Knot theories and root systems Part III

Tammo tom Dieck

February 9, 2009

We continue our study of knot theory from the point of view of root systems. The presentation is partly expository, in preparation for later parts of this series.

The first part of this note is concerned with the Coxeter-Dynkin series of type D. It turns out that some relevant algebra can be reduced to the corresponding algebra for the series A and B. We demonstrate this for the Temperley-Lieb algebra T D_{n} and the Kauffman functor for D-tangles.

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, he weight m(s, t) = 3 is omitted in 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} — the simply laced case. 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. We verify that φ respects the defining relations of the Hecke algebra. Certainly φ(x_{s}x_{t}) = φ(x_{t}x_{s}), whenever m(s, t) = 2. Moreover,

(pe_{s} − 1)(pe_{s} − 1)
= p^{2}e_{s}^{2} − 2pe_{s} + 1
= p^{2}(p + p^{−1})e_{s} − 2pe_{s} + 1
= (p^{2} − 1)(pe_{s} − 1) + p^{2},

hence compatibility with the quadratic relations of the Hecke algebra. Finally,

(pe_{s} − 1)(pe_{t} − 1)(pe_{s} − 1)
= p^{3}e_{s}e_{t}e_{s} − p^{2}(e_{s}e_{t} + e_{t}e_{s} + e_{s}^{2}) + p(2e_{s} + e_{t}) − 1
= −p^{2}(e_{s}e_{t} + e_{t}e_{s}) + pe_{s} + pe_{t} − 1,

if m(s, t) = 3. The result is symmetric in s, t; hence compatibility with the remaining 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).

We have to verify compatibility with (1.2). The first relation follows from the quadratic relation in the Hecke algebra, the second one is obviously satisfied. For the third one we compute modulo I

p^{−1}(x_{s} + 1)p^{−1}(x_{t} + 1)p^{−1}(x_{s} + 1)
= p^{−3}(x(s, t) + x_{s}^{2} + x_{s})
= p^{−3}((p^{2} − 1)x_{s} + p^{2} + x_{s})
= p^{−1}(x_{s} + 1).

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

A similar computation shows 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

(1.5) \begin{matrix}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. \end{matrix}

We define a linear map X_{s}: V → V by

(1.6) 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

(1.7) 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

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

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

(1.9) 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

(1.10) 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}.

(1.11) 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. With (1.8) 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} ⟩

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. □

(1.12) Remark. The determinant of the bilinear form B for the graph D_{n} in the basis (v_{s} | s ∈ S) equals (q + 1)(q^{n−1} + 1). Therefore the form is regular if q is not a root of unity. ♡

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

(1.13) \begin{matrix}Z_{s}^{2} & = & a_{ss}Z_{s} \\ Z_{s}Z_{t}Z_{s} & = & a_{st}a_{ts}Z_{s}. \end{matrix}

Then a simple verification from the definitions give:

(1.14) 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 Z(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}.

(1.15) 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

T D_{n}/I ^{α \overline{}}T A_{n−1} ^{α_{1} \overline{}}T D_{n}/I

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

(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}.

T. tom Dieck 2. The structure of T D_{n} 7

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} ^{\overline{}(α_{1}, 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

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

(2.8) \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 connecting 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). The diagram

HD_{n}  ^{α̃ \overline{}}H′D_{n}

[figure: φ; φ′]
T D_{n}  ^{α \overline{}}T′D_{n}

is commutative. □

For use in other situations we formalize the preceding discussion. Suppose we have two Coxeter graphs which differ in a similar way as B_{n} and D_{n} do. The following figure illustrates this situation:

[figure: κ_{2}; κ_{1}; κ_{0}; 4; x_{2}; x_{1}; x_{0}; S; T]

The box stands for the same subgraph in both cases and is connected to x_{2} (resp. κ_{2}) by the same weighted edges. The graph T carries the involution τ which interchanges κ_{0} and κ_{1}. Therefore we can form the semi-direct product of the braid group

10 T. tom Dieck

(2.13) ZT ×_{τ} ZZ/2. We also have the quotient (2.14) Z′T = ZS/(κ_{0}^{2} = 1). The same proof as for (.??) shows: (2.15) Proposition. The groups ZT ×_{τ} ZZ/2 and Z′T are isomorphic. □

Also (??) has its analogue in this case.

T. tom Dieck 3. The reduced Temperley-Lieb algebra 11

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:

(1) 1 − \frac{1}{2}ε_{0}
(2) 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
(3) ε_{j}f_{k} = f_{k}ε_{j} = 0,  0 ≤ k ≤ k
(4) g_{0} = \frac{1}{2}ε_{0}
(5) 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
(6) ε_{j}g_{k} = g_{k}ε_{j} = 0,  1 ≤ j ≤ k
(7) ε_{0}g_{k} = g_{k}ε_{0},  0 ≤ k ≤ n − 1
(8) g_{k}f_{k} = f_{k}g_{k} = 0,  0 ≤ k ≤ n − 1
(9) η(f_{k}) = 1 − \frac{1}{2}ε_{0}
(10) 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 V̄ 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 ≅ ¯V̸ (resp. V ≅ V̄). 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.

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

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

(3) Suppose U is a simple 𝔄-module of type I. Moreover, resV ≅ resV̄ ≅ 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 M̄_{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 M̄_{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. □

T. tom Dieck 4. Braids and tangles of type D_{n} 13

4. Braids and tangles of type D_{n}

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 group W D_{n} has order 2^{n−1} · n! and is a semi-direct product

1 → (ZZ/2)^{n−1} → W D_{n} → S_{n} → 1

with the symmetric group S_{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) 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 intersection 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. 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 D-tangle 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.

T. tom Dieck 5. Categories of bridges 15

5. Categories of bridges

This section introduces some general terminology for certain graphical categories.

A free involution σ: P → P of a set P is calles a P -bridge. A free involution of P is a partition of P into 2-element subsets {i, σ(i)}, called the arcs or strings of the bridge. A bridge is called oriented if its arcs are ordered sets {a_{1}, a_{2}}.

We study bridges with a geometric terminology. Suppose σ: P → P is a bridge. The geometric realization |σ| of σ is the one-dimensional simplicial complex with P as set of 0-simplices and a 1-simplex for each arc {i, σ(i)} with i and σ(i) as boundary points. We say that the arc connects its boundary points. The arcs are the components of |σ|.

