On tensor representation of knot algebras

Tammo tom Dieck

The main purpose of this work is to communicate certain R-matrix relations. They are designed to yield tensor representations of (extended) Hecke algebras of Coxeter type B. We also define various Birman-Wenzl algebras of type B and derive the corresponding R-matrix identities. These identities are also the basis for representations of tangle categories of type B. The tensor representations yield, via quantum traces, Markov traces of the algebras and therefore B-type analogues of Jones polynomials, Kauffman polynomials and the like. The tangle theory will be the subject of another paper. We also collect some material for later use in this series of papers. This is Part V of the series Knot Algebra and Root Systems.

1. The four braid relation

Let W be a module over the integral domain 𝔎. We study automorphisms X and Y of W which satisfy the four braid relation

(1.1) XY XY = Y XY X.

We are particularly interested in the following case:

(1)The automorphism X is an R-matrix; this means W = V ⊗ V for a free module V with basis v_{1}, …, v_{n} and X satisfies the Yang-Baxter equation

(1.2) (X ⊗ 1)(1 ⊗ X)(X ⊗ 1) = (1 ⊗ X)(X ⊗ 1)(1 ⊗ X).

(2) The automorphism Y has the form F ⊗ 1 for an automorphism F of V.

The interest in this case comes from the representation theory of braid groups. Recall that the braid group ZB_{n} associated to the Coxeter graph B_{n}

(1.3) [figure: t; g_{1}; g_{2}; g_{n−1}; 4; B_{n}]

with n vertices has generators t, g_{1}, …, g_{n−1} and relations:

(1.4) \begin{matrix}tg_{1}tg_{1} & = & g_{1}tg_{1}t \\ tg_{i} & = & g_{i}t & i > 1 \\ g_{i}g_{j} & = & g_{j}g_{i} & |i − j| ≥ 2 \\ g_{i}g_{j}g_{i} & = & g_{j}g_{i}g_{j} & |i − j| = 1. \end{matrix}

Given automorphisms F and X as above, we obtain a tensor representation of ZB_{n} on the n-fold tensor power V^{⊗n} of V by setting:

(1.5) \begin{matrix}t & = & F ⊗ 1 ⊗ ⋯ ⊗ 1 \\ g_{i} & = & 1 ⊗ ⋯ ⊗ X ⊗ ⋯ ⊗ 1. \end{matrix}

The X in g_{i} acts on the factors i and i+1. Later we will study the cases when such representations of ZB_{n} induce representations of suitable Hecke algebras and Birman-Wenzl algebras. Moreover, we extend this to representations of tangle categories.

We point out that any two automorphisms X, Y: W → W which satisfy (1.1) yield a representation of ZB_{2} on W.

We begin with a prototype computation which is used later on several occasions. We think of X and Y as given by (2, 2)-matrices of the following type:

(1.6)  X = ( \begin{matrix}Z & 0 \\ 0 & qI \end{matrix} ) ,  Y = ( \begin{matrix}A & B \\ C & D \end{matrix} ) .

Here q ∈ 𝔎^{∗} (the units of 𝔎) and I is the identity matrix. The matrices Z, A (and qI, D) are square matrices of the same size, respectively. A computation of (1.1) in block form yields:

(1.7) Proposition. The relation (1.1) holds if and only if the following equalities hold: (I) ZAZA + qZBC = AZAZ + qBCZ (II) Z(AZB + qBD) = q(AZB + qBD) (III) (CZA + qDC)Z = q(CZA + qDC). □

Equation (II) means that the columns of AZB+qBD are eigenvectors of Z for the eigenvalue q (if they are nonzero). Equation (III) has a similar interpretation for the row vectors of CZA + qDC (or consider the transpose).

(1.8) Corollary. If A and D are zero, then (1.1) holds if and only if BC commutes with Z. □

As a very special case, we think of X and Y in (1.6) as ordinary (2, 2)-matrices over 𝔎. Then Z ∈ 𝔎^{∗} and det(Y) = AD − BC ∈ 𝔎^{∗}. From (1.7) we obtain:

(1.9) Proposition. Let X and Y in (1.6) be (2, 2)-matrices. Then (1.1) holds in the trivial cases Z = q or B = C = 0. If Z ≠ q and B (or C) are nonzero, then (1.1) holds if and only if AZ = −qD. □

T. tom Dieck 2. R-matrices of type A_{n} 3

Suppose we are still in the situation of (1.9). The element ζ = XY XY generates the center of ZB_{2}. Under the assumption AZ = −qD, the matrix of ζ is

(1.10) ζ = − det(X) det(Y)I.

If Y has the characteristic polynomial (Y − b_{1})(Y − b_{2}), then we compute (with Tr(Y) = b_{1} + b_{2})

(1.11) A = −\frac{Z}{q − Z} Tr(Y), D = −\frac{q}{Z − q} Tr(Y), b_{1}b_{2} = AD − BC.

The group ZB_{2} has the presentation ⟨ X, Y | XY XY = Y XY X ⟩. Another presentation is Z′B_{2} = ⟨ X, C | C^{2}X = XC^{2} ⟩. An isomorphism Z′ → Z is given by X ↦ X, C ↦ XY. The element C generates the center. The computations above yield the following result:

(1.12) Proposition. Let 𝔎 be a field. Suppose given (a_{1}, a_{2}), (b_{1}, b_{2}) ∈ 𝔎^{∗2} with a_{1} ≠ a_{2} and b_{1} ≠ b_{2}. Then there exists an irreducible twodimensional representation of ZB_{2}, unique up to isomorphism, such that X, Y have the characteristic polynomial (t − a_{1})(t − a_{2}), (t − b_{1})(t − b_{2}), respectively.

Proof. The characteristic polynomials are invariants of the isomorphism type. We have shown above the existence of a representation where

X = ( \begin{matrix}a_{1} & 0 \\ 0 & a_{2} \end{matrix} ) Y = ( \begin{matrix}A & B \\ C & D \end{matrix} )

and (1.11) determines A, D and BC. Within the isomorphism type we can change B, C to BU, CU^{−1}. Hence we can normalize to B = 1. □

2. R-matrices of type A_{n}

We now turn our attention to the standard R-matrices of quantum group theory. We use the basis v_{ij} = v_{i} ⊗ v_{j} with lexicographical ordering. In order to get some insight, we begin with the simplest non-trivial R-matrix

(2.1)  X = ( \begin{matrix}q \\ & δ & 1 \\ & 1 & 0 \\ & & & q \end{matrix} )  δ = q − q^{−1} ≠ 0.

As explained in section 1, we look for matrices

F = ( \begin{matrix}a & b \\ c & d \end{matrix} )

such that with Y = F ⊗ 1 the four braid relation holds.

