is necessary. It also shows what reduction has to do with the defining relations (1.1) of a Temperley-Lieb algebra.

(2.5) Example. We use the notation

[figure: = e_{1},; = e_{0}.]

Then by (2.4) e_{0}e_{1}e_{0} = e_{0} and by (2.3) e_{1}e_{0}e_{1} = fe_{1}. We use this to compute (e_{0}e_{1}e_{0})e_{1} = e_{0}(e_{1}e_{0}e_{1}) in two ways and see that f = 1 is necessary. The requirement c = d is unimportant and can always be achieved by a suitable parameter transformation (in an extension ring). ♡

Reduction can be defined globally as follows. It is easy to see that the domains of a bridge are 2-cells. Let D be a pillarless domain of a bridge which has at least 2 adjacent pillar domains P_{1}, …, P_{k}. The intersection of the closures \overline{D} ∩ \overline{P}_{j} is an interval I_{j}. Choose a point x ∈ D and connect it by an arc w_{j} in D to I_{j}. For i ≠ j the arcs w_{i} and w_{j} should only intersect in x, so that w = ⋃ w_{j} is a T. tom Dieck 3. Examples 7

star-like contractible complex. Let W be a closed regular neighbourhood of w in \overline{D} such that \overline{P}_{j} ∩ W ⊂ I_{j}. Then the interior of W ∪ \overline{P}_{1} ∪ … ∪ \overline{P}_{k} is a new pillar domain. We apply this process successively to all domains of type D as above. The result is the reduced pillar bridge. The same result is obtained by a succession of moves (2.4) in any order.

The reader should draw a figure and follow the prescription above in the following example. The (12,12)-bridge

\frac{•|(•())| • ((•)(•))}{|(•)|((•((•))))}

has 4 domains D_{1}, D_{2}, D_{3}, D_{4} of the type D just considered, adjacent to 4,3,2,2 pillar domains, respectively. The reduction process yields

\frac{•()()()()()()}{()()()()()()}.

The algebra T E_{n,k}. It is here where the actual value of k matters. We consider the subalgebra T E_{n,k} of T E(n, k) which is generated by ordinary bridges (i. e. bridges without pillars) and a single further bridge e_{0} with only vertical arcs and with a single pillar between the kth and k + 1st arc. Additively, T E_{n,k} is the free 𝔎-module with basis a certain subset E_{n,k} of E(n, k). This definition is taylored to give the following result. The algebra T_{d}E_{n}(k) has been defined in the introduction.

(2.6) Theorem. There is a canonical surjective homomorphism of algebras γ_{n,k}: T_{d}E_{n}(k) → T E_{n,k}.

Proof. We have to specify the image of the generators e_{j}. They will be denoted with the same symbol. For j ≥ 1 we use the ordinary bridge with HE = 1, which connects j and j + 1 on top and on the bottom. The image of e_{0} is the vertical bridge with a pillar between the kth and k + 1st string. (Example (2.5) displays the case n = 2, k = 1.) One has to verify that γ_{n,k} respects the relations (1.1). This is easy. As far as e_{0} is involved, (2.3) and (2.4) are relevant. □

We show in section 5 that γ_{6,3} is an isomorphism. It is conjectured that γ_{n,k} is always an isomorphism. It is known that γ_{n,1} and γ_{n,2} are isomorphisms. For the latter see [3].

The algebra T E_{n,k}^{∗}. For completeness we define T E_{n,k}^{∗} to be the subalgebra of T E^{∗}(n, k) generated by ordinary bridges and the bridge e_{0} with only vertical arcs and a pillar at position k. Additively, it is the free 𝔎-module with basis a certain subset E_{n,k}^{∗} of E^{∗}(n, k).

3. Examples

We present some examples of (n, n)-bridges with basic number k for small values of n and k.

(3.1) n = 2, k = 1. The pillar bridges are:

\begin{matrix}\frac{||}{||} & \frac{| • |}{||} & \frac{()}{()} & \frac{(•)}{()} & \frac{()}{(•)} & \frac{(•)}{(•)} & . \\ 1 & e_{0} & e_{1} & e_{0}e_{1} & e_{1}e_{0} & e_{0}e_{1}e_{0} \end{matrix}

The right most element is not reduced. The algebra T E^{∗}(2, 1) is generated by e_{0} and e_{1} with relations

e_{0}^{2} = ce_{0}, e_{1}^{2} = de_{1}, e_{1}e_{0}e_{1} = fe_{1}.