A (P, Q)-bridge is a bridge on the disjoint union P ∐ Q. An arc of a (P, Q)- bridge σ is called horizontal if its boundary points are either contained in P or in Q. The other arcs are called vertical.

We use a graphical notation for (P, Q)-bridges σ. We think of P ⊂ IR × 0, Q ⊂ IR × 1 and we draw an arc in IR × [0, 1] from i to σ(i). This is illustrated by the next figure.

??

The crossings of the arcs have no significance right now. The notation horizontal and vertical is evident in this context. The horizontal arcs with endpoints in P are called the lower part of the bridge, the horizontal arcs with endpoints in Q the upper part.

(5.1) Remark. Suppose P has 2n elements. The number of P -bridges is

(2n − 1) · (2n − 3) ⋯ 3 · 1.

Proof. There are 2n − 1 possibilities to connect a fixed element of P. Having fixed this connection, a set with 2n − 2 elements remains. Now use induction.□

We will use bridges with further properties.

Let G be a group and suppose P and Q are G-sets. A G-equivariant (P, Q)- bridge is a G-equivariant free involution σ of P ∐ Q. Equivariant means: σ(gi) = gσ(i) for g ∈ G and i ∈ P ∐ Q.

Suppose the bridge σ: P → P is G-eqivariant. We have an induced G-action on |σ|. The action on the 0-simplices is given. If {i, σ(i)} is a 1-simplex, then, by equivariance, {gi, gsigma(i)} is a 1-simplex. It can happen that these simplices coincide. This is the case if g is in the isotropy group G_{i} of i. If g ∈ G acts non-trivially on {i, σ(i)}, then

gi = σ(i), gσ(i) = i, g^{2}i = i

and hence g^{2} ∈ G_{i}. Geometrically, g acts as reflection in the barycentre of the 1-simplex {i, σ(i)} in this case.

In the sequel we only consider G-sets P with the following additional properties:

(1) The isotropy groups are finite.

(2) The orbit set is finite.

(3) G acts effectively on each orbit. Under these hypotheses we have:

(5.2) Proposition. Let σ be a G-equivariant (P, Q)-bridge. Then the following holds:

(1) The G-action respects lower, upper and horizontal arcs.

(2) The G-action on |σ| is proper.

(3) The orbit space |σ|/G is a compact one-dimensional CW-complex.

Proof. (1) is clear from the definitions.

(2) The G-action on the barycentric subdivision |σ|′ of |σ| is a cellular action with finite isotropy groups. Now use [??,I(3.22)].

(3) This follows since G acts cellularly on |σ|′ and the orbit space has a finite number of cells [??,II(1.15),(1.17)]. □

Suppose σ is a G-equivariant (P, Q) − bridgeandτ a G-equivariant (Q, R)- bridge. Consider the G-space |τ| ∪_{Q} |σ|. The components of this space can be if different type. Consider the G-orbit B = Gx of a component x. Let H denote the isotropy group of the component x. Then B/G is homeomorphic to x/H. The orbit space of |τ| ∪ |σ is compact. Hence x/H is compact. We use:

(5.3) Lemma. There is no proper action of a discrete group on [0, 1[ with compact orbit space. □

This Lemma tells us that the components of |τ| ∪ |σ| are not homeomorphic to [0, 1[. Since the components are one-dimensional manifolds (with or without boundary), there are three cases:

(5.4) A componente of |τ| ∪ |σ| is homeomorphic to [0, 1], S^{1}, or ]0, 1[. □

(5.5) Proposition. The components of |τ| ∪ |σ| which are homeomorphic to [0, 1] define a G-equivariant (P, R)-bridge.

Proof. If the component is homeomorphic to [0, 1], then the boundary points are contained in P ∐ R.

For each point in P ∐ R there exists a component of |τ| ∪ |σ| with this point as boundary point. Since components of type [0, 1[ do not exist, the component has a second boundary point in P ∐ R. □

We denote the bridge in (5.5) by τ ∧sigma. The components of |τ|∪|σ| which are homeomorphic to S^{1} are called cycles, the components which are homeomorphic to ]0, 1[ are called snakes.

(5.6) Remark. Let H = G_{x} be the subgroup of elements which map the component x into itself. Then H acts effectively and properly on the one-dimensional manifold x. Therefore we have, up to H-homeomorphism, the following possibilities:

(1) Suppose x ≅ S^{1}. Then H ≅ ZZ/m or H ≅ D_{2m}, m ≥ 1, and the action is by the usual action of a subgroup of O(2).

(2) Suppose x ≅ IR. Then H ≅ ZZ or H ≅ D_{∞} and the action is by the usual action as a subgroup of the group of affine transformations. ♡

Let Z/τ, σ) denote the orbit set of the components of |τ|∪|σ| which are cycles or snakes. The G-orbits of components in Z(τ, σ) are counted according to types. The type of a component x consists of the conjugacy class of G_{x} together with the G_{x}-homeomorphism type of the G_{x}-action. The group ZZ/2 has to different actions on S^{1}, by rotation or by reflection. (In the latter case it is the group D_{2}.) It is an observation of H. Reich [??] that these two actions should be distinguished.

Let C denote the set of possible types. We denote by k(c, τ, σ) the number of elements in Z(τ, σ) of type c.

After these preparations we define the category F (G) of G-bridges. The objects of F (G) are the G-sets as above, i. e. with finite isotropy groups, finte orbit set and effictive action on orbits.

We fix a ground ring 𝔎. The morphism set Mor(P, Q) is the free 𝔎-module on the set of G-equivariant (P, Q)-bridges.

In order to define the composition of morphisms we fix a map d: C → 𝔎, called the parameter function. The composition of morphisms Mor(Q, R) × Mor(P, Q) → Mor(P, R) is assumed to be 𝔎-bilinear. The composition of bridges is defined to be

τ ∘ σ ∏_{c∈C} d(c)^{k(c,τ,σ}τ ∧ σ.

The identity P → P is represented by the bridge ι: P ∐ P → P ∐ P which connects i ∈ P vertically with i ∈ P. We have |σ| ∪ |ι| ≅ |σ| and |ι| ∪ |σ| ≅ |σ|, if defined.

