Temperley-Lieb algebras associated to the root system D

Tammo tom Dieck

Abstract In this note we define and compute the Temperley-Lieb algebras associated to the Coxeter–Dynkin graphs of type D_{n}. The computation relates these algebras to those corresponding to the root systems of type A and B. We also show the connection to braid theory and to the Kauffman bracket and describe a related graphical calculus.

AMS classification: 57M15; 57M20; 20F36

Keywords: Temperley-Lieb algebras; Hecke algebras; Graphical calculus; Knot algebra; Root systems; Braid groups; Towers of algebras.

1. Hecke algebras and Temperley-Lieb algebras

This section collects some general results. Let S be a finite set. A Coxeter matrix is a symmetric mapping m: S×S → IN∪{∞} such that m(s, s) = 1 and m(s, t) ≥ 2 for s ≠ t. A Coxeter matrix (S, m) is often specified by its Coxeter graph Γ(S, m). It has S as its set of vertices and an edge with weight m(s, t) whenever m(s, t) ≥ 3. Usually, the weight m(s, t) = 3 is omitted from the notation.

The standard Hecke algebra H_{q}(S, m) associated to a Coxeter matrix (S, m) is the associative algebra with 1 over the commutative ring K with generators (x_{s} | s ∈ S) and relations

(1.1) \begin{matrix}x_{s}^{2} & = & (q − 1)x_{s} + q, & q ∈ K^{∗} \\ x_{s}x_{t}x_{s} … & = & x_{t}x_{s}x_{t} …, & m(s, t) ≥ 2 \end{matrix}

(m(s, t) factors on each side, alternating). Here K^{∗} denotes the unit groups of K.

Suppose m takes values in {1, 2, 3}. Then the Temperley-Lieb algebra T_{d}(S, m) is the associative algebra with 1 over K with generators (e_{s} | s ∈ S) and relations

(1.2) \begin{matrix}e_{s}^{2} & = & de_{s} & d ∈ K^{∗} \\ 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}

We shall obtain the Temperley-Lieb algebra as a quotient of a Hecke algebra. For this purpose we assume (1.3) p ∈ K^{∗}, q = p^{2}, d = p + p^{−1}.

(1.4) Proposition. Under the hypothesis (1.3) the assignment x_{s} ↦ pe_{s} − 1 defines a surjective homomorphism φ: H_{q}(S, m) → T_{d}(S, m). The kernel of φ is the twosided ideal generated by the elements x(s, t) = x_{s}x_{t}x_{s} + x_{s}x_{t} + x_{t}x_{s} + x_{s} + x_{t} + 1; here (s, t) runs over the pairs (s, t) with m(s, t) = 3.

Proof. (Compare [??, 2.11].) One verifies easily that φ respects the defining relations of the Hecke algebra. Certainly, φ is surjective. Let I ⊂ H_{q}(S, m) denote the ideal generated by the x(s, t) for (s, t) with m(s, t) = 3. We define a homomorphism ψ: T_{d}(S, m) → H_{q}(S, m)/I by ψ(e_{s}) = p^{−1}(x_{s} + 1). One verifies that this is compatible with (1.2) and that x(s, t) is contained in the kernel of φ. Hence φ induces φ: H_{q}/I → T_{d}. By construction, φ and ψ are inverse homomorphisms. □

The preceding construction can, in particular, be applied to Coxeter matrices of ADE-type. The resulting algebras are then finite dimensional. The structure of T A_{n−1} associated to the linear graph A_{n−1} with n vertices is well known, see [??]; this is the classical Temperley-Lieb algebra. In the following sections 2 and 3 we study the algebras related to the graph D_{n} with n vertices (n ≥ 4). In section 4 we briefly discuss D-tangles and the associated Kauffman functor.

But first we present one general result: By way of example we show that T_{d}(S, m) is non-zero. This is done by constructing a standard module which arises from the reflection representation of the Hecke algebra.