This is an algebra of type T B_{2}; it was studied in [1, 2, 3]. In the reduced case T E(2, 1) we have the same generators but the relations are now

e_{0}^{2} = de_{0}, e_{1}^{2} = de_{1}, e_{1}e_{0}e_{1} = e_{1}, e_{0}e_{1}e_{0} = e_{0}.

This is the algebra T_{d}A_{2}. In this case we have equalities T E^{∗}(2, 1) = T E_{2,1}^{∗} and T E(2, 1) = T E_{2,1}.

(3.2) n = 2, k = 0. The pillar bridges are:

\begin{matrix}\frac{||}{||} & \frac{•||}{||} & \frac{||•}{||} & \frac{()}{()} & \frac{•()}{()} & \frac{•||•}{||} & . \\ 1 & e_{0} & f_{0} & e_{1} & e_{0}e_{1} & e_{0}f_{0} \end{matrix}

They form a basis of T E^{∗}(2, 0). The algebra T E(2, 0) = T E_{2,0} has the basis 1, e_{0}, e_{1}, e_{0}e_{1} = e_{1}e_{0} and is isomorphic to T_{d}A_{1} ⊗ T_{d}A_{1}.

(3.3) n = 3, k = 1. The algebra T E_{3,1} has the following basis.

\frac{|||}{|||}  \frac{()|}{()|}  \frac{|()}{|()}  \frac{()|}{|()}  \frac{|()}{()|}
1  e_{1}  e_{2}  e_{1}e_{2}  e_{2}e_{1}
\frac{| • ||}{|||} \frac{(•)|}{()|}  \frac{( )|}{(•)|}  \frac{(•)|}{|()}  \frac{()|}{| • ()}
e_{0}  e_{0}e_{1}  e_{1}e_{0}  e_{0}e_{1}e_{2} e_{1}e_{2}e_{0}
\frac{| • ()}{()|} \frac{|()}{(•)|}  \frac{| • ( )}{|()}  \frac{()|•}{()|}
e_{0}e_{2}e_{1} e_{2}e_{1}e_{0} e_{0}e_{2} = e_{2}e_{0} e_{1}e_{0}e_{2}e_{1}

This are 14 elements. The generators e_{0}, e_{1}, e_{2} satisfy the relations of T_{d}A_{3}, hence T A_{3} ≅ T E_{3,1}.

The unreduced algebra T E^{∗}(3, 1) has the additional elements

\frac{| • ||•}{|||}  \frac{(•)|}{(•)|}  \frac{| • ()}{(•)|}  \frac{(•)|}{| • ()}
e_{0}e_{1}e_{0}  e_{0}e_{2}e_{1}e_{0} e_{0}e_{1}e_{2}e_{0}
\frac{(•)|•}{()|}  \frac{()|•}{(•)|}  \frac{(•)|•}{(•)|}  .
e_{0}e_{1}e_{0}e_{2}e_{1} e_{1}e_{0}e_{2}e_{1}e_{0} e_{0}e_{1}e_{0}e_{2}e_{1}e_{0}

T. tom Dieck 4. The algebra T E_{6,3} 9

The first element only appears in T E^{∗}(3, 1). Hence there are six further elements in T E_{3,1}^{∗}. The generators e_{0}, e_{1}, e_{2} satisfy the relations

e_{0}^{2} = ce_{0}, e_{1}^{2} = de_{1}, e^{2} = de_{2}, e_{1}e_{0}e_{1} = fe_{1}, e_{0}e_{2} = e_{2}e_{0}.

This are relations for an algebra T B_{3} which was studied in [3].

(3.4) n = 4, k = 2. We display the reduced elements in the following manner. There are 14 pillarless (4, 4)-bridges. For each bridge we draw the possible pillars, but assemble all possibilities in a single figure. This yields all bridges with one pillar. Under the figure we write the number of the corresponding basis elements.