Associativity of composition follows from a geometrical consideration: The cycles and snakes of |τ| ∪ |σ| ∪ |ρ| are those of |τ| ∪ |σ|, plus those of |σ| ∪ |ρ|, plus those of |τ ∧ σ| ∪ |ρ| (equal to those of |τ| ∪ |σ ∧ ρ|).

We shall mostly work with suitable subcategories of F (G). For instance, we could use only free G-sets. Or we restrict the morphisms; this will be the case in the Temperley-Lieb categories.

The composition of bridges with only vertical strings is again a bridge of this form. No cycles or snakes appear. The vertical (P, P)-bridges under composition can be identified with the group of G-equivariant permutations of P. We describe this group.

6. Braid groups of type B

The braid group ZB_{n} associated to the Coxeter graph B_{n} is, by definition, the group generated by t, g_{1}, …, g_{n−1} with relations

(6.1) \begin{matrix}(1) & g_{i}g_{j}g_{i} & = & g_{j}g_{i}g_{j}, & |i − j| = 1 \\ (2) & g_{i}g_{j} & = & g_{j}g_{i}, & |i − j| ≥ 2 \\ (3) & tg_{i} & = & g_{i}t, & i ≥ 2 \\ (4) & tg_{1}tg_{1} & = & g_{1}tg_{1}t. \end{matrix}

For certain applications we need other presentations of this group.

Let Z′B_{n} be the group with generators c, g_{1}, …, g_{n−1} and relations

(6.2) \begin{matrix}(1) & g_{i}g_{j}g_{i} & = & g_{j}g_{i}g_{j}, & |i − j| = 1 \\ (2) & g_{i}g_{j} & = & g_{j}g_{i}, & |i − j| ≥ 2 \\ (3) & cg_{i} & = & g_{i−1}c, & i ≥ 2, \\ (4) & c^{2}g_{1} & = & g_{n−1}c^{2}. \end{matrix}

We abbreviate g = g_{n−1}g_{n−2} ⋯ g_{1}.

(6.3) Proposition. The assignment φ(g_{i}) = g_{i}, 1 ≤ j ≤ n−1, and φ(t) = g^{−1}c defines an isomorphism φ: ZB_{n} → Z′B_{n}.

Proof. The relations (1) and (2) yield in both groups

(6.4)  g_{i−1}g = gg_{i},  i > 1.

We define in ZB_{n} (resp. Z′B_{n}) an element c (resp. t) by gt = c. From (1), (2) and (2.4) we see that the relations cg_{i} = g_{i−1}c and g_{i}t = tg_{i} are equivalent for i > 1.

We set h = g_{n−1} ⋯ g_{2}, k = g_{n−2} ⋯ g_{1} and infer from (2.4)

(6.5) gh = kg.

We use this to show that c^{2}g_{1} = g_{n−1}c^{2} and tg_{1}tg_{1} = g_{1}tg_{1}t are equivalent, provided (1), (2), and (3) hold. We compute

g_{n−1}^{−1}c^{2}g_{1} = g_{n−1}^{−1}g_{n−1}kthg_{1}tg_{1} = khtg_{1}tg_{1}
c^{2} = gthg_{1}t = ghtg_{1}t = kgtg_{1}t = khg_{1}tg_{1}t

and see the equivalence. □

The braid group ZÃ_{n−1} of the Coxeter graph with n vertices Ã_{n−1} has, by definition, generators g_{1}, …, g_{n} and relations

(6.6) \begin{matrix}g_{i}g_{j}g_{i} & = & g_{j}g_{i}g_{j}, & m(i, j) = 3 \\ g_{i}g_{j} & = & g_{j}g_{i}, & m(i, j) = 2. \end{matrix}

T. tom Dieck  6. Braid groups of type B  19

Indices will be considered mod n in this case. We have m(i, j) = 3 if and only if i ≡ j ± 1 mod n. All this holds for n ≥ 3. For n = 2, the group is the free group generated by g_{1} and g_{2}.

The graph Ã_{n−1} has an automorphism which permutes the vertices cyclically. We have an induced automorphism s of ZÃ_{n−1} given by

s(g_{i}) = g_{i−1}, i mod n.

The n-th power of s is the identity.

We use s to form the semi-direct product

(6.7) ZÃ_{n−1} → G_{n} → ZZ;

the generator 1 ∈ ZZ acts through s on ZÃ_{n−1}. The semi-direct product is the group structure on the set ZÃ_{n−1}×ZZ defined by (x, m)·(y, n) = (x·s^{m}(y), m+n). The group G_{n} has the following description by generators and relations. Let G′_{n} denote the group with generators s, g_{1}, …, g_{n} and relations (2.6) together with

(6.8)  sg_{i} = g_{i−1}s,  i mod n.

(6.9) Proposition. The assignment ψ(g_{i}) = (g_{i}, 0) and ψ(s) = (e, 1) yields an isomorphism ψ: G′_{n} → G_{n} (neutral element e).

Proof. One verifies that ψ is compatible with relations (2.6) and (2.8). This is obvious for (2.6). The relation (e, 1)(x, 0)(e, 1)^{−1} = (s(x), 0) is used to show compatibility with (2.8).

An element x ∈ ZÃ_{n−1} has an image x′ ∈ G′_{n}, induced by g_{i} ↦ g_{i}. This assignment has the property (s(x))′ = sx′s^{−1}. We have the Homomorphism G_{n} → G′_{n}, (x, m) ↦ x′s^{m} by (1.4). It is inverse to psi. □

(6.10) Proposition. The assignment α(g_{i}) = g_{i}, 1 ≤ i ≤ n − 1, and α(c) = s defines an isomorphism α: Z′B_{n} → G′_{n}.

Proof. The assignment is compatible with the relations of Z′B_{n}, since

α(c^{2}g_{1}c^{−2}) = s^{2}g_{1}s^{−2} = sg_{n}s^{−1} = g_{n−1}.

An inverse to α is induced by the assignment β(g_{i}) = g_{i}, β(g_{n}) = cg_{1}g^{−1}, and β(s) = c. In order to see that β is well defined, one has to check, in particular, the relations

g_{n−1}g_{n}g_{n−1} = g_{n}g_{n−1}g_{n}, g_{1}g_{n}g_{1} = g_{n}g_{1}g_{n}.