(2.2) Proposition. Suppose F is not a multiple of the identity. Then (1.1) holds for the matrices X and F above if and only if a = 0.

Proof. We reorder the basis v_{12}, v_{21}, v_{11}, v_{22}. Then X has the form (1.6) with

Z = ( \begin{matrix}δ & 1 \\ 1 & 0 \end{matrix} ) ,

and Y has the form (1.6) with

A = D = ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) , B = C = ( \begin{matrix}0 & b \\ c & 0 \end{matrix} ) .

We check the conditions (1.7). Equation (I) holds if and only if a^{2}δ = adδ. Hence either a = 0 or a = d. We compute AZB + qBD to be

( \begin{matrix}ac & abδ + qbd \\ qac & bd \end{matrix} ) .

Since (q, 1)^{t} is the eigenvector for the eigenvalue q of Z, we must have, by (II), that ac = 0 and ab = 0. In case a ≠ 0, we arrive at a multiple of the identity. If a = 0, then (II), and dually (III), are satisfied. □

We now study the for braid relation for the standard R-matrix X associated to the root system A_{n−1}, see e. g. [21, p. 171]. It is given by the linear map

(2.3) Xv_{ij} = \{ \begin{matrix}qv_{ij} & i = j \\ v_{ji} & i > j \\ v_{ji} + δv_{ij} & i < j, \end{matrix}

where 1 ≤ i, j ≤ n. (The case n = 2 is displayed in (2.1).)

Proposition (2.2) is a (weak) motivation for considering only matrices F = (f_{ij}) which are bottom-right triangular, i. e. f_{ij} = 0 for i + j ≤ n. As a further motivation for later assumptions, we mention the following computational results.

(2.4) Proposition. Suppose F = (f_{ij}) is bottom-right triangular. Then, in the cases n = 3, 4, the relation (1.1) for Y = F ⊗ 1 and X as in (2.3) implies that all elements of F which do not lie on one of the diagonals are zero. Moreover, f_{33} = f_{44} in the case n = 4. □

If a reader wants to check this: It suffices to consider the matrix positions (5,6) in the case n = 3 and the positions (2,16) and (3,16) in the case n = 4. A similar inspection should yield such a result for all n.

Because of (2.2) and (2.4), we only consider maps F of the following form

(2.5) \begin{matrix}F (v_{j}) & = & α_{j}v_{j} + β_{j}v_{n+1−j} & j ≠ \frac{n+1}{2} \\ F (v_{j}) & = & av_{j} & j = \frac{n+1}{2} \end{matrix}

where α_{j} = 0 for 1 ≤ j < \frac{n+1}{2} and α_{j} = w for \frac{n+1}{2} < j ≤ n. The a-term does not appear for odd n.

(2.6) Theorem. Suppose F has the form (2.5) and X is given as in (2.3). Then the four braid relation (1.1) holds in the case n = 2m. If n = 2m − 1, then (1.1) holds if and only if a^{2} = β_{j}β_{n+1−j} + aw for all j ≠ m.

Proof. For the proof we need a bit of organisation. We take advantage of the fact that X and F have many zeros and repetitions. We have two involutions σ and τ on the set of indices J = {(i, j) | 1 ≤ i, j ≤ n}, namely σ(i, j) = (n+1−j, j) and τ(i, j) = (j, i). Since στστ = τστσ, they formally generate the dihedral group D_{8} of order 8. We decompose J into the orbits under this D_{8}-action. We have to consider 4 orbit types. Set n + 1 − i = i′.

Let n = 2m. Then we have orbits of type (i, i), (i′, i), (i, i′), (i′, i′) of length 4. This is the orbit of (i, j) if i = j or i = j′. If i ≠ j, j′, then the orbit of (i, j) has length 8.

Let n = 2m − 1. Then we have the fixed point (m, m). There is another orbit type of length 4, namely (m, j), (j, m), (j′, m), (m, j′) for j ≠ m. The subspace spanned by an orbit is invariant under X and Y. Therefore it suffices to verify the four braid relation on these subspaces. The matrices involved depend only on the isomorphism type of the orbit. Therefore we need only consider the cases n = 3 and n = 4.

We present some details of the computation.

Let n = 3. We consider the subspace generated by v_{12}, v_{21}, v_{23}, v_{32}. The corresponding matrices have the following form (we set β_{1} = v, β_{3} = u):

X = ( \begin{matrix}δ & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & δ & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ) , Y = ( \begin{matrix}0 & 0 & 0 & u \\ 0 & a & 0 & 0 \\ 0 & 0 & a & 0 \\ v & 0 & 0 & w \end{matrix} ) .

We compute the product

XY = ( \begin{matrix}0 & a & 0 & δu \\ 0 & 0 & 0 & u \\ v & 0 & aδ & w \\ 0 & 0 & a & 0 \end{matrix} ) .

and its square

(XY)^{2} = ( \begin{matrix}0 & 0 & aδu & au \\ 0 & 0 & au & 0 \\ vaδ & va & a^{2}δ^{2} + aw & δvu + aδw \\ va & 0 & a^{2}δ & aw \end{matrix} ) .

If we transpose this matrix and interchange u and v we obtain (Y X)^{2}. On the other hand, we obtain by this procedure the same matrix if and only if a satisfies a^{2} = aw + vu.

Let n = 4. We use the subspace generated by v_{12}, v_{21}, v_{13}, v_{31}, v_{24}, v_{42}, v_{34}, v_{43}. Then X has the block diagonal matrix

X = ( \begin{matrix}H \\ & H \\ & & H \\ & & & H \end{matrix} ) , H = ( \begin{matrix}δ & 1 \\ 1 & 0 \end{matrix} )

and Y the matrix (with notation β_{1} = v_{1}, β_{2} = v_{2}, β_{3} = u_{2}, β_{4} = u_{1})

Y = ( \begin{matrix}& \begin{matrix}0 & 0 \\ 0 & u_{2} \end{matrix} & \begin{matrix}0 & u_{1} \\ 0 & 0 \end{matrix} \\ \begin{matrix}0 & 0 \\ 0 & v_{2} \end{matrix} & \begin{matrix}0 & 0 \\ 0 & w \end{matrix} & & \begin{matrix}0 & u_{1} \\ 0 & 0 \end{matrix} \\ \begin{matrix}0 & 0 \\ v_{1} & 0 \end{matrix} & & \begin{matrix}0 & 0 \\ 0 & w \end{matrix} & \begin{matrix}u_{2} & 0 \\ 0 & 0 \end{matrix} \\ & \begin{matrix}0 & 0 \\ v_{1} & 0 \end{matrix} & \begin{matrix}v_{2} & 0 \\ 0 & 0 \end{matrix} & \begin{matrix}w & 0 \\ 0 & w \end{matrix} \end{matrix} ) .