\frac{|| • ||}{||||}  \frac{() • ||•}{( )||}  \frac{•|(•)|•}{|(•)|}  \frac{•|| • ( )}{||()}
2  3  5  3
\frac{() • ||•}{|(•)|}  \frac{•|(•)|•}{( )||}  \frac{•|(•)|•}{||()}  \frac{•|| • ( )}{|(•)|}
4  4  4  4
\frac{() • ||•}{||()}  \frac{•|| • ()}{()||}  \frac{() • ()}{()()}  \frac{((•))}{() • ()}
3  3  2  3
\frac{() • ()}{((•))}  \frac{((•))•}{((•))}
3  4 + 1

There exists a single reduced element with 2 pillars. This accounts for the +1 in the last row. Altogether we obtain 48 elements. The generators

\frac{∥ • ∥}{∥∥} = e_{0}, \frac{()∥}{()∥} = e_{1}, \frac{|()|}{|()|} = e_{2}, \frac{∥()}{∥()} = e_{3}

satisfy the relations of the Temperley-Lieb-Algebra T_{d}D_{4}. The algebra T_{d}D_{4} has rank

rank T A_{3} + \frac{1}{2}rank T B_{4} − 1 = 14 + 35 − 1 = 48,

as was shown in [3]. One checks that the displayed bridges are contained in T E_{4,2}. Hence T E_{4,2} ≅ T_{d}D_{4}.

4. The algebra T E_{6,3}

As an application of the calculus we study the algebra T E_{6,3} and its module structure. Unfortunately, a lot of case by case checking is involved. It is also assumed that the reader knows how to deal graphically with the ordinary Temperley-Lieb algebra and its module theory. This will not be reviewed here. The algebra T E_{6,3} is generated by the ordinary (6, 6)-bridges and the vertical pillar bridge ||| • ||| = e_{0}. We use the following tools.

(4.1) Invariants of bridges.

(1) The horizontal edge number HE. In our case it is contained in {0, 1, 2, 3}.

(2) The number V P (b) of vertical pillar domains of a bridge b. Recall that a domain is called vertical if it has boundary points in both IR × 0 and IR × 1. The element e_{0} is vertical.

(3) The filtration of a bridge b is defined to be F (b) = HE(b) + V P (b). We see in a moment that 0 ≤ F (b) ≤ 3.

(4) The separation numbers SNT, SNB, SNL, SNR of a domain. We shall

see that 0 ≤ SNB + SNT ≤ 3 and SNL, SNR ∈ {1, 3} holds. ♡ We state properties of these invariants.

(4.2) The separation numbers mod 2 are topological invariants. Under multiplication SNR and SNL cannot increase. Since e_{0} has left and right separation number 3, we see that SNL, SNR ∈ {1, 3}. ♡

(4.3) For each domain of a bridge either SNT = 0 or SNB = 0. A separation of a domain from IR × 0 is achieved by an upper horizontal arc. Therefore SNT equals at most the maximal possible HE. In our case SNT can assume the values 0, 1, 2, 3. ♡

(4.4) The HE cannot decrease under multiplication. If a vertical pillar domain exists, then it is separated from left and right infinity by one or three vertical arcs. Hence there exist 6, 4, 2, or 0 vertical arcs. These cases lead to the distribution of pillars as displayed in the following table.

\begin{matrix}Pillar & HE & F \\ ||| • ||| & 0 & 1 \\ | • ||| & 1 & 2 \\ ||| • | & 1 & 2 \\ | • | & 2 & 3 \end{matrix}

The reduction condition says that further pillars are impossible. ♡

In table 1 we display in bracket notation the possible configurations of pillar domains and vertical arcs. The configurations have to be filled up with ordinary horizontal arcs. Under each symbol we have specified the number of fillings. Each such number is the product of the upper and lower fillings. They are obtained from well known combinatorics of bridges. Certain cases in the table appear twice, namely if interchange of top and bottom yields different configurations. This accounts for the 2 in front of a symbol. As an example, consider the left most entry in the first row. There are 5 6-bridges with 3 horizontal arcs; they are the possible “denominators”. There are 9 possibilities to fill these 5 bridges with a pillar such that SNL = 1; they are the possible “numerators”. The next entry in this line has 9 denominators and 5 numerators. The last one has 5 denominators. The right most column in the table gives the total number of cases in the corresponding row. We have to add the 132 ordinary bridges without pillars. Altogether we obtain

132 + 190 + 107 + 110 + 44 + 68 + 12 + 30 = 703.