In the first case, this amounts to the equalitiy of

g_{n−1}cg_{1}c^{−1}g_{n−1} = c^{2}g_{1}c^{−1}g_{1}cg_{1}c^{−2}

and

cg_{1}c^{−1}g_{n−1}cg_{1}c^{−1} = cg_{1}cg_{1}c^{−1}g_{1}c^{−1}.

We compute

cg_{1}g_{2}g_{1}c^{−1} = cg_{2}g_{1}g_{2}c^{−1} = cg_{2}c^{−1}cg_{1}c^{−1}cg_{2}c^{−1} = g_{1}cg_{1}c^{−1}g_{1}

and hence

c(g_{1}cg_{1}c^{−1}g_{1})c^{−1} = c^{2}g_{1}g_{2}g_{1}c^{−2}.

On the other hand, g_{1}c^{−1}g_{1}cg_{1} = g_{1}g_{2}g_{1}. This yields the desired equality. The second relation above leads to the same situation. □

If we combine the foregoing, we obtain a semi-direct product

(6.11) ZÃ_{n−1} → ZB_{n} → ZZ.

In terms of the original generators, the inclusion ZÃ_{n−1} ⊂ ZB_{n} is given by

(6.12) g_{n} ↦ gtg_{1}t^{−1}g^{−1}; g_{i} ↦ g_{i}, 1 ≤ i ≤ n − 1.

The homomorphism ZB_{n} → ZZ in (2.14) is given by g_{i} ↦ 0 and t ↦ 1.

Different types of Weyl groups (= Coxeter groups) are related to these braid groups. We have the Coxeter groups W Ã_{n−1} and W B_{n} associated to the graphs Ã_{n−1} and B_{n}. In addition, we will also use a group W^{∞}B_{n}. It is obtained from ZB_{n} by adding the relations g_{j}^{2} = 1, but no relation for t. The reason for introducing this group is a semi-direct product in analogy to (2.14). The arguments which lead to (2.14) also give a semi-direct product

W Ã_{n−1} → W^{∞}B_{n} → ZZ.

We give another interpretation and describe these groups as groups of permutations.

Let t_{n}: ZZ → ZZ, x ↦ x+n be the translation by n. Let P_{n} denote the group of t_{n}-equivariant permutations σ: ZZ → ZZ. Equivariance means σ(i + n) = σ(i) + n. Hence σ induces \overline{σ}: ZZ/n → ZZ/n, and σ ↦ \overline{σ} is a homomorphism π: P_{n} → S_{n} onto the symmetric group S_{n}.

(6.13) Proposition. The kernel of π is isomorphic to ZZ^{n}. The group P_{n} is isomorphic to the semi-direct product ZZ^{n} → P′_{n} → S_{n} in which S_{n} acts on ZZ^{n} by permutations.

Proof. Let σ_{1} ∈ P_{n}. Then there exists a permutation α of {1, …, n} and an ntuple (k_{1}, …, k_{n}) ∈ ZZ^{n} such that σ(i+tn) = α(i)+(k_{i}+t)n. We denote this map by σ_{1} = σ(α; k_{1}, …, k_{n}). Suppose σ_{2} = σ(β; l_{1}, …, l_{n}) is another permutation written in this form. Then

σ_{2} ∘ σ_{1} = σ(βα; l_{α(1)} + k_{1}, …, l_{α(n)} + k_{n}).

If we think of P′_{n} = S_{n} × ZZ^{n} as sets, then the desired isomorphism is given by (α; k_{1}, …, k_{n}) ↦ σ(α; k_{1}, …, k_{n}). □

The semi-direct product P′_{n} has a normal subgroup Q′_{n} which is given as a semi-direct product

(6.14) N → Q′_{n} → S_{n} with N = {(x_{1}, …, x_{n}) | ∑ x_{i} = 0} ⊂ ZZ^{n}. The homomorphism

ε: P′_{n} → ZZ, (α; k_{1}, …, k_{n}) ↦ ∑ k_{i}

is a surjection with kernel Q′_{n}. The canonical sequence

(6.15) Q′_{n} → P′_{n} → ZZ

is itself a semi-direct product; the assignment 1 ↦ (id; 1, 0, …, 0) gives a splitting of ε. Under the isomorphism (2.13) the subgroup Q′_{n} corresponds to the subgroup

Q_{n} = {σ ∈ P_{n} | 1 + 2 + ⋯ + n = σ(1) + ⋯ + σ(n)}.

(6.16) Proposition. The groups W^{∞}B_{n} and P_{n} are isomorphic. The groups W Ã_{n−1} and Q_{n} are isomorphic. The element g_{i} is mapped to the transposition (i, i + 1), i ∈ nZZ. The element t is mapped to σ(i) = i + n for i ≡ 1 mod n and σ(j)j otherwise.

The proof is given after the proof of (2.21). In the proof of (2.17) we use the following:

(6.17) Lemma. The elements

t_{0} = t, t_{1} = g_{1}tg_{1}, …, t_{n−1} = g_{n−1} … g_{2}g_{1}tg_{1}g_{2} … g_{n−1}

of the braid group ZB_{n} pairwise commute.

Proof. We set

g(i, j) = g_{i}g_{i+1} … g_{j}, i ≤ j
g(i, j) = g_{i}g_{i−1} … g_{j}, i ≥ j.

The braid relations imply immediately

g(1, j)g_{j+1}g(j, 1) = g(j + 1, 2)g_{1}g(2, j + 1)

and (2.5)

g(2, j + 1)g(1, j + 1) = g(1, j + 1)g(1, j).

By commutativity of g_{j}-elements, it suffices to show t_{i}t_{i+1} = t_{i+1}t_{i}. We compute

t_{j}t_{j+1} = g(j, 1)tg(1, j)g_{j+1}g(j, 1)tg(1, j + 1)
= g(j, 1)tg(j + 1, 2)g_{1}g(2, j + 1)tg(1, j + 1)
= g(j, 1)g(j + 1, 2)tg_{1}tg(2, j + 1)g(1, j + 1)
= g(j, 1)g(j + 1, 2)[tg_{1}tg_{1}]g(2, j + 1)g(1, j).