Empty places contain, as always, a zero. These data yield the product:

XY = ( \begin{matrix}& \begin{matrix}0 & u_{2} \\ 0 & 0 \end{matrix} & \begin{matrix}0 & δu_{1} \\ 0 & u_{1} \end{matrix} \\ \begin{matrix}0 & v_{2} \\ 0 & 0 \end{matrix} & \begin{matrix}0 & w \\ 0 & 0 \end{matrix} & & \begin{matrix}0 & δu_{1} \\ 0 & u_{1} \end{matrix} \\ \begin{matrix}v_{1} & 0 \\ 0 & 0 \end{matrix} & & \begin{matrix}0 & w \\ 0 & 0 \end{matrix} & \begin{matrix}δu_{2} & 0 \\ u_{2} & 0 \end{matrix} \\ & \begin{matrix}v_{1} & 0 \\ 0 & 0 \end{matrix} & \begin{matrix}δv_{2} & 0 \\ v_{2} & 0 \end{matrix} & \begin{matrix}wδ & w \\ w & 0 \end{matrix} \end{matrix} )

and its square

( \begin{matrix}0 & 0 & 0 & u_{1}u_{2}H \\ 0 & 0 & v_{2}u_{1}H & u_{1}wH \\ 0 & v_{1}u_{2}H & zδH & u_{2}w(δH + I) \\ v_{1}v_{2}H & v_{1}wH & v_{2}w(δH + I) & zδH + w^{2}(δH + I) \end{matrix} ) .

This matrix does not change if we transpose it and interchange u and v. This finishes the case n = 4. □

T. tom Dieck 3. A three-dimensional example 7

(2.7) Remark. For n = 2m − 1, the matrix Y satisfies the equation Y^{2} = wY + z with z = β_{j}β_{n+1−j} for j ≠ m. For n = 2m the matrix Y satisfies ∏_{j=1}^{m}(Y^{2}−wY −β_{j}β_{n+1−j}) = 0. Thus, in the generic case, the minimal polynomial of Y has degree n. A similar result as (2.6) holds for the slightly more general R-matrices in [21, p. 171]. Suppose q +q^{−1} is invertible in 𝔎. Then the eigenspace S^{2}(V) of X for the eigenvalue q has the basis v_{ij} + q^{−1}v_{ji} for i < j and v_{ii}; and the eigenspace ∧^{2}(V) for the eigenvalue −q^{−1} has the basis v_{ij} − qv_{ji}, i < j. ♡

3. A three-dimensional example

We aim at results for the root systems B_{n} and C_{n} which are analogous to those in section two. The case dim V = 3 seems to be exceptional and we study it in detail. In any case, explicit computations in higher dimensions are difficult.

We use the following R-matrix X.

q
δ  1
μ λ q^{−1}
1  0
λ  1
δ  1
q^{−1}  0
1  0
q

It uses

δ = q − q^{−1}, μ = δ(1 − q^{−1}), λ = −p^{−1}δ, q^{1/2} = p.

The inverse of X is obtained by reflection in the skew diagonal and replacement of p by p^{−1}. From this, the matrix

E_{1} = \frac{X − X^{−1}}{δ} − 1

is computed to be:

\begin{matrix}−q^{−1} & −p^{−1} & −1 \\ −p^{−1} & −1 & −p \\ −1 & −p & −q \end{matrix}

We see that E_{1} has rank one. We have

(3.1) E_{1}^{2} = −(q + 1 + q^{−1})E_{1}.

As in the previous section, we look for a (3,3)-matrix F such that with Y = F ⊗1 the four braid relation (1.1) holds. We specify F in bottom-right triangular form

F = ( \begin{matrix}0 & 0 & b_{2} \\ 0 & a & a_{2} \\ b_{1} & a_{1} & b \end{matrix} ) .

The product Y X is computed to be the following matrix.

q^{−1}b_{2}
b_{2}
b_{2}q
a a_{2}q^{−1}
aλ  a a_{2}
aδ  a a_{2}q
b_{1} a_{1} bq^{−1}
b_{1}δ a_{1}λ b_{1} a_{1} b
b_{1}μ  b_{1}λ a_{1}δ b_{1}q^{−1} a_{1} bq

In order to test the braid relation we note: Since X is symmetric, XY XY is obtained from Y XY X by transposition and interchange of the indices {1, 2}. We decompose the matrix Y XY X into (3, 3)-blocks. It turns out that each block is a bottom-right triangular matrix. We display the blocks B_{ij}, 1 ≤ i, j ≤ 3.

B_{11} = ( \begin{matrix}0 \\ & 0 \\ & & b_{1}b_{2}μq \end{matrix} )

B_{12} = ( \begin{matrix}0 \\ & 0 & aδb_{2} \\ & b_{1}b_{2}qλ & a_{1}b_{2}qδ \end{matrix} ) , B_{21} = ( \begin{matrix}0 \\ & 0 & a^{2}λ \\ & ab_{1}δ & a_{2}qb_{1}μ + aa_{1}λ \end{matrix} )

B_{22} = ( \begin{matrix}0 & & ab_{2} \\ & 0 & aa_{2} + aa_{2}δ \\ ab_{1} & a_{2}b_{1}qλ + aa_{1} & a_{1}a_{2}qδ + ab + a^{2}δ^{2} \end{matrix} )

B_{13} = ( \begin{matrix}0 & & b_{2}^{2} \\ & ab_{2} & a_{2}b_{2}q \\ b_{1}b_{2} & a_{1}b_{1}q & b_{2}bq^{2} \end{matrix} )

B_{31} = ( \begin{matrix}0 & & b_{1}b_{2} \\ & b_{1}a & b_{1}a_{2}q^{−1} + a_{1}aλ \\ b_{1}^{2} & b_{1}a_{1}q^{−1} + a_{1}b_{1}δ & ab_{1}λ^{2} + b_{1}bq^{−2} + a_{1}^{2}λ + b_{1}bqμ \end{matrix} )

B_{23} = ( \begin{matrix}0 & & a_{2}b_{2} \\ & aa_{2} & ab_{2}qλ + a_{2}^{2}a \\ a_{2}b_{1} & a_{1}a_{2}q + a^{2}δ & a_{2}bq^{2} + aa_{2}qδ \end{matrix} )

B_{32} = ( \begin{matrix}0 & & a_{1}b_{2} \\ & aa_{1} & b_{1}b_{2}δ + a_{1}a_{2} + baδ \\ a_{1}b_{1} & ab_{1}λ + a_{1}^{2} + b_{1}bqλ & a_{2}b_{1}λ + aa_{1}δ^{2} + a_{1}b + a_{1}bqδ \end{matrix} )