We work with a field K. Let V denote the free K-module with basis {v_{s} | s ∈ S}. We define a symmetric bilinear form B on V by

B(v_{s}, v_{s}) = q + 1
B(v_{s}, v_{t}) = p  m(s, t) = 3
B(v_{s}, v_{t}) = 0  m(s, t) = 2.

We define a linear map X_{s}: V → V by X_{s}(v) = qv_{s} − B(v_{s}, v)v. Then X_{s}(v_{s}) = −v_{s}, and X_{s}(v) = v for v in the orthogonal complement of v_{s}. We assume q +1 ∈ K^{∗}. Then V is the orthogonal direct sum of Kv_{s} and (Kv_{s})^{⊥}. On the latter, X_{s} acts as multiplication by q. Hence X_{s} satisfies the quadratic equation X_{s}^{2} = (q − 1)X_{s} + q of the Hecke algebra.

The determinant d_{s,t} of B on the submodule ⟨ v_{s}, v_{t} ⟩ generated by v_{s} and v_{t} equals

d_{s,t} = \{ \begin{matrix}(q + 1)^{2} & m(s, t) = 2 \\ q^{2} + q + 1 & m(s, t) = 3. \end{matrix}

We therefore also assume q^{2} + q + 1 ∈ K^{∗}. Then V is the orthogonal direct sum of ⟨ v_{s}, v_{t} ⟩ and ⟨ v_{s}, v_{t} ⟩^{⊥}. On the latter subspace, X_{s} and X_{t} act as multiplication by q. The action of X_{s} and X_{t} on ⟨ v_{s}, v_{t} ⟩ in the basis v_{s}, v_{t} is given by

X_{s} = ( \begin{matrix}−1 & p \\ 0 & q \end{matrix} ) , X_{t} = ( \begin{matrix}q & 0 \\ p & −1 \end{matrix} ) ,

T. tom Dieck 1. Hecke algebras and Temperley-Lieb algebras 3

in the case m(s, t) = 3. A simple computation shows

X_{s}X_{t}X_{s} = X_{t}X_{s}X_{t} = ( \begin{matrix}0 & −pq \\ −pq & 0 \end{matrix} ) .

Thus we have constructed the reflection representation V of H_{q}(S, m).

The assignment ω: H_{q}(S, m) → H_{q}(S, m), x_{s} ↦ −qx_{s}^{−1} is an involutive automorphism of the Hecke algebra. It transforms V into a new module W = V^{ω}.

(1.5) Proposition. The module W factors over the homomorphism φ of (1.4).

Proof. We set Y_{s} = −qX_{s}^{−1}. We have to show that the operator

Y_{s,t} = Y_{s}Y_{t}Y_{s} + Y_{s}Y_{t} + Y_{t}Y_{s} + Y_{s} + Y_{t} + 1

acts on V as the zero map. We compute that Y_{s} + 1 and Y_{t} + 1 act on ⟨ v_{s}, v_{t} ⟩ in the basis v_{s}, v_{t} through the matrices

Z_{s} = ( \begin{matrix}q + 1 & −p \\ 0 & 0 \end{matrix} ) , Z_{t} = ( \begin{matrix}0 & 0 \\ −p & q + 1 \end{matrix} ) .

This is used to verify on ⟨ v_{s}, v_{t} ⟩ the relation Z_{s}Z_{t}Z_{s} = qZ_{s}. A formal calculation, using the quadratic equation for Y_{s}, yields

(Y_{s} + 1)(Y_{t} + 1)(Y_{s} + 1) − q(Y_{s} + 1) = Y_{s,t}.