But not all cases are possible in T E_{6}(3)! For one, the right separation numbers have to be at most 3. This eliminates the right most cases in the first two rows, hence 11 cases.

Moreover, the cases in the last row are impossible. This has the following reason. We have to investigate which figures are generated by ordinary bridges and e_{0}. The cases in question have to use e_{0} at least twice. (Recall: Left multiplication is placed graphically on top of the figure.) From a lower pillar domain we can obtain at most a vertical one by multiplication. In the first case (left in the last row of the table) we have, in the course of the multiplication with generators, to multiply a bridge with an upper pillar domain with e_{0}. But then the reduction condition is not satisfied, and therefore a second pillar domain does not occur.

Therefore a second pillar domain can only appear, if the upper foot

[figure]

has distance at least 4 from the already existing pillar domain. This can only be a lower pillar domain. In cases 2 and 3 of the last row the HE can be at most 1. Hence a fourfold separation is impossible. What we have seen until now is

(4.5) The algebra T E_{6,3} has rank at most 662. □

In order to show that the rank is exactly 662 one could try to check that all remaining figures are actually possible. This is a matter of patience, if one does not use further structural investigations. But it turns out that the module and ideal structure of T E_{6,3} gives better insight. We will show in the remaining part of this section:

(4.6) Theorem. The algebra T E_{6,3} is semi-simple for generic parameters d in a field 𝔎 and has in that case simple modules of rang 1, 6, 20, and 15.

We point out 662 = 1^{2} + 6^{2} + 20^{2} + 15^{2}. Note that the ranks are binomial coefficients \binom{6}{n}. In the non generic case we too have modules of the specified rank, since the modules are constructed with help of the graphical calculus. We assume from now on that the parameter d ∈ 𝔎 is invertible. We also assume known a geometric treatment of the module theory for the ordinary Temperley-Lieb algebra T_{d}A_{5}.

Again we use bracket notation for the upper pillar bridges. A bullet indicates the pillar domain. The standard bridges with horizontal edge number 0,1, 2, 3 are denoted β_{0}, β_{1}, β_{2}, β_{3}; the upper and lower bridges of these pillarless bridges are, by definition, ||| |||, ||()||, |(())|, ((())). The simple modules will be the left ideals generated by the β_{j} modulo bridges of higher filtration. This uses:

(4.7) Lemma. Multiplication cannot decrease the filtration.

Proof. A vertical pillar domain can only be removed by multiplication with an ordinary bridge if its vertical boundary points become connected. But then the HE increases.

If multiplication by e_{0} decreases the HE, then this happens through a reduction process which produces a vertical pillar domain. □

(4.8) Theorem. The element β_{3} generates a left ideal M(3) of rank 15 and a twosided ideal L(3) of rank 15^{2}. The ideal L(3) is the direct sum of 15 left ideals isomorphic to M(3). The upper bridges of a basis of M(3) are displayed in the following table.

\begin{matrix}((())) & ()(()) & (())() & ()()() & (()()) \\ (((•))) & ()(•()) & (()•)() & ()(•)() & (() • ()) \\ (•(())) & (•)(()) & (())(•) & (•)()() & ()()(•) \end{matrix}

Proof. The first row contains the upper bridges of the pillarless elements of M(3). The second row contains the elements which are obtained from the first row by left multiplication with e_{0}. The third row contains the elements

e_{3}z, e_{4}e_{3}x, e_{2}e_{3}y, e_{3}x, e_{3}y,

where x, y, and z are the second, third, and fifth element of the second row. Further elements are impossible; this is seen by considering reduction and separation number.

By (4.7), the bridges with maximal filtration 3 generate a twosided ideal which contains M(3).