B_{33} = ( \begin{matrix}0 & & bb_{2} \\ & ab & a_{2}bq + a_{1}b_{2}qλ \\ b_{1}b & aa_{1}δ + a_{1}bq & b_{1}b_{2}qμ + a_{1}a_{2}qδ + b^{2}q^{2} \end{matrix} )

In order to test the four braid relation, we note the following simplifications: In most places the corresponding matrix entries coincide right away. The other places appear pairwise: interchange the indices. We display only one member of each such pair, the one arising from B_{ij} with i ≤ j. Altogether, we obtain 9 conditions:

\begin{matrix}(1) & a^{2} = b_{1}b_{2}q & B_{12} \\ (2) & a_{1}b_{2}qδ = a_{1}b_{2}qμ + aa_{2}λ & B_{12} \\ (3) & aa_{2}(1 + δ) = aa_{2} + a_{1}b_{2}qλ & B_{22} \\ (4) & a_{1}b_{2}q = b_{2}a_{1}q^{−1} + a_{2}aλ & B_{13} \\ (5) & b_{2}bq^{2} = ab_{2}λ^{2} + b_{2}bq^{−2} + a_{2}^{2}λ + b_{2}bqμ & B_{13} \\ (6) & a_{1}a_{2}q + a^{2}δ = b_{1}b_{2}δ + a_{1}a_{2} + δba & B_{23} \\ (7) & ab_{2}qλ + a_{2}^{2}q = ab_{2}λ + a_{2}^{2} + b_{2}bqλ & B_{23} \\ (8) & a_{2}bq^{2} + aa_{2}qδ = b_{2}a_{1}λ + aa_{2}δ^{2} + a_{2}b + a_{2}bqδ & B_{23} \\ (9) & a_{2}qb + a_{1}b_{2}qλ = aa_{2}δ + a_{2}bq & B_{33} \end{matrix}

We discuss the conditions. (1) is a condition on the diagonal elements of F. We can multiply F by a scalar without changing the four braid relation; we thus could normalize to a = 1. (Since F is required to be invertible, we have ab_{1}b_{2} ≠ 0.) Condition (2) can be rewritten as

(3.2) pa_{1}b_{2} = −aa_{2}.

Therefore a_{1}, a_{2} are both zero or non-zero. Inspection shows that (3), (4), (8), and (9) lead to the same condition (3.2). Equation (5) can be rewritten as

(3.3) a_{2}^{2} = b_{2}(p + p^{−1})(ap^{−1}(p − p^{−1}) − b).

Again (6) and (7) lead to (3.3). It is remarkable that from the potentially 81 conditions only 2 remain.

We normalize a = 1. It is seen that, without essentially restricting the generality, we can assume F to be symmetric. Then b_{1} = b_{2} = p^{−1}. The characteristic polynomial of F is

(3.4) (t − q^{−1})(t^{2} + (1 + q^{−1} − b)t + 1).

We formally split the quadratic factor (t + ρ)(t + ρ^{−1}). Then

ρ + ρ^{−1} − 2 = q^{−1} − 1 − b.

In the case ρ = 1 we have a_{1}a_{2} = 0 and F satisfies a quadratic equation. If we write ρ = γ^{2} and introduce α = (p + p^{−1})^{1/2}(γ − γ^{−1}), then F obtaines the form

(3.5) F = ( \begin{matrix}0 & 0 & p^{−1} \\ 0 & −1 & p^{−1/2}α \\ p^{−1} & p^{−1/2}α & 1 + q^{−1} − (ρ + ρ^{−1}) \end{matrix} ) .

The inverse of this matrix F is obtained by reflection in the skew diagonal and application of the involution p ↦ p^{−1}, γ ↦ −γ^{−1}. The matrix

Ẽ = F + F^{−1} + (ρ + ρ^{−1})I

is computed to be

(3.6)  Ẽ = ( \begin{matrix}1 + p^{2} & p^{1/2}α & p + p^{−1} \\ p^{1/2}α & (γ − γ^{−1})^{2} & p^{−1/2}α \\ p + p^{−1} & p^{−1/2}α & 1 + p^{−2} \end{matrix} ) .

It satisfies the relation

(3.7)  ẼF = F Ẽ = q^{−1}Ẽ.

We set E = (ρ + ρ^{−1})Ẽ and compute

(3.8) E^{2} = (\frac{q + q^{−1}}{ρ + ρ^{−1}} + 1) E.

(Since Ẽ has rank one, the coefficient ν in the equality Ẽ^{2} = νẼ is computed to be the scalar product of the first row with itself.) Later we also need to know the following identity. It is used for the representation theory of tangle categories.

(3.9) E_{1}Y XY = E_{1}.

Since E_{1} has rank one, there must hold identities of the form E_{1}Y^{±1}E_{1} = λ_{±}E_{1}. The coefficients are determined by applying both sides to a suitable vector, e. g. (0, 0, 1, 0, 0, 0, 0, 0, 0). The result is as follows:

(3.10) λ_{±} = q^{±1}(ρ + ρ^{−1} − 1).

From this, one obtaines finally

(3.11) E_{1}EE_{1} = (q + 1 + q^{−1})E_{1}.

T. tom Dieck 4. R-matrices of type B_{n} and C_{n} 11

(3.12) Remark. The equation (3.9) also determines the matrix Y. Since E_{1} has many zeros, this simplifies the computation. Also, since E_{1} has rank one, it suffices to determine a single row. Since (3.9) is inhomogeneous in Y, the normalization of Y is determined up to sign, once the other data are fixed. ♡ For tangle theory it is important to decompose E

[figure: E: V; φ; 𝔎; κ; V,  V = 𝔎^{3}.]

We aim at a symmetric decomposition and set (using the standard basis e_{i})

(3.13) \begin{matrix}φ: V → 𝔎, & (e_{1}, e_{2}, e_{3}) ↦ (p^{1/2}\sqrt{p + p^{−1}}, γ − γ^{−1}, p^{−1/2}\sqrt{p + p^{−1}}) \\ κ: 𝔎 → V, & 1 ↦ p^{1/2}\sqrt{p + p^{−1}}e_{1} + (γ − γ^{−1})e_{2} + p^{−1/2}\sqrt{p + p^{−1}}e_{3}. \end{matrix}

These maps satisfyE = κφ. Moreover, we have

(3.14) F κ(1) = q^{−1}κ(1).

4. R-matrices of type B_{n} and C_{n}

In this section we establish a four braid relation for R-matrices of Coxeter-Dynkin type B_{n} and C_{n}. We begin by specifying the relevant R-matrices. See [28] and [29] for the use of these matrices in knot and tangle theory.