Therefore Y_{s,t} acts as zero on ⟨ v_{s}, v_{t} ⟩. Since X_{s} is multiplication by q on ⟨ v_{s}, v_{t} ⟩^{⊥}, we see that −qX_{s}^{−1} + 1 is the zero map. □

We give a more direct construction of a T_{d}(S, m)-module which does not use the reflection representation of the Hecke algebra. Let A = (a_{st}) denote a symmetric S × S-matrix over K. We consider the associative algebra T (A) over K with generators (Z_{s} | s ∈ S and relations

\begin{aligned}Z_{s}^{2} & = & a_{ss}Z_{s} \\ Z_{s}Z_{t}Z_{s} & = & a_{st}a_{ts}Z_{s}. \end{aligned}

Then a simple verification from the definitions gives:

(1.6) Proposition. Let V be the K-module with basis (v_{s} | s ∈ S. The operators Z_{s}(v_{t}) = a_{st}v_{s} make V into a T (A)-module. (Hence each Z_{s} has rank at most one on V.) □

The matrix A = (a_{st}) is called indecomposable, if there is no partition S = S_{1} ∐ S_{2} with a_{uv} = 0 for u ∈ S_{1}, v ∈ S_{2}.

(1.7) Proposition. Let K be a field. Suppose A is indecomposable and det(A) ≠ 0. The the module V of the previous proposition is a simple T (A)- module.

Proof. We have Z_{s}(∑_{j} a_{j}v_{j} = (∑_{j} a_{j}a_{sj})v_{s}. Suppose v = ∑_{j} a_{j}v_{j} ≠ 0. Since det(A) ≠ 0, not all Z_{s}v are zero. If 0 ≠ M ⊂ V is a T (A)-submodule, then there exists s ∈ S with v_{s} ∈ M. Suppose v_{t} ∉ M. Since Z_{t}v_{s} = a_{ts}v_{t} ∈ M, we must have a_{ts} = 0. This contradicts the indecomposability of A. Hence all v_{t} are containes in M. □

In the case of a Coxeter graph, we set a_{ss} = d, a_{st} = 1 for m(s, t) = 3, and a_{st} = 0 for m(s, t) = 2. Then V becomes a module over T_{d}(S, m). Also, det(A) is a non-trivial monic polynomial in d, hence in general not zero.

2. The structure of T D_{n}

The algebra T D_{n} with parameter d ∈ K is associated to the graph D_{n}. In the following figure we have specified the names of the generators.

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

The algebra T D_{n} will be decomposed into an algebra which belongs to the linear graph A_{n−1} and another algebra which is related to the graph B_{n}. Here A_{n−1} is the linear Coxeter graph with n − 1 vertices e_{1}, …, e_{n−1} and m(e_{j}, e_{j+1}) = 3. We use the same notation for the generators of T D_{n} and T A_{n−1}. The following is easily verified.

(2.1) Proposition. The assignment e_{0} ↦ e_{1} and e_{j} ↦ e_{j} (j ≥ 1) defines a surjective homomorphism α: T D_{n} → T A_{n−1}. □

We remark that the automorphism of the graph D_{n} which interchanges e_{0} and e_{1} and fixes e_{j} for j ≥ 2 induces an involution τ: T D_{n} → T D_{n}. We have ατ = α.

(2.2) Proposition. The kernel of α is the twosided ideal I generated by the difference e_{0} − e_{1}. The homomorphism α_{1}: T A_{n−1} → T D_{n}, e_{1} ↦ e_{j} is right inverse to α. We therefore have a splitting of modules T D_{n} = I ⊕ T A_{n−1}.

Proof. The inclusion I ⊂ kernel α follows from the definitions. The relation αα_{1} = id is obvious. The composition α_{1} ∘ α is easily seen to be be the identity on generators. Hence I = kernel α. □

We now use the following algebra T′D_{n} of Temperley-Lieb type: It has generators ε_{0}, …, ε_{n−1} and relations

T. tom Dieck 2. The structure of T D_{n} 5

(2.3) \begin{matrix}ε_{j}^{2} & = & dε_{j} & j ≥ 1 \\ ε_{0}^{2} & = & 2ε_{0} \\ ε_{i}ε_{j} & = & ε_{j}ε_{i} & |i − j| ≥ 2 \\ ε_{1}ε_{0}ε_{1} & = & dε_{1} \\ ε_{i}ε_{j}ε_{i} & = & ε_{i} & |i − j| = 1; i, j ≥ 1. \end{matrix}

This is a variant of the algebra of B_{n} type which has been studied in [??],[??].