We reflect the basis bridges of M(3) in IR× \frac{1}{2} (interchange of top and bottom). The left ideals generated by these reflected elements are isomorphic to M(3). An isomorphism is obtained by right multiplication with a suitable element: If b is a basis element, then there exists a bridge c such that bc = λβ_{3} with invertible λ ∈ 𝔎; moreover β_{3}b = μβ_{3} with invertible μ ∈ 𝔎. The sum of these left ideals contains L(3). The sum is direct, since the basis sets of the ideals are pairwise disjoint. The element β_{3} generates L(3), since it generates the above left ideals.□

(4.9) Theorem. Let M(2) denote the left ideal generated by β_{2} modulo the submodule generated by bridges of higher filtration. The module M(2) has rank 20 and is generated by bridges with lower bridge |(())| and filtration 2. The twosided ideal generated by β_{2} modulo L(3) is an algebra of rank 20^{2} and is the direct sum of 20 ideals isomorphic to M(2).

Proof. By definition, the module M(2) is generated by the bridges as stated in the theorem. One has to check that there 20 of them. The possible pillar bridges without vertical pillar domain are in condensed notation displayed in the following table.

|((•))|  |()()|  |()|(•) (•)|()| (•)||(•)
||(•)(•) (•)(•)|| (•())|| ||(()•)

Condensed notation means: all possible occurances of pillars are assembled in the same figure. It remains to verify that they are generated by the 9 ordinary bridges among them and e_{0}. Left multiplication of ordinary bridges with e_{0} produces

|((•))|, ||(•)(), ()(•)||, (()•)||, ||(•()).

Finally, we multiply the last two with e_{3}, e_{4}, and e_{5}.

We now finish the proof as for (4.8). □

Finally, we use a similar procedure for β_{0} and β_{1}. The first one yields the trivial module M(0), the second one the 6-dimensional module M(1). We mention in passing that M(1) is obtainable from the 6-dimensional reflection representation of the Hecke algebra (see [3]).

This finishes the discussion of the module theory. We do not discuss in detail conditions for semi-simplicity. By (2.6) and the homomorphism φ of the introduction, T E_{n,k} is a quotient of the Hecke algebra H_{q}E_{n}(k), provided q = p^{2}, d = p + p^{−1}. Thus T E_{n,k} is certainly semi-simple when this holds for the Hecke algebra. The simplicity of the modules M(j) follows from the next theorem if one assumes known the theory of T_{d}A_{n}-modules. The argument for M(3) is as follows (notation as in (4.10)). Suppose M(3) has a decomposition A ⊕ B into submodules. Then resA or resB contains the uniquely determined submodule M_{3}, say β_{3} ∈ A. But β_{3} generates M(3). The other modules are handled similarly. □

We have the subalgebra T_{d}A_{5} ⊂ T E_{6,3} generated by the ordinary bridges. From the structure theory of Temperley-Lieb algebras we know (see e. g. [4, 2.8]): The algebra T_{d}A_{5} has in the generic case simple modules M_{0}, M_{1}, M_{2}, and M_{3} of rank 1, 5, 9, and 5, respectively. We denote the restriction of a T E_{6,3}-module M to T A_{5} simply by resM.

(4.10) Theorem. The following isomorphisms hold:

resM(0) ≅ M_{0}
resM(1) ≅ M_{1} ⊕ M_{0}
resM(2) ≅ M_{2} ⊕ 2M_{1} ⊕ M_{0}
resM(3) ≅ M_{3} ⊕ M_{2} ⊕ M_{0}.

Proof. The case M(0) is trivial. We start with M(3) and consider the bridges of filtration 3 described above. The ordinary bridges yield a submodule M_{3} of rank 5. We study M(3)/M_{3} and consider in it the element

z = \frac{()(•)()}{()()()}.

If we multiply this element from the right by e_{0} we obtain β_{2}. Multiplication from the left with ordinary bridges yields all ordinary upper (2, 2)-bridges, and there are 9. Therefore z generates in M(3)/M_{3} a 9-dimensional module isomorphic to M_{2}. Its quotient module has a basis (((•))).