A similar computation works for t_{j+1}t_{j}. □

The semi-direct product relation (2.13, (2.17) between W^{∞}B_{n} and W A_{n−1} has a counterpart for the braid groups. The homomorphism

λ: K_{n} → ZA_{n−1}, g_{j} ↦ g_{j}, t ↦ 1

splits by g_{j} ↦ g_{j}. Therefore we have a semi-direct product

(6.18) ZP_{n} → ZB_{n} → ZA_{n−1}.

The elements

y_{0} = t, , y_{1} = g_{1}tg_{1}^{−1}, …, y_{n−1} = g_{n−1} … g_{1}tg_{1}^{−1} … g_{n−1}^{−1}

are contained in the kernel K_{n} of λ.

(6.19) Lemma. The elements y_{j} have the following conjugation properties with respect to ZA_{n−1}:

(1) g_{k}^{−1}y_{j}g_{k} = y_{j}, k > j, k < j − 1

(2) g_{k}^{−1}y_{k}g_{k} = y_{k−1},

(3) g_{k}^{−1}y_{k−1}g_{k} = y_{k−1}y_{k}y_{k−1}^{−1}. Proof. (2) follows directly from the definitions.

(1) If k > j, then g_{k} commutes with every generator in the definition of y_{j}. In the case k < j − 1 one uses the commutation relation between generators and g_{k+1}g_{k}g_{k+1}^{−1} = g_{k}^{−1}g_{k+1}g_{k} (and the inverse) to cancel g_{k}^{−1} and g_{k}.

(3) is proved by induction on k. The verification for k = 0 is easy. We calculate with (1) and (2)

g_{k}^{−1}y_{k}y_{k+1}y_{k}^{−1}g_{k} = y_{k−1}y_{k+1}y_{k−1}^{−1} = g_{k+1}y_{k−1}y_{k}y_{k−1}^{−1}g_{k+1}^{−1}.

On the other hand, by (1) and (2)

g_{k+1}^{−1}g_{k}^{−1}g_{k+1}^{−1}y_{k}g_{k+1}g_{k}g_{k+1} = g_{k}^{−1}g_{k+1}^{−1}g_{k}^{−1}y_{k}g_{k}g_{k+1}g_{k}
= g_{k}^{−1}g_{k+1}^{−1}y_{k−1}g_{k+1}g_{k}
= g_{k}^{−1}y_{k−1}g_{k}.

This yields the induction step. □

(6.20) Proposition. The group K_{n} is the free group generated by y_{0}, …, y_{n−1}.

Proof. By the previous Lemma, the group K_{n}^{0} generated by the y_{0}, …, y_{n−1} is invariant under conjugation by elements of ZA_{n−1}. Since t ∈ K_{n}^{0} and t together with ZA_{n−1} generates ZB_{n}, we must have equality K_{n}^{0} = K_{n}.

Let F_{n} denote the free group generated by y_{0}, …, y_{n−1}. We define homomorphisms γ_{1}, …, γ_{n−1}: F_{n} → F_{n} by immitating (2.20):

(1) γ_{k}(y_{j}) = y_{j}, k > j, k < j − 1

(2) γ_{k}(y_{k}) = y_{k−1},

(3) γ_{k}(y_{k−1}) = y_{k−1}y_{k}y_{k−1}^{−1}. We claim:

(6.21) Lemma. The γ_{j} are automorphisms and satisfy the braid relations

γ_{i}γ_{j}γ_{i} = γ_{j}γ_{i}γ_{j},  |i − j| = 1, and γ_{i}γ_{j} = γ_{j}γ_{i},  |i − j| ≥ 2.

Proof. First we check that the homomorphism δ_{k}: F_{n} → F_{n} (1) δ_{k}(y_{j}) = y_{j}, k > j, k < j − 1

(2) δ_{k}(y_{k−1}) = y_{k},

(3) δ_{k}(y_{k}) = y_{k}^{−1}y_{k−1}y_{k} is inverse to γ_{k}. Hence γ_{k} is an isomorphism. Since γ_{k} fixes y_{j} for j ∉ {k − 1, k}, the second braid relation is obviously satisfied. For the first relation, the reader may check the following values of γ_{1}γ_{2}γ_{1} and γ_{2}γ_{1}γ_{2} on y_{0}, y_{1}, y_{2}:

y_{0} ↦ y_{0}y_{1}y_{2}y_{1}^{−1}y_{0}^{−1}, y_{1} ↦ y_{0}y_{1}y_{1}^{−1}, y_{2} ↦ y_{0}.

We use this Lemma to define a semi-direct product

(6.22) F_{n} → Γ_{n} → ZA_{n−1},

in which g_{j} ∈ ZA_{n−1} acts on F_{n} through δ_{j}. By (2.19) and K_{n}^{0} = K_{n}, we have a canonical epimorphism μ: Γ_{n} → ZB_{n}. We show that μ is an isomorphism. As a set, Γ_{n} = F_{n} × ZA_{n−1}. An inverse to μ has to send g_{j} ↦ (1, g_{j}) and t ↦ (y_{0}, 1). We have to check that this assignment is compatible with the relations of ZB_{n}. This is obvious for the g_{j}. Moreover:

tg_{1}tg_{1} ↦ (y_{0}, 1)(1, g_{1})(y_{0}, 1)(1, g_{1})
= (y_{0}, g_{1})(y_{0}, g_{1})
= (y_{0}δ_{1}(y_{0}), g_{1}^{2})
= (y_{0}y_{1}, g_{1}^{2})

g_{1}tg_{1}t ↦ (1, g_{1})(y_{0}, 1)(1, g_{1})(y_{0}, 1)
= (y_{1}, g_{1})(y_{1}, g_{1})
= (y_{1}δ(y_{1}), g_{1}^{2})
= (y_{0}y_{1}, g_{1}^{2}).