(2.4) Proposition. The assignment β(e_{0}) = (ε_{0} − 1)ε_{1}(ε_{0} − 1) and β(e_{j}) = ε_{j} for j ≥ 1 defines a homomorphism β: T D_{n} → T′D_{n}.

Proof. For j ≥ 1, the e_{j} and ε_{j} satisfy the same relations; we consider the remaining ones. We use (ε_{0} − 1)^{2} = 1, ε_{1}(ε_{0} − 1)ε_{1} = 0, and verify easily β(e_{0}^{2}) = dβ(e_{0}) and β(e_{0}e_{2}e_{0}) = β(e_{2}e_{0}e_{2}). Moreover β(e_{0}e_{1}) = 0 = β(e_{1}e_{0}). □

The proof of the previous proposition shows that the twosided ideal J generated by e_{0}e_{1} is contained in the kernel of β. The image of β will turn out to be half of T′D_{n}, and J is equal to the kernel of β. In order to prove these statements we introduce the crossed product of T D_{n}/J =: A with the algebra K[τ]/(τ^{2} − 1) where τ acts via the previously defined involution τ on A. Formally, this crossed product B is defined as the free A-module with basis 1, τ and multiplication

(a + bτ) · (c + dτ) := (ac + bd^{τ}) + (bc^{τ} + ad)τ,

where x^{τ} denotes the action of τ on x. Note that the ideal J is τ-stable.

(2.5) Proposition. The assignment ε_{j} ↦ ε_{j} (j ≥ 1) and ε_{0} ↦ 1 + τ defines an isomorphism β_{1}: T′D_{n} → B. The image of β corresponds to the subalgebra A. The kernel of β is equal to J.

Proof. The relation ε_{0}^{2} = 2ε_{0} corresponds to (1 + τ)^{2} = 1 + 2τ + 1 = 2(1 + τ). The element ε_{1}ε_{0}ε_{1} is mapped to e_{1} · (1 + τ) · e_{1} = (e_{1} + e_{1}τ) · e_{1} = e_{1}^{2} + e_{1}e_{0}τ and this equals de_{1} modulo J. We see that β_{1} is well-defined and surjective. The inverse homomorphism is given by β and τ ↦ ε_{0} − 1. □

The algebras T′D_{n} and T A_{n−1} have augmentation homomorphisms to K which map the generators ε_{j} and e_{j} to zero. Let T′′D_{n} denote the image of β. We have a sequence

(2.6) 0 → T D_{n} ⟶^{α,β} T A_{n−1} ⊕ T′′D_{n} → K → 0;

the map to K is the difference of the augmentations. The structure of T D_{n} will be obtained from the next result.

(2.7) Theorem. The sequence (2.6) is exact.

Proof. We show that α maps the kernel J of β isomorphically onto the kernel of the augmentation. For this purpose we recall a basis of T A_{n−1}, see [??]. We set

e(i, j) = e_{i}e_{i−1} … e_{j}, i ≥ j.

Then a basis of T A_{n−1} is given by the products

(2.8) (i_{1}, j_{1}) · . . . · e(i_{p}, j_{p})

with

\begin{matrix}1 ≤ i_{1} < … < i_{p} ≤ n − 1, & 1 ≤ j_{1} < … < j_{p} ≤ n − 1 \\ j_{s} ≤ i_{s}, & 0 ≤ p ≤ n − 1. \end{matrix}

We exhibit a basis of J which is mapped onto this basis. We use the notation

ē(i, j) = e_{i}e_{i−1} … e_{1}e_{0}e_{2} … e_{j}, j ≥ 2.

This element is mapped under α to de(i, j). Recall that d ∈ K^{∗} is a unit.