The matrix X_{n} = X(B_{n}) describes an automorphism of V ⊗ V in the lexicographical basis v_{ij}, 1 ≤ i, j ≤ 2n + 1 = m. If i + j ≠ m + 1, then X_{n} coincides with a matrix of A-type, as specified in (2.3). The subspace of V ⊗ V generated by v_{ij}, i + j = m + 1 is invariant under X_{n}. The corresponding matrix block will be denoted by Z_{n}. We describe Z_{n} inductively. Again we use δ = q − q^{−1} and p = q^{1/2}. We let Z_{0} denote the unit matrix of size 1. The matrix Z_{n} is a symmetric matrix with central matrix Z_{n−1}, i. e. we adjoin to Z_{n−1} new rows and columns in the positions 1 and 2n + 1. The (2n + 1)-row is (q^{−1}, 0, …, 0). The first row is

−δ(q^{−(2n−1)} − 1, q^{−(2n−3)}, …, q^{−n}, p^{−(2n−1)}, q^{−n+1}, …, q^{−1}, 1) + (0, …, 0, q).

We need information about the eigenspaces of Z_{n}. We set e_{j} = v_{j,m+1−j}. (For (4.1) and (4.2), we assume that (q + q^{−1})(q − q^{−m+1})(q^{−1} + q^{−m+1}) is invertible in 𝔎. There analogous assumptions in (4.4) and (4.5). See section 6 for these conditions.)

(4.1) Proposition. The matrix Z_{n} has eigenvalues q, −q^{−1}, q^{−m+1}. (1) The q-eigenspace has the basis

z_{j} = qe_{j} + q^{−1}e_{m+1−j} − e_{j+1} − e_{m+1−(j+1)},  1 ≤ j ≤ n − 1
z_{n} = qe_{n−1} − (p + p^{−1})e_{n} + q^{−1}e_{n+1}.

(2) The (−q^{−1})-eigenspace has the basis

y_{j} = (e_{j} − e_{m+1−j}) − (q^{−1}e_{j+1} − qe_{m+1−(j+1)})  1 ≤ j ≤ n − 1
y_{n} = e_{n−1} + (p − p^{−1})e_{n} − e_{n+1}.

(3) An eigenvector for q^{−m+1} is

(1, q, …, q^{n−1}, p^{2n−1}, q^{n}, …, q^{2n−1}).

Proof. We prove (1) by induction on n. The case n = 1 is a simple verification. For the induction step it remains to check:

(1) z_{1} is an eigenvector of Z_{n}.

(2) The scalar product of the first row of Z_{n} with z_{2}, …, z_{n} is zero. For (1), we compute the scalar product of z_{1} with the first row to be

δ(1 − q^{−2n+1})q + δq^{−2n+2} + δq^{−1} + q^{−1}q^{−1} = q^{2},

and with the second row to be

−δq^{−2n+2}q − δ(1 − q^{−2n+3}) − q^{−1} = q.

These values are correct. The scalar product with rows 3 to m gives trivially the correct result. For (2), we compute the scalar product with z_{2} to be

−δq^{−2n+2}q + δq^{−2n+3} + δp^{−2} − δq^{−2} = 0

and similarly for z_{3}, …, z_{n−1}. For z_{n} we have

−δq^{−n}q + δ(p + p^{−1})p^{−2n+1} − δq^{−n+1}q^{−1} = 0.

The verification for the other eigenspaces is similar. □

Let now F_{n} denote a (2n + 1, 2n + 1)-matrix as in (2.5) with w = p^{−1} − p, a = −p, β_{j} = 1 and set Y_{n} = F_{n} ⊗ 1.

(4.2) Theorem. The matrices X(B_{n}) and Y_{n} satisfy the four braid relation.

Proof. As before, we decompose X(B_{n}) and Y_{n} into suitable blocks. The subspace W of V ⊗ V generated by v_{ii}, v_{i,m+1−i}, 1 ≤ i ≤ m is invariant under X and Y. The remaining basis elements generate a subspace where the four braid relation is satisfied by the results of section 2. We order the basis of W as follows:

v_{1,m}, v_{2,m−1}, …, v_{m,1}, v_{11}, …, v_{mm}.

We assume that v_{n,n} occurs among the v_{j,m+1−j}. In that case, we are in the formal situation of (1.6) with Z = Z_{n} and A = Dia(0, …, 0, −p, w, …, w), D = Dia(0, …, 0, w, …, w) with w appearing n times in A and D. Moreover B^{t} = C and

B = ( \begin{matrix}0 & J \\ 0 & 0 \\ J & 0 \end{matrix} ) , J = ( \begin{matrix}& 1 \\ 1 \end{matrix} ) ,

i. e. J is the co-unit matrix. In order to check the conditions of (1.7), we use:

(4.3) Proposition. The column vectors of AZB + qBD generate the qeigenspace of Z.

Let us assume this for the moment. Then (1.7, II) holds. Since Z, A, D are symmetric and B^{t} = C, the condition (1.7, III) follows by transposition. The matrix BC is the unit matrix, but with the central diagonal element replaced by zero. In order to verify (1.7, I), one notes the following facts which are simple consequences of the general structure of the matrix blocks:

qBCZ is qZ with the n-th column replaced by zero;

qZBC is qZ with the n-th row replaced by zero;

AZAZ is the n-th row of Z, multiplied by q, and the rest replaced by zero; ZAZA is the n-th column of Z, multiplied by q, and the rest replaced by zero.

(1.7, I) is a direct consequence of these facts. □

Proof of (4.3). By inspection, one sees that the non-zero columns S_{1}, …, S_{n} of AZB + qBD are given in terms of the eigenvectors (4.1) as follows: wz_{j} = S_{j} − qS_{j+1} for 1 ≤ j ≤ n − 1 and wz_{n} = S_{n}. □

We now consider the R-matrices X(C_{n}) = X′_{n} which act on V ⊗ V with dim V = 2n = m. Again, the v_{ij} for i+j ≠ 2n+1 are mapped as for X(B_{n}). The subspace of V ⊗ V generated by the v_{ij} with i + j = 2n + 1 is invariant under X′_{n} and the corresponding matrix block Z′_{n} is defined inductively, beginning with

Z′_{1} = ( \begin{matrix}δ(1 + q^{−2}) & q^{−1} \\ q^{−1} & 0 \end{matrix} ) .

The matrix Z′_{n} is a symmetric matrix with central matrix Z′_{n−1}. Its 2n-th row is (q^{−1}, 0, …, 0). The first row is

δ(1 + q^{−2n}, q^{−2n+1}, …, q^{−n−1}, −q^{−n+1}, …, −q^{−1}, −1) + (0, …, 0, q).

Again we need the eigenspace structure of Z′_{n}.

(4.4) Proposition. Z′_{n} has eigenvalues q, −q^{−1}, −q^{−m−1}.