This finishes the proof of Proposition (2.21). □

Proof of (2.17). The elements t_{j} of (2.18) and the elements y_{j} coincide in W^{∞}B_{n}, since g_{j} = g_{j}^{−1} in this group. Lemma (2.20) shows that conjugation y ↦ g_{k}^{−1}yg_{k} acts on the set (y_{0}, …, y_{n−1}) by interchanging y_{k−1} and y_{k}. The proof of (2.21) is now easily adapted to show the isomorphism W^{∞}B_{n} ≅ P′_{n}. This isomorphism restricts to an isomorphism W Ã_{n−1} ≅ Q′_{n}. □

We now apply the previous results to Hecke algebras. We have the Hecke algebras HA_{n−1}, HÃ_{n−1}, and HB_{n} associated to the corresponding Coxeter graphs. We consider algebras over the ground ring 𝔎. The first one is given by generators g_{1}, …, g_{n−1}, the braid relations between them and the quadratic relations g_{j}^{2} = (q − 1)g_{j} + q with a parameter q ∈ 𝔎. The second one has generators g_{1}, …, g_{n}, the braid relations (2.8) and the same quadratic relations. The algebra HB_{n} has generators t, g_{1}, …, g_{n−1}, the braid relations (2.1), the quadratic relations above for the g_{j} and t^{2} = (Q − 1)t + Q with another parameter Q ∈ 𝔎.

If we omit the quadratic relation for Q, then we obtain the definition of H^{∞}B_{n}. This is not a Hecke algebra in the formal sense, i. e. associated to a Coxeter graph. It is a deformation of the group algebra of W^{∞}B_{n}.

We know from Hecke algebra theory that an additive basis of the Hecke algebra is in bijective correspondence with the elements of the Coxeter group. There is a similar relation between W^{∞}B_{n} and H^{∞}B_{n}. In order to derive it, we relate HÃ_{n−1} and H^{∞}B_{n}.

The algebra HÃ_{n−1} has an automorphism τ given by τ(g_{i}) = g_{i−1} (indices mod n). We define the twisted tensor product over the ground ring 𝔎

(6.23) HÃ_{n−1} ⊗ 𝔎[τ, τ^{−1}] =: H_{n}^{∞}

by the multiplication rule (x ⊗ τ^{k}) · (y ⊗ τ^{l}) = (x · τ^{k}(y), τ^{k+l}) for k, l ∈ ZZ and x, y ∈ HÃ_{n−1}.

(6.24) Proposition. The algebra (2.24) is canonically isomorphic to H^{∞}B_{n}.

Proof. We use the isomorphism (2.3) to redefine the algebra H^{∞}B_{n} by generators c, g_{1}, …, g_{n−1} relations (2.2) and the quadratic relations for the g_{j}. The assignment g_{j} ↦ g_{j}⊗1, c ↦ 1⊗τ induces a homomorphism H^{∞}B_{n} → HÃ_{n−1}⊗H_{n}^{∞}. We have a homomorphism HÃ_{n−1} → H^{∞}B_{n}, x ↦ x′ induced by g_{j} ↦ g_{j} with g_{n} = gtg_{1}t^{−1}g^{−1} in H^{∞}B_{n} (see (2.12)). This extends to a homomorphism H_{n}^{∞} → H^{∞}B_{n} by x ⊗ τ^{k} ↦ x′ · c^{k}, since τ(y)′ = cy′c^{−1}. These homomorphisms are inverse to each other. □

(6.25) Corollary. Suppose (b_{j} | j ∈ J) is a 𝔎-basis of HÃ_{n−1}. Then (b′_{j}c^{k} | j ∈ J, k ∈ ZZ) is a 𝔎-basis of H^{∞}B_{n}. □

T. tom Dieck  7. Braids of type B  25

7. Braids of type B

We use a theorem of Brieskorn [??] to derive some geometric interpretations of the braid group ZB_{n}. The starting point is the reflection representation of the Weyl group W B_{n}. This group is a semi-direct product

(7.1) (ZZ/2)^{n} → W B_{n} → S_{n}.

It acts on complex n-space C^{n} as follows:

(1) S_{n} acts by permuting the coordinates.

(2) (ZZ/2)^{n} act by sign changes (z_{1}, …, z_{n}) ↦ (ε_{1}z_{1}, …, ε_{n}z_{n}), ε_{i} ∈ {±1}. This group contains the reflections in the hyperplanes

z_{i} = ±z_{j}, i ≠ j; and z_{j} = 0.

Let X denote the complement of these hyperplanes. From the theory of finite reflection groups it is known, that W = W B_{n} acts freely on X. Brieskorn [??] shows:

(7.2) Theorem. The braid group ZB_{n} is isomorphic to the fundamental group π_{1}(X/W) of the orbit space X/W. □

If we think of W B_{n} as the Coxeter group with generators t, g_{1}, …, g_{n−1}, then g_{j} acts as the transposition (j, j + 1) and t as z_{1} ↦ −z_{1}.

We use (5.2) to give several interpretations of ZB_{n} by braids.

We remove the hyperplanes z_{j} = 0 from C^{n}. It remains the n-fold product C^{∗} × ⋯ × C^{∗} = C^{∗n}. Removal of the remaining reflection hyperplanes yields the space X of n-tuples (z_{j}) ∈ C^{∗n} with pairwise different squares z_{j}^{2}.

The configuration space C^{n}(C^{∗}) is the space of subsets of C^{∗} with cardinality n. As topological space it is defined as Y/S_{n} where Y ⊂ C^{∗n} is the set of n-tuples (y_{j}) with pairwise different components.

(7.3) Proposition. X/W is homeomorphic to C^{n}(C^{∗}).

Proof. We arrive at X/W in two steps: First we form Y′ = X/(ZZ/2)^{n} and then we divide out by the S_{n}-action. The map (z_{j}) ↦ (z_{j}^{2}) yields an S_{n}-equivariant homeomorphism Y′ → Y. □

By (5.2) and (5.3), ZB_{n} ≅ π_{1}(C^{n}(C^{∗})). The elements of π_{1}(C^{n}(C^{∗})) will be interpreted as braids in the cylinder (cylindrical braids). We take (1, ω, ⋯, ω^{n−1}), ω = exp(2πi/n), as base point in C^{n}(C^{∗}). A loop in C^{n}(C^{∗}) lifts to a path

w: I → Y, t ↦ (w_{1}(t), …, w_{n}(t))

with this initial point. Thus we have (1) w(0) = (1, ω, …, ω^{n−1}).