We show that J is spanned by the elements of the form (2.8) where e(i_{1}, j_{1}) is replaced by ē(i_{1}, j_{1}); this finishes the proof of (2.7).

We consider words in the symbols e_{0}, …, e_{n−1}. An elementary reduction of a word is one of the following replacements: e_{j}e_{j} by de_{j}, e_{i}e_{j}e_{i} by e_{i}, e_{i}e_{j} by e_{j}e_{i}. Note that the length of the word is not increased. A coefficient of the form d^{r} may appear. A word is in reduced form, if it cannot be shortened by an elementary reduction. The words (2.8) are reduced. Certainly, J is generated by reduced words.

We claim that J is generated by reduced words of the form ae_{0}e_{1}b in which a and b do not involve e_{0} and e_{1}.

We know already that J is the ideal generated by words ce_{0}e_{1}d. If d contains e_{1}, say, then the word contains a string of the form e_{1}xe_{1} in which x involves only e_{j}, j ≥ 2. A word of this type is never reduced; this follows easily by using (2.8) for x. This shows the claim.

We next consider normal forms of reduced words in J by induction on n. Suppose a reduced word contains two factors e_{n−1}, say a string e_{n−1}ye_{n−1} with y not involving e_{n−1} and of shortest length. Then, by induction, this string must equal ē(n−1, n−1). If a word contains z = ē(n−1, n−1), it is not reduced, unless it is equal to z. Therefore z is the only reduced word in J with two appearances of e_{n−1}. Next, consider reduced words which have the form w = xe_{n−1}y. By interchanging elements, if necessary, we assume that y has minimal length. Then y necessarily has the form e(n − 2, j) or ē(n − 2, j). Since x does not contain e_{n−1}, we can apply the induction hypothesis to x. Since w is reduced, it is easily seen that w has the form (2.8) with e(i_{1}, j_{1}) replaced by ē(i_{1}, j_{1}). □

We assume known the structure of T A_{n−1} in the generic case (q not a root of unity) [??]. It remains to study the algebra T′′D_{n}. This is the subject of the next section.

We conclude this section with some remarks concerning the algebra of the graph D_{n}: braid groups and Hecke algebras.

Each Coxeter matrix (S, m) has associated to it a braid group Z(S, m) with generators (x_{s} | s ∈ S) and relations x_{s}x_{t}x_{s} … = x_{t}x_{s}x_{t} … with m(s, t) factors on each side. For the graph D_{n} we define another braid group Z′D_{n} with generators κ_{0}, …, κ_{n−1} and relations

(2.9) \begin{matrix}κ_{i}κ_{j}κ_{i} & = & κ_{j}κ_{i}κ_{j} & |i − j| = 1; i, j ≥ 1 \\ κ_{0}κ_{1}κ_{0}κ_{1} & = & κ_{1}κ_{0}κ_{1}κ_{0} \\ κ_{i}κ_{j} & = & κ_{j}κ_{i} & |i − j| ≥ 2 \\ κ_{0}^{2} & = & 1. \end{matrix}

This is a quotient of the group ZB_{n} for which the last relation is not present.

(2.10) Proposition. The group Z′D_{n} is the semidirect product of ZD_{n} with ZZ/2. The generator τ of ZZ/2 acts on ZD_{n} by the automorphism induced by the graph automorphism.

Proof. Let G denote the semi-direct product. We define inverse homomorphisms f: G → Z′D_{n} and g: Z′D_{n} → G by

f: τ, x_{0}, x_{1}, …, x_{n−1} ↦ κ_{0}, κ_{1}κ_{0}κ_{1}, κ_{1}, …, κ_{n−1}
g: κ_{0}, κ_{1}, …, κ_{n−1} ↦ τ, x_{1}, …, x_{n−1}.
□

We remark that conjugation by κ_{0} corresponds to τ.