(1) The q-eigenspace has the basis

z′_{j} = qe_{j} + q^{−1}e_{m+1−j} − e_{j+1} − e_{m+1−(j+1)}  1 ≤ j ≤ n − 1
z′_{n} = qe_{n} + q^{−1}e_{n+1}.

(2) The (−q^{−1})-eigenspace has the basis

y′_{j} = (e_{j} − e_{m+1−j}) − (q^{−1}e_{j+1} − qe_{m+1−(j+1)}), 1 ≤ j ≤ n − 1.

(3) An eigenvector for the eigenvalue −q^{−m−1} is

(1, q, …, q^{n−1}, −q^{n+1}, …, −q^{2n}).

Proof. The proof is by induction on n as for (4.1). □

Let now F′_{n} be a (2n, 2n)-matrix as in (2.5) with β_{j} = β and set Y′_{n} = F′_{n} ⊗ 1.

(4.5) Theorem. The matrices X(C_{n}) and Y′_{n} satisfy the four braid relation.

Proof. We use the same method as for Theorem (4.2). We have A = D = Dia(0, …, 0, w, …, w) with w appearing n times and B = C = βJ. It is easy to see that the non-zero columns S′_{1}, …, S′_{n} of AZ′B + qBD have the form S′_{j} − q^{−1}S′_{j+1} = βwz′_{j} for 1 ≤ j ≤ n − 1 and S′_{n} = βwz′_{n}. Therefore (1.7, II and III) hold. The matrix BC is a multiple of the identity, hence commutes with Z′. The matrix Z′AZ′A is zero. Therefore (1.7, I) is satisfied too. □

The inverse of the matrices X(B_{n}) and X(C_{n}) is obtained by reflection in the skew diagonal and replacement of p by p^{−1}. The matrix X = X(B_{n}) satisfies

0 = (X − q)(X + q^{−1})(X − q^{−m+1}), m = 2n + 1

and the matrix X = X(C_{n})

0 = (X − q)(X + q^{−1})(X + q^{−2n−1}).

The matrices E, defined by (q −q^{−1})(I +E) = X −X^{−1} are in both cases of rank one. Their entries are nonzero only in places (ij, kl) with i + j = k + l = m + 1. This essential part of E is symmetric and therefore determined by its first row. This row is in the case B_{n}

−(q^{−2n+1}, q^{−2n+2}, …, q^{−n}, p^{−2n+1}, q^{−n+1}, …, 1)

and in the case C_{n}

(q^{−2n}, …, q^{−n−1}, −q^{−n+1}, …, −1).

The matrices E satisfy the following equations

(4.6) \begin{matrix}X(B_{n})E(B_{n}) = E(B_{n})X(B_{n}) = q^{−2n}E(B_{n}) \\ X(C_{n})E(C_{n}) = E(C_{n})X(C_{n}) = −q^{−2n−1}E(C_{n}) \\ E(B_{n})^{2} = ([2n]_{q} + 1)E(B_{n}), E(C_{n})^{2} = ([2n + 1]_{q} − 1)E(C_{n}). \end{matrix}

Here we have used the quantum numbers [h]_{q} = (q^{h} − q^{−h})/(q − q^{−1}). For tangle theory and Birman-Wenzl algebras we need relations between E- and Y -matrices. It turns out that other normalizations of the Y -matrices have better properties. From now on we use the previously defined matrices multiplied by p^{−1}. In this case, F^{−1} is obtained from F by reflection in the skew diagonal and replacement of p by p^{−1}. It satisfies (F − q^{−1})(F + 1) = 0.

(4.7) Theorem. The matrices Y just defined satisfy in the cases B_{n} and C_{n} the identity Y XY E = E.

T. tom Dieck 5. The example H^{∞}B_{2} 15

Proof. As in previous proofs we decompose V ⊗ V into invariant subspaces. On the D_{8}-orbits the matrices E are zero. For the remaining part we use the notations of (4.2) and (4.5). The identity in question is then equivalent to the two equations

(AZA + qBC)E = E, (AZB + qBD)E = 0.

Inspection shows that with our new normalization of Y the matrix AZA + qBC is actually the identity, thus the first equation holds. (This seems also a better explanation for (1.7, I).) The eigenvalue relation (1.7, II) yields by the very definition of E the second equation. □

(4.8) Remark. The decomposition of V ⊗ V into eigenspaces is the decomposition into irreducible representations of the relevant quantum group. Except for the one-dimensional trivial representation the second symmetric power S^{2}V and second alternating power Λ^{2}V are relevant. ♡

(4.9) Remark. One can also establish a four braid relation for automorphisms of the form 1 ⊗ F. This is obtained from the previously considered form by conjugation with the interchange automorphism τ: V ⊗V → V ⊗V, v⊗w ↦ w⊗v. This is due to the fact that τ ∘ X(q) ∘ τ = X(q^{−1})^{−1}. ♡

(4.10) Remark. A tedious computation shows that for X(B_{2}) and a bottomright triangular matrix F_{2} the four braid relation holds if and only if the matrix is the one used in (4.2). In this sense, the case X(B_{1}) seems to be exceptional. I am grateful to R. Häring [15] for an independent verification of this computation.

5. The example H^{∞}B_{2}

As an example for later investigations we communicate some computations for the generalized Hecke algebra H^{∞}B_{2}. This is the associative algebra with 1 over 𝔎 with generators t, g and relations tgtg = gtgt and g^{2} = (v−1)g+v with v ∈ 𝔎^{∗}. The following commutation rule is basic for many computations.

(5.1) Theorem. For k ∈ ZZ and l ∈ IN_{0} the following commutation rule holds:

gt^{k}gt^{l} − t^{l}gt^{k}g = (1 − v) ∑_{j=1}^{l}(t^{j}gt^{k+l−j} − t^{k+l−j}gt^{j}).

We can write this rule more symmetrically as a commutator rule

[gt^{k}g + (v − 1) ∑_{j=1}^{l} t^{j}gt^{k−j}, t^{l}] = 0.

For 0 ≤ k ≤ l the same relation holds for the sum from 0 to k; thus we have an element that commutes with t^{l}. By multiplication from right and left with t^{−l} one obtains similar results for l < 0.

16 T. tom Dieck

Proof. By induction over k and l for k, l ∈ IN. For details see [1]. □

The commutation rules are used to derive a basis for the algebra (compare section 8). (5.2) Theorem. The algebra H^{∞}B_{2} has the following bases: Either t^{l}gt^{k}, t^{l}gt^{k}g^{−1} for k, l ∈ ZZ or t^{l}gt^{k}, t^{l}gt^{k}g for k, l ∈ ZZ. □

We use these results to determine all traces on the algebra. (5.3) Theorem. The traces T on H^{∞}B_{2} with T (1) = 1 correspond bijectively to pairs of families (τ(n) | n ∈ ZZ) and (σ(k, l) = σ(l, k) | (k, l) ∈ ZZ^{2}). Proof. By (5.2), we can define a linear form T on the algebra by

T (1) = 1, T (t^{k}gt^{l}) = τ(k + l), T (t^{k}gt^{l}g) = σ(k, l).