(2) w(1) = (σ(1), …, σ(ω^{n−1})), with a permutation σ of the set ZZ/n = {1, ω, …, ω^{n−1}}.

(3) For j ≠ k we have w_{j}(t) ≠ w_{k}(t).

These data yield a braid z_{w} with n strings in C^{∗} × [0, 1] from ZZ/n × 0 to ZZ/n × 1

z_{w}(t) = {w_{1}(t), …, w_{n}(t)} × t.

Homotopy classes of loops correspond to isotopy classes of such braids. Multiplication of loops corresponds to concatenation of braids, as usual. Thus we have:

(7.4) Theorem. The braid group ZB_{n} is the group of n-string braids in the cylinder C^{∗} × [0, 1]. □

A second interpretation is by symmetric braids in C × [0, 1]. This was already used in [??]. We take the base point (1, 2, …, n) ∈ X. We lift a loop in X/W to a path

w: I → X, t ↦ (w_{1}(t), …, w_{n}(t)).

Then we have:

(1) w(0) = (1, 2, …, n).

(2) w(1) = (±σ(1), …, ±σ(n)) with a permutation σ of {1, …, n}.

(3) For j ≠ k we have w_{j}(t) ≠ ±w_{k}(t).

(4) w_{j}(t) ≠ 0. Let [±n] = {−n, …, −1, 1, …, n}. The data yield a braid with 2n strings in C × [0, 1] from [±n] × 0 to [±n] × 1, namely

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

These braids are ZZ/2-equivariant with respect to (z, t) ↦ (−z, t) and are therefore called symmetric. The theorem of Brieskorn thus gives:

(7.5) Theorem. The group ZB_{n} is isomorphic to the group of symmetric braids with 2n strings. □

Symmmetric braids are drawn as ordinary braids but with additional symmetry with respect to the axis 0 × [0, 1]. Here are figures for the generators t and g_{j}.

??

The symmetry is not the reflection in the axis, but corresponds to a spacial rotation about this axis. The relation tg_{1}tg_{1} = g_{1}tg_{1}t appears in this context as a generalized Reidemeister move.

Braids in the cylinder with n strings can be visualized as ordinary braids with n + 1 strings — the axis of the cylinder is the additional string. This method has been used by Lambropoulou [??]. It allows the reduction of B_{n}-type braids to ordinary Artin braids, also with respect to proofs. The theorem of Brieskorn is then not used.

The twofold covering, ramified along the axis, of the cylinder produces a symmetric braid from a cylindrical one — and vice versa.

The cylinder C^{∗} ×[0, 1] has the universal covering C×[0, 1]. Lifting cylindrical braids with n strings produces n-periodic infinite braids in C × [0, 1] from ZZ × 0 to ZZ × 1. They are invariant with respect to the translation (z, t) ↦ (z + n, t). This gives yet another interpretation of ZB_{n} by n-periodic braid pictures.

The relation between ZB_{n} and ZÃ_{n−1} has the following geometric origin or counterpart. The map

C^{∗n} → C^{∗}, (z_{1}, …, z_{n}) ↦ z_{1} · . . . · z_{n}

is S_{n}-equivariant and induces therefore a map from the configuration space

α: C^{n}(C^{∗}) → C^{∗}.

(7.6) Lemma. The map α is a fibre bundle.

Proof. Let

H = {(z_{1}, …, z_{n}) ∈ C^{∗n} | ∏ z_{j} = 1}.

This is an S_{n}-invariant subset. The map

γ: C^{∗} ×_{ZZ/n} H → C^{∗n}, (z, z_{1}, …, z_{n}) ↦ (zz_{1}, …, zz_{n})

is an S_{n}-equivariant homeomorphism. Thus γ is the fibre bundle with fibre H assoziated to the ZZ/n-principal bundle C^{∗} → C^{∗}, z ↦ z^{n}. In C^{∗n} we have to remove the subset

C = {(z_{1}, …, z_{n}) | there exists i ≠ j such that z_{i} = z_{j}}.

Let D = H ∩ C. Then γ induces an S_{n}-equivariant homeomorphism

γ: C^{∗} ×_{ZZ/n} (H \ D) → C^{∗n} \ S.

This yields the fibre bundle description

C^{∗} ×_{ZZ/n} (H \ T)/S_{n} → C^{∗}

for the configuration space. □

We apply the fundamental group to this fibration and obtain the exact sequence

1 → kernel α_{∗} → ZB_{n} → ZZ → 0.

It can be shown that this is the sequence (2.11), i. e. ZÃ_{n−1} is the fundamental group of the fibre of α.

Our next aim is to describe an additive basis of the Hecke algebra H^{∞}B_{n} by geometric means, i. e. by specifying a certain canonical set of basic braids.

A cylindrical braid with n strings is called descending, if for i < j the i-th string is always overcrossing the j-th string. The i-th string is the one starting at ω^{i}, 0 ≤ i ≤ n − 1. Overcrossing means the following: We look radially and orthogonally from infinity onto the axis. The braid is in general position if we only see transverse double points. The first string we meet, coming from infinity, is the overcrossing one.

(7.7) Theorem. The descending braids form a 𝔎-basis of the algebra H^{∞}B_{n}. The descending braids with winding number zero form a 𝔎-basis of the algebra HÃ_{n−1}.

We use (2.11) to reduce the first statement to the second. For the latter Hecke algebra we have the canonical basis related to the elements of reduced form in the Weyl group, and elements of the Weyl group will be shown to correspond to descending braids. We use the description of the Weyl group elements as nperiodic permutations of ZZ. We represent such a permutation by n straight lines c_{1}, …, c_{n} in the strip IR × [0, 1] starting at {1, …, n} × 0 such that c_{i} and c_{j} have at most one crossing, and then repeat with period n. By slightly moving the endpoints of the c_{j} we can assume that the curves are in general position. The resulting crossings are used to write the permutation as a product of reflections. This product is reduced in the sense of Coxeter group theory (see (??)). It is geometrically obvious that the same configuration of crossings can be realized by a descending braid.

(7.8) Proposition. The set

ℭ = {y_{n−1}^{k}g_{n−1}g_{n−2} … g_{j} | k ∈ ZZ, 1 ≤ j ≤ n}

is a system of representatives for the left cosets of the inclusion W^{∞}B_{n−1} ⊂ W^{∞}B_{n}.