We define the Hecke algebra H′D_{n} as the associative algebra with 1 generated by κ_{0}, …, κ_{n−1} with braid relations as above and quadratic relations κ_{0}^{2} = 1 and κ_{j}^{2} = (q − 1)κ_{j} + q for j ≥ 1. This is a Hecke algebra of B_{n}-type where the parameter Q belonging to κ_{0} has been specialized to 1. We have an embedding α̃: HD_{n} → H′D_{n}, x_{0} ↦ κ_{0}κ_{1}κ_{0}, x_{j} ↦ κ_{j} for j ≥ 1. As in the case of the Temperley-Lieb algebras we see:

(2.11) Proposition. The algebra H′D_{n} is the crossed product of HD_{n} with K[τ]/(τ^{2} − 1). □

There is a connection between Hecke algebras and Temperley-Lieb algebras as follows.

(2.12) Proposition. The algebra T′D_{n} is a quotient of H′D_{n} under the homomorphism φ′: κ_{0} ↦ ε_{0} − 1, κ_{j} ↦ pε_{j} − 1 (j ≥ 1). Moreover α ∘ φ = φ′ ∘ α̃. □

3. The reduced Temperley-Lieb algebra

This section presents the structure of T′D_{n} and T′′D_{n} for generic parameters (p not a root of unity). The algebra T′D_{n} is of the type B_{n} but not exactly the same. Therefore we have to extend some of results in [??] to the present situation.

There exists idempotent elements f_{k} and g_{k} in T′D_{n} with the following properties:

f_{0} = 1 − \frac{1}{2}ε_{0}
f_{k} = f_{k−1} + \frac{p^{k−1}+p^{−k+1}}{p^{k}+p^{−k}} f_{k−1}ve_{k}f_{k−1}, 1 ≤ k ≤ n − 1
g_{0} = \frac{1}{2}ε_{0}
g_{k} = g_{k−1} + \frac{p^{k−1}+p^{−k+1}}{p^{k}+p^{−k}} g_{k−1}ε_{k}g_{k−1}, 1 ≤ k ≤ n − 1
ε_{j}f_{k} = f_{k}ε_{j} = ε_{j}g_{k} = g_{k}ε_{j} = 0,  1 ≤ j ≤ k
ε_{0}g_{k} = g_{k}ε_{0},  0 ≤ k ≤ n − 1
g_{k}f_{k} = f_{k}g_{k} = 0,  0 ≤ k ≤ n − 1
η(f_{k}) = 1 − \frac{1}{2}ε_{0}, η(g_{k}) = \frac{1}{2}ε_{0}.

The map η is the augmentation which sends e_{j}, j ≥ 1, to zero.

The proof for these assertions is as for [??], Satz 5.2, by induction on k. With the help of the central orthogonal idempotents f_{n−1} and g_{n−1} it is shown as in [??], Satz (7.1), that the Bratteli diagram of the inclusion T′D_{n−1} ⊂ T′D_{n} is the same as for the inclusion T B_{n−1} ⊂ T B_{n}. In particular, T′D_{n} has n + 1 simple modules M_{0}(n), M_{1}(n), …, M_{n}(n) with M_{j}(n) = N_{j} of dimension \binom{n}{j}.

The simple modules of T′′D_{n} are determined via restriction from T′D_{n}.

(3.1) Theorem. The algebra T′′D_{n} has the following irreducible modules:

(1) Suppose n = 2k + 1. The restrictions resM_{j} for j ≤ k. Moreover resM_{j} ≅ resM_{n−j}.

(2) Suppose n = 2k. The restrictions resM_{j}, j < k. In this case resM_{j} ≅ resM_{n−j}. The module resM_{k} is the direct sum of two simple T′′D_{n}-module of the same dimension.

The proof of (3.1) is by induction on n. One uses the structure of the Brattelidiagram for T′D_{n−1} ⊂ T′D_{n} and the following general fact about the crossed product construction of T′D_{n} from T′′D_{n}.