We have to show that any such linear form satisfies T (xy) = T (yx) for all basis elements x and y. We make a series of deductions.

(1) We have that T (t^{k+l}g) = T (t^{k}gt^{l}) = τ(k + l) depends only on k + l.

(2) We must have T (t^{k}gt^{l}) = T (t^{l}gt^{k}) for any trace T. The commutation rule (=CR) (7.1) shows that σ(a, b) has to be symmetric in a, b.

(3) T (t^{u}gt^{k}gt^{l}) = T (t^{u+l}gt^{k}g). Proof. We apply the CR to gt^{k}gt^{l} and obtain

t^{u}gt^{k}gt^{l} = t^{u}t^{l}gt^{k}g + (1 − v) ∑_{j}(t^{u+j}gt^{k+l−j} − t^{u+k+l−j}gt^{j}).

Because of (1) the sum terms cancel.

(4) T ((t^{u}gt^{v})(t^{k}gt^{l})) = T ((t^{k}gt^{l})(t^{u}gt^{v})). Proof. By (3), the left, right side equals T (t^{u+l}gt^{v+k}g), T (t^{v+k}gt^{l+u}g), respectively, and both are equal to σ(l + u, v + k) by (2).

(5) T (gt^{a}gt^{b}g) = (v − 1)σ(a, b) − vτ(a + b). Proof. We apply the CR to (gt^{a}gt^{b})g and obtain

gt^{a}gt^{b}g = t^{b}gt^{a}g^{2} + (1 − v) ∑_{j=1}^{b}(t^{j}gt^{a+b−j}g − t^{a+b−j}gt^{j}g).

The sum terms cancel, by (2). Now use the quadratic relation for g^{2}.

(6) T (t^{m}gt^{k}gt^{l}g) = T (t^{l}gt^{m}gt^{k}g).

Proof. We apply the CR on both sides to t^{m}gt^{k}g. The resulting sum terms cancel by (3). Then one uses (5) to show

T (gt^{k}gt^{m+l}g) = T (t^{l}g^{2}t^{k}gt^{m}).

(7) The trace of t^{a}gt^{b}gt^{c}g is invariant under permutations of a, b, c.

Proof. The CR is applied in t^{l}gt^{m}gt^{k}g to t^{m}gt^{k}g in order to show that the trace is invariant under permutation of k, l. The CR is applied to t^{l}gt^{m}g in oder to show that the trace is invariant under permutation of m, k.

(8) T (t^{u}gt^{v}gt^{k}gt^{l}) = T (t^{k}gt^{l}gt^{u}gt^{v}g).

Write gt^{v}gt^{k}g as a linear combination of the basis and use (3) in order to show that the left hand side of the claim has the trace T (t^{u+l}gt^{v}t^{k}g). Then use (7).

(9) T (t^{a}gt^{b}gt^{c}gt^{d}g) = T (t^{d}gt^{a}gt^{b}gt^{c}g).

Proof. Write gt^{b}gt^{c}g in terms of the basis and use the invariance properties already known. □

The left hand side of (9) is invariant under all permutations of a, b, c, d. Does a similar invaraince hold for longer products of this form?

The algebra H^{∞} = H^{∞}B_{2} is isomorphic to the group algebra of the corresponding infinite Weyl group. The group algebra W^{∞} of this Weyl group has generators T, G and relations G^{2} = 1 and GT GT = T GT G.

(5.4) Theorem. Suppose v + 1 ∈ 𝔎^{∗}. The following assigments define inverse isomorphisms H^{∞} ≅ W^{∞}:

G ↦ \frac{1}{v + 1}(2g + (1 − v)), T ↦ \frac{1}{v + 1}(1vt + (v − 1)gt),

g ↦ \frac{1}{2}((v + 1)G + (v − 1)), t ↦ ((1 − v)G + (1 − v))T.

Proof. One has to show that the assignments yield well defined maps. The elements

e_{1} = \frac{g + 1}{v + 1}, e_{2} = \frac{v − q}{v + 1}

are orthogonal idempotens with sum 1 in H^{∞}. Therefore

e_{1} − e_{2} = \frac{1}{v + 1}(2g + (1 − v))

satisfies T^{2} = 1. The compatibility relations are verified more easily from other presentations of the algebras involved. We state these presentations but do not translate the results to the present situation. □

The algebra H^{∞} has the presentation (see section 8) H^{′∞} with generators f, g, c, c^{−1} and relations g^{2} = (v − 1)g + v, f^{2} = (v − 1)f + v, cgc^{−1} = f, cfc^{−1} = g, cc^{−1} = c^{−1}c = 1. The correspondence is given by the following table:

\begin{matrix}H^{∞} & H^{′∞} \\ g & g \\ t & g^{−1}c \\ gt & c \\ gtgt^{−1}g^{−1} & f \end{matrix}

The algebra W^{′∞} has a similar presentation with generators G, F, C, C^{−1} and relations above for v = 1.

(5.5) Theorem. The following assignments define inverse isomorphisms H^{′∞} ≅ W^{′∞}:

G ↦ \frac{1}{v + 1}(2g + 1 − v), F ↦ \frac{1}{v + 1}(2f + 1 − v), C ↦ c

g ↦ \frac{1}{2}((v + 1)G + v − 1), f ↦ \frac{1}{2}((v + 1)F + v − 1), c ↦ C.

□

By combining the isomorphisms, one sees that the elements

\frac{1}{v + 1}(2vt + (v − 1)gt), \frac{1}{v + 1}(2vgtg^{−1} + (v − 1)gt)

commute in H^{∞}; these elements correspond to CG and GC, respectively.

Suppose v = p^{2} and set d = p + p^{−1}. Let T^{∞}B_{2} denote the algebra over 𝔎[D] with generators e, τ, τ^{−1} and relations ττ^{−1} = τ^{−1}τ = 1, e^{2} = de, τ^{2}e = e = eτ^{2}, and eτ^{±1}e = De. This is a kind of Temperley-Lieb algebra. We use the presentation of H^{∞}B_{2} with generators g and c, c^{−1}. The assignment φ(g) = pe−1 and φ(c) = τ yields a homomorphism φ: H^{∞}B_{2} → T^{∞}B_{2}. We also have the Temperley-Lieb algebra T Ã with generators e, f and relations e^{2} = de, f^{2} = df, efe = D^{2}e, fef = D^{2}f. The assignment e ↦ e, f ↦ τeτ^{−1} yields an embedding ψ: T Ã → T^{∞}B_{2}. The algebra T Ã has the automorphism τ which interchanges e and f. In analogy to (8.23) we have an isomorphism

T Ã ⊗ 𝔎[τ, τ^{−1}] ≅ T^{∞}B_{2}.