We treat M(2) in a similar manner. We have the submodule M_{2} of ordinary bridges. The remaining 11 basis elements are displayed in the following table.

||(•)() ||()(•) ||(()•) ()||(•) |()|(•)
()(•)|| (•)()|| (•())|| (•)||() (•)|()|
|((•))|

We claim that the rows yield modules M_{1}, M_{1}, M_{0}, respectively. If we use the lower bridge ||()() in the first row and multiply with e_{0} from the right we obtain the upper bridges with HE = 1. Similarly for the second row by using the lower bridge ()()||.

The case M(1) finally is clear, since we have the submodule of ordinary bridges M_{1}. □

\begin{matrix}2\underline{(•)} & 2\frac{(•)||}{||} & 2\frac{(•)||||}{||||} & 190 \\ 45 & 45 & 5 \\ \frac{| • |}{||} & \frac{| • |||}{||||} & \frac{| • |||||}{||||||} & 107 \\ 81 & 25 & 1 \\ 2\underline{(((•)))} & 2\frac{||(•)}{||} & 2\frac{||(•)||}{|| ||} & 110 \\ 5 & 45 & 5 \\ 2\frac{|((•))}{|} & \frac{||| • |}{||||} & \frac{||| • |||}{||| |||} & 44 \\ 9 & 25 & 1 \\ 2\frac{(((•)))}{(•)} & 2\frac{|((•))|}{(•)||} & 2\frac{||(•)}{(•)||} & 68 \\ 9 & 5 & 25 \\ \frac{(((•)))}{(((•)))} & 2\frac{||(•)}{|((•))|} & \frac{|((•))|}{|((•))|} & 12 \\ 1 & 5 & 1 \\ 2\frac{(•)||(•)}{||} & 2\frac{(•)||| • |}{||||} & 2\frac{||(•)||}{(•)||} & 30 \\ 9 & 5 & 1 \end{matrix} Table 1

5. Comparison with the algebra T_{d}E_{6}(3)

The algebra T_{d}E_{6}(3) is the Temperley-Lieb algebra associated to the Coxeter graph E_{6}(3) defined by generators and relations, see the introduction. By construction, we have a surjective homomorphism T_{d}E_{6}(3) → T E_{6,3}, see (2.6). We already know the rank of T E_{6,3}. In order to show that the homomorphism is injective we derive a normal form for words in e_{0}, …, e_{5} and verify that there are 662 normal forms. We have different cases according to the appearances of e_{0} in the words.

(5.1) We have the subalgebra T_{d}A_{5} of rank 132 generated by e_{1}, …, e_{5}.

(5.2) We determine the normal form of monomials of the form αe_{0}β with α, β ∈ T_{d}A_{5}. Monomial means: an arbitrary word in the symbols e_{j}, possibly with a further coefficient from 𝔎. We only consider monomials which cannot be shortened by the defining relations and which have α of minimal length in its equivalence class modulo relations. The minimality of α means:

(1) α = 1 or

(2) α contains e_{3} and finishes on the right with e_{3}. Left to α there are no

generators commuting with e_{3}, i. e. no e_{1} and e_{5}. We use the fact that T_{d}A_{5} can be described by bridges. Thus, we talk about bridges instead of words or monomials. Bridges with α satisfying these conditions have lower bridges ||()||, |(())|, ((())). The number of α′s with these lower bridges is 5, 9, 5, respectively. They are displayed in table 2. The second column in this table gives the word in the generators, with 243 as shorthand for e_{2}e_{4}e_{3}. The third column gives the number of possible β′s. We explain this in a moment. Elements α which are related by the reflection e_{j} ↔ e_{6−j} yield the same number. We now go through the cases of table 2 and derive an upper bound for the possible β’s. In the following discussion we eliminate those β’s which obviously lead to words which can be shortened.

α = e_{3}. The β which begin with e_{3} lead to words which can be shortened by the relation e_{3}e_{0}e_{3} = e_{3}. The β’s which begin with e_{3} have an upper bridge which contains ||()||. These are in bijection with the 42 upper 10-bridges. There remain 132 − 42 = 90 cases.

α = e_{4}e_{3}. No beginning with ||()|| or |||()|. These are disjoint cases. Hence there remain 132 − 2 · 42 = 48.

α = e_{5}e_{4}e_{3}. No beginning with ||()||, |||()| or ||||(). The first and third case have as intersection the bridges which begin with ||()(). These are 14 in number. There remain 132 − 3 · 42 + 14 = 20.