Proof. This is an immediate consequence of the semi-direct product description. The powers of y_{n−1} are representatives for cosets of V_{n−1} ⊂ V_{n}, and the products g_{n−1} … g_{j} are representatives for the cosets of S_{n−1} ⊂ S_{n}. □

We use this Proposition to derive the following result of Lambropoulou and Przytycki which was proved by them in a purely algebraic manner. The relation to standard Hecke algebra bases and the interpretation by descending braids seems more transparent, though.

(7.9) Theorem. Let 𝔅 be the canonical basis of H^{∞}B_{n−1}. Then {bc | b ∈ 𝔅, c ∈ ℭ} is a basis of H^{∞}B_{n}.

Proof. Represent a basis element of H^{∞}B_{n} by a descending braid. □

Recall the construction and definition of a Markov trace in section 2. The last Theorem gives immediately the uniqueness of a Markov trace with given parameters.

(7.10) Corollary. There exists a unique Markov trace on H^{∞}B_{n} with given parameters (s(k) | k ∈ ZZ) and z. □

From a Markov trace (U^{n}) on ZA one obtains a link invariant. Let x̂ denote the Alexander closure of the braid x ∈ ZA_{n−1}. Write x as a product of symbols (crossings) g_{1}, g_{1}^{−1}, …, g_{n−1}, g_{n−1}^{−1}, and let w(x) denote the resulting sum of exponents (writhe of x). Then a link invariant P is obtained by setting

P (x̂) := α^{−w(x)}β^{n}U^{n}(x)

for x ∈ ZA_{n−1}. Related are Markov traces T r = (T r_{n}) on Hecke algebras HA = (HA_{n}). These are 𝔎-linear maps T r_{n}: HA_{n−1} → 𝔎 such that

(1) T r_{n+1}|HA_{n−1} = T r_{n},

(2) T R_{n+1}(xx_{n}) = zT r_{n}(x), x ∈ HA_{n−1} with a parameter z ∈ 𝔎. Here we use the names x_{1}, …, x_{n−1} for the standard generators of the Hecke algebra because we want to distinguish them from the g_{j}. The Hecke algebras are defined with a parameter q ∈ 𝔎^{∗} which enters the quadratic relation x_{j}^{2} = (q − 1)x_{j} + q. The relation between the two notions of Markov traces is the following.

(7.11) Proposition. Let q = p^{2} and β(p − p^{−1}) = α − α^{−1} with p^{2} ≠ 1. Let U = (U^{n}) be a Markov trace on ZA with parameters α, β, as defined in section 3. Let ι: ZA_{n} → (HA_{n})^{∗} be the homomorphism g_{j} ↦ p^{−1}x_{j}. Then there exists a unique Markov trace T r on HA such that T r_{n} ∘ ι = U^{n}. It has parameter z = p^{−1}αβ^{−1}. The corresponding link invariant satisfies the skein relation αP (L+) − α^{−1}P (L_{−}) = (p − p^{−1})P (L_{0}). □

Lambropoulou [??] has proved a Markov theorem for links of type B (symmetric links). The statement is exactly as in the classical case, here called of type A. A Markov trace (T^{n}: T B_{n} → 𝔎) therefore yields an invariant of B-links by setting

P (x̂) = α^{−w(x)}β^{n}T^{n}(x)

for x ∈ ZB_{n}. Here w(x) still counts the exponent sum in terms of the generators g_{j}.

8. Traces on groups

Let G be a group and 𝔎 a commutative ring. A (𝔎-valued) trace on G is a function T: G → 𝔎 such that for all g, h ∈ G

(8.1) T (gh) = T (hg).

Equivalently, a trace is a function constant on conjugacy classes. A trace extends to a 𝔎-linear map T: 𝔎G → 𝔎 from the group algebra 𝔎G such that (1.1) holds for any two elements g, h in the group algebra.

Suppose τ: G → G is an automorphism and T a trace. We call T (strongly) τ-invariant if for all g, h ∈ G the relation

(8.2) T (g · τ(h)) = T (g · h)

holds. If we set g = 1, we have the ordinary τ-invariance T (τ(h)) = T (h). If T is τ_{1}- and τ_{2}-invariant, then also τ_{1}^{−1}- and τ_{1}τ_{2}-invariant. If Γ is a group of automorphisms of G, then a trace is called Γ-invariant, if T is τ-invariant for each τ ∈ Γ. It suffices to check Γ-invariance for a generating set of Γ.

Suppose T_{i} is a trace on G_{i} (i = 1, 2). Then (g_{1}, g_{2}) ↦ T_{1}(g_{1})T_{2}(g_{2}) is a trace on G_{1} × G_{2}. We want to generalize this to semi-direct products.

Let α: Γ → Aut G be a group of G-automorphisms. The semi-direct product G ×_{α} Γ is a group structure on the set G × Γ defined by

(g, σ)(h, τ) := (g · σ(h), στ).

In this group structure we have

(8.3) (1, σ)(g, 1)(1, σ)^{−1} = (σ(g), 1).

We will use the following fact several times.

(8.4) Lemma. A pair of group homomorphisms λ: G → H and μ: Γ → H defines via (g, σ) ↦ λ(g)μ(σ) a homomorphism φ: G ×_{α} Γ → H if and only if for all g ∈ G and σ ∈ Γ the relation λ(σ(g)) = μ(σ)λ(g)μ(σ)^{−1} holds. Each homomorphsm φ has this form for a unique pair (λ, μ). □

Is α: Γ → Aut (G) is an antihomorphism, we define the semi-direct product Γ_{α}× G with multiplication (σ, g)(τ, h) = (στ, τ(g)h).

The following is immediately verified from the definitions.

(8.5) Proposition. Let S be a Γ-invariant trace on G and U a trace on Γ. Then

T: G ×_{α} Γ → 𝔎, (g, σ) ↦ S(g)U(σ)

is a trace on G ×_{α} Γ. □

If φ: G → H is a group homomorphism and T a trace on H, then T ∘ φ is a trace on G. Any function T: G → 𝔎 on an abelian group G is a trace. Characters of finite dimensional representations are traces.