Let A be a semi-simple algebra with an involutive automorphism τ over the field K of characteristic zero and let B denote the crossed product algebra as described in section 2. If U is an A-module, let U^{τ} denote the same vector space with the A-action twisted by τ. The map a + bτ ↦ a − bτ is an automorphism of B. If V is a B-module, then \overline{V} is obtained from V by twisting with this automorphism (conjugate module). A simple A-module U is called of type I (resp. type II) if U ≅ U^{τ} (resp. U ≅ U̸^{τ}). A simple B-module V is called of type I (resp. type II) if V ≅ \overline{V̸} (resp. V ≅ \overline{V}). If U is an A-module, we call B ⊗_{A} U the induced B-module ind U. These notations are used in the statement of the following result.

(3.2) Proposition.

T. tom Dieck 4. Braids and tangles of type D 9

(1) Suppose V is a simple B-module of type I. Then resV = U is simple of type I and ind U ≅ V ⊕ \overline{V}.

(2) Suppose V is a simple B-module of type II. Then resV ≅ U ⊕ U^{τ} and V ≅ ind U ≅ ind U^{τ}. Moreover, U, U^{τ} of type II.

(3) Suppose U is a simple A-module of type I. Then ind U ≅ V ⊕\overline{V}. Moreover, resV ≅ res\overline{V} ≅ U.

(4) Suppose U is a simple A-module of type II. Then ind U = V is a simple B-module of type II and resV ≅ U ⊕ U^{τ}.

Proof. The proof of this proposition is by an adaption of the argument in [??], Ch. VI for the proof of Theorem (7.3). □

Proof of (3.1). By (3.2) it suffices to determine the modules \overline{M}_{j}(n). Let res_{n−1} denote the restriction via T′D_{n−1} ⊂ T′D_{n}. From the Bratteli diagram we know

(3.3) \begin{matrix}res_{n−1}M_{j}(n) = M_{j−1}(n − 1) ⊕ M_{j}(n), 1 ≤ j ≤ n − 1 \\ res_{n−1}M_{0}(n) = M_{0}(n − 1), res_{n−1}M_{n}(n) = M_{n−1}(n − 1). \end{matrix}

The isomorphism type of a simple T′D_{n}-module M is therefore determined by res_{n−1}M. We show by induction on n that \overline{M}_{j}(n) = M_{n−j}(n). Since restriction is compatible with conjugation, the induction step follows from (3.3). The induction starts with the irreducible representations of the group ZZ/2 generated by τ. □

4. Braids and tangles of type D

The permutations σ of [±n] = {−n, …, −1, 1, …, n} with the property σ(−i) = −σ(i) form the Weyl group W B_{n} of the root system B_{n}. The subgroup of even permutations in W B_{n} is the Weyl group W D_{n} of the root system D_{n}. The reflection representation of W D_{n} on C^{n} is given as follows: The subgroup S_{n} acts by permutation of coordinates and (ZZ/2)^{n−1} by sign changes (z_{j}) ↦ (±z_{j}) with an even number of minus signs. The reflection hyperplanes are given by z_{i} = z_{j} and z_{i} = −z_{j} for all pairs (i, j) with i ≠ j. Let X be the complement of the reflection hyperplanes and X/W the orbit space of the free W = W D_{n} action. Brieskorn [??] has shown that the fundamental group π_{1}(X/W) is the braid group ZD_{n}.

We translate this result and obtain a description of ZD_{n} by planar braid pictures.

A loop [w] in X/W with base point (1, …, n) can be lifted to X with (1, …, n) as starting point. Let w: [0, 1] → X, t ↦ (w_{j}(t)) be the resulting path from (1, …, n) to (±σ(1), …, ±σ(n)). Here σ ∈ S_{n}, and the number of minus signs is even. We consider the braid in C × [0, 1] with 2n strings given by

t ↦ {−w_{n}(t), …, −w_{1}(t), w_{1}(t), …, w_{n}(t)} × {t}.