α = e_{2}e_{5}e_{4}e_{3}. No beginning with |()|||, ||()||, |||()| or ||||(). The first and third case have intersection |()()|; the second and fourth case have intersection ||()(); the first and fourth case have intersection |()()|(). There are no further intersections. There remain 132 − 4 · 42 + 3 · 14 = 6.

T. tom Dieck 5. Comparison with the algebra T_{d}E_{6}(3) 17

The remaining α′s contain all symbols e_{1}, …, e_{5} and therefore β has to be 1. The sum of all these cases is 440.

(5.3) Monomials which contain two e_{0}: αe_{0}βe_{0}γ. In order that βe_{0}γ be minimal with minimal β, the element β has, by (5.2), to terminate with β = ||()||, |(())|, or ((())). Hence β has to be one of the bridges β_{2} and β_{3} of section 2. In the second case α = γ = 1.

In the first case therefore, by (5.2), either α = 1 and 20 cases for γ, or γ = 1 and 20 cases for α, minus the intersection case α = γ = 1. Altogether we obtain 40 cases.

If α ≠ 1 and γ ≠ 1, then γ has to begin with e_{1}, e_{5} and α has to end with e_{1}, e_{5}; and these cases have to be different. Hence we obtain altogether 2N^{2} possibilities, where N is the number of γ which begin with e_{1} but not with e_{2}, e_{3}, e_{4} and which have no presentation beginning with e_{5}. These are the five bridges which begin with ()|||. Thus we obtain 50 further cases.

Altogether we now have the upper bound 132+440+40+50 = 662 on normal forms of words. Since e_{0} can occur at most twice, there are no further cases. We already have the surjection T_{d}E_{6}(3) → T E_{6,3} onto an algebra of rank 662. Thus we found normal forms of words.

Upper bridge  Word  Number of β

||()||  3  90
|||()||  43  48
|()|||  23  48
|| ||()  543  20
()||||  123  20

|()()|  243  20
|()|()  2543  6
()|()|  1243  6
|(())|  3243  20
||()()  32543  6
()()||  13243  6
()||()  12543  1
||(())  432543  6
(())||  213243  6

()()()  132543  1
()(())  1432543  1
(())()  2132543  1
(()())  21432543  1
((()))  321432543  1

Table 2

T. tom Dieck 6. Concluding remarks 19

6. Concluding remarks

As one might guess from the shape of the Coxeter graphs, the following isomorphisms hold T E_{n,1} ≅ T A_{n} and T E_{n,2} ≅ T D_{n}. The structure of T D_{n} was determined in [3]. The calculus of the present paper adds a geometric interpretation to the algebra of [3]. In [1] we defined and studied the Temperley-Lieb algebra T B_{n} associated to the Coxeter graph B_{n}. With suitable parameters, the algebras T E_{n,1}^{∗} and T B_{n} are isomorphic. The algebra T B_{n} uses bridges which are symmetric with respect to a reflection in 0 × IR. One can also consider pillar bridges which have this symmetry. A certain algebra of this type is related to the graph F_{4}. Finally, one could use pillars of different type; this has applications to affine root systems. The usual closing procedure (braids to links) yields when applied to pillar bridges a semi-geometric definition of Markov traces on the family T E_{n,k} for fixed k.

References

1. tom Dieck, T.: Symmetrische Brücken und Knotentheorie zu den Dynkin-Diagrammen vom Typ B. J. reine angew. Math. 451, 71 – 88 (1994).

2. tom Dieck, T.: Knotentheorien und Wurzelsysteme. Teil II. Mathematica Gottingensis 44 (1993).

3. tom Dieck, T.: Knot theories and root systems. Part III. Mathematica Gottingensis 22 (1994).

4. Goodman, F. M., de la Harpe, P., and V. F. R. Jones: Coxeter graphs and towers of algebras. New York – Berlin, Springer 1989.

5. Kauffman, L. H.: Knots and physics. Singapore, World Scientific 1992.

Version of February 9, 2009

Tammo tom Dieck Mathematisches Institut Bunsenstraße 3/5 D – 37073 Göttingen tammo@uni-math.gwdg.de