The braid is symmetric with respect to the symmetry C → C, z ↦ −z. the strings are ζ_{±j}: t ↦ (±w_{j}(t), t). Since w maps into X, the strings have the following property: The string pairs (ζ_{j}, ζ_{−j}) and (ζ_{k}, ζ_{−k}) never meet for j ≠ k; an intersection would correspond to a point w_{j}(t) = ±w_{k}(t) on a reflection hyperplane. A value w_{j}(t) = 0 is not excluded, though. In this case, the strings ζ_{j} and ζ_{−j} intersect. Therefore we are not dealing with a braid in the usual sense. Of course, we can always choose representing paths w such that no intersection of ζ_{j} with ζ_{−j} occurs.

As usual, we consider planar generic projections of braids in the strip IR×[0, 1] from [±n] × 0 to [±n] × 1 which are symmetric with respect to the axis 0 × [0, 1]. The transverse intersections on the axis are ordinary crossings, and the other crossings are over- and undercrossings which appear in symmetric pairs.

In the geometric picture, the extended braid group Z′D_{n} has generators κ_{0}, …, κ_{n−1} with κ_{0} given by

[figure: κ_{0}; ⋯; ⋯; 1; −1; 2; n]

and κ_{j} (j ≥ 1) given by the symmetrized crossing of the j-th and 1 + j-th string (see the next figure for κ_{1}). The braid group ZD_{n} has generators x_{j} = κ_{j} (j ≥ 1) and x_{0} = κ_{0}κ_{1}κ_{0} represented by

[figure: ⋯; ⋯; x_{0}; −2; −1; 1; 2]

The relation κ_{0}^{2} = 1 corresponds to a standard Reidemeister move of type II. It would also be possible to use over- and under-crossing on the axis, but then allow for an interchange of over-crossing and under-crossing on the axis.

Elements in the subgroup ZD_{n} have in their geometric picture an even number of crossings on the axis.

The geometric braid groups Z′D_{n} and ZD_{n} are included in tangle categories S′D and SD (in the sense of [??], [??]). The category S′D has objects [±n], n ∈ IN_{0}. The morphisms from [±m] to [±n] are tangle pictures in IR × [0, 1] from [±m] × 0 to [±n] × 1 which are symmetric with respect to the axis 0 × [0, 1]. The crossings on the axis are ordinary crossings. Composition is defined by placing one tangle above the other and shrinking of [0, 2] to [0, 1]. The subcategory SD of S′D consists of tangles with an even number of points on the axis. There are similar categories S_{0}D and S′_{0}D of oriented tangles. Also, one may consider banded (framed) tangles by not allowing Reidemeister type I moves. The Dtangle categories are analogous to the B-tangle categories [??], except for the special treatment of the crossings on the axis. The categories are tensor module categories over the appropriate categories of ordinary tangles. Ordinary tangles are included by symmetrizing.

There is a Kauffman functor from S′D to the category T′D of bridges. In this context one chooses a parameter A with p = −A^{2}. The Kauffman functor resolves a symmetrized ordinary crossing as usual in the definition of the Kauffman bracket [??]. A crossing on the axis is treated as in the following figure.

[figure: ↦; −]

There is also a forgetful functor to A-tangles which maps

[figure: ↦]

and takes the ZZ/2-quotient of the resulting tangle. These two functors correspond to the splitting of the algebra T D_{n} in section 2.

References

1. Brieskorn, E.: Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe. Invent. Math. 12, 57 – 61 (1971).

2. Bröcker, Th., and T. tom Dieck: Representations of compact Lie groups. New York – Heidelberg, Springer 1985.

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

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

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

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

7. Kauffman, L. H.: State models and the Jones polynomial. Topology 26, 395 – 407 (1987).

8. Reshetikhin, N., and V. G. Tuarev: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys. 127, 1 – 26 (1990).

9. Turaev, V. G.: Operator invariants of tangles and R-matrices. USSR Izvestia 35, 411 – 444 (1990).