(The tensor product is twisted by τ.) The homomorphism φ above is related to the Kauffman calculus of B_{2}-braids. The algebra T^{∞}B_{2} has a geometrically defined trace which comes from an interpretation by ZZ-equivariant bridges (section 10). The parameter D counts snakes, in the sense of section 10.

T. tom Dieck 6. Some formulas 19

6. Some formulas

Let R be an integral domain. for use in the next section, we collect some formulas for the algebra A = R[X]/I, where I is the ideal generated by

p = (X − a_{1})(X − a_{2}) ⋯ (X − a_{n}), a_{j} ∈ R.

We consider the generic case when the a_{j} are pairwise different. We assume that the a_{j} are units in R. Then we have: (6.1) Proposition. The element X ∈ A is invertible. □

We set

E′_{j} = ∏_{i,i≠j}(X − a_{i}).

From p = 0 we see that the E′_{j} are pairwise orthogonal (6.2) E′_{j}E′_{k} = 0 j ≠ k and satisfy (6.3) XE′_{j} = E′_{j}X = a_{j}E′_{j}. The E′-elements are almost idempotent (6.4) E′_{j}^{2} = α_{j}E′_{j}, with

(6.5) α_{j} = ∏_{i,i≠j}(a_{j} − a_{i}).

If we assume that

(6.6) Δ = ∏_{i<j}(a_{i} − a_{j})

is invertible in R, then the elements (6.7) E_{j} = α_{j}^{−1}E′_{j} are pairwise orthogonal idempotents. The sum ∑_{i} E_{i} is formally a polynomial of degree n − 1 in R[X] which assumes the value 1 at the a_{j} and is therefore the constant 1. Hence: (6.8) Proposition. If Δ ∈ R^{∗}, then the E_{1}, …, E_{n} are an R-basis of A and a decomposition of 1 into orthogonal idempotents. □

The R-module A is always free of rank n with basis 1, X, …, X^{n−1}. The E′_{j} have with respect to this basis, up to sign, the determinant Δ.

We have

(6.9) ∏(X^{−1} − a_{j}^{−1}) = ∏(a_{j}X)^{−1} ∏(a_{j} − X).

If we define A over R = ZZ[a_{1}, a_{1}^{−1}, …, a_{n}, a_{n}^{−1}], then the assignment a_{j} ↦ a_{j}^{−1}, X ↦ X^{−1} defines an involution ι on A. The elements ι(E_{j}) are therefore also an orthogonal decomposition of 1 (in A[Δ^{−1}]). From the defining equation ι(E′_{j}) = ∏_{i,i≠j}(X^{−1} − a_{j}^{−1}) we see that ι(E_{j})E_{i} = 0 for i ≠ j. Therefore we have:

(6.10) ι(E_{j}) = E_{j}, ι(E′_{j}) = (−1)^{n−1} ∏_{i,i≠j}(a_{j} − a_{j}^{−1})E′_{j}.

What we have done so far just makes explicit the decomposition of A provided by the Chinese remainder theorem.

7. Birman-Wenzl algebras

We recall the definition of Birman-Wenzl algebras for simply laced Coxeter graphs, see [2] and [33].

Given a Coxeter graph (S, m) with values m(s, t) ∈ {1, 2, 3}. We choose parameters q, x, λ ∈ 𝔎 and assume 𝔎^{∗}. The Birman-Wenzl algebra

BW (S, m) = BW (S, m; q, x, λ)

is the associative algebra with 1 with generators (G_{s}, G_{s}^{−1}, E_{s} | s ∈ S) and the following relations (m = m(s, t) always):

(1)  G_{s}G_{s}^{−1} = G_{s}^{−1}G_{s} = 1
(2)  G_{s}G_{t} = G_{t}G_{s},  m = 2
(3)  G_{s}G_{t}G_{s} = G_{t}G_{s}G_{t},  m = 3
(4) G_{s} − G_{s}^{−1} = (q − q^{−1})(E_{s} + 1)
(5)  E_{s}E_{t} = E_{t}E_{s},  m = 2
(6)  E_{s}E_{t}E_{s} = E_{t}E_{s}E_{t},  m = 3
(7)  E_{s}^{2} = (x − 1)E_{s}
(8) E_{s}G_{t}^{±1}E_{s} = λ^{±1}E_{s},  m = 3
(9)  G_{s}E_{s} = E_{s}G_{s} = λ^{−1}E_{s}.
(10)  G_{s}G_{t}E_{s} = E_{t}G_{s}G_{t},  m = 3
(11)  G_{s}G_{t}E_{s} = E_{t}E_{s},  m = 3
(12)  E_{t}G_{s}G_{t} = E_{t}E_{s},  m = 3
(13)  G_{s}E_{t}G_{s} = G_{t}^{−1}E_{s}G_{t}^{−1},  m = 3
(14)  G_{s}E_{t}E_{s} = G_{t}^{−1}E_{s},  m = 3
(15)  E_{s}E_{t}G_{s} = E_{s}G_{t}^{−1},  m = 3

For q ≠ 1, the parameter x is determined by the identity (q − q^{−1})x = λ − λ^{−1}. The original Birman-Wenzl algebra [2] belongs to the graph A_{n−1}. There are

T. tom Dieck 7. Birman-Wenzl algebras 21

corresponding generalized Brauer centralizer algebras; they are the special cases q = 1, λ = 1. The Brauer centralizer algebras for the graph B_{n} have a different definition. They are studied in [27]. If q − q^{−1} ∈ 𝔎^{∗}, then E_{s} can be computed from X_{s}; hence we can eliminate these generators. This yields:

(7.1) Proposition. Suppose q − q^{−1} ∈ 𝔎^{∗}. Then the relations (10)-(15) are consequences of the remaining ones. Another consequence of the relations is (X_{s}− λ^{−1})(X_{s} + q^{−1})(X_{s} − q) = 0. □

This proposition can be used to redefine the algebra BW (S, m), in the case that q − q^{−1} ∈ 𝔎^{∗}, by generators (X_{s} | s ∈ S) and relations (2), (3), (8) and the cubic relation of the last proposition. Going back and forth uses section 6.

The algebra BW (S, m) has the Hecke algebra H(S, m) as a quotient: Set E_{s} = 0. The (E_{s} | s ∈ S) generate a subalgebra T (S, m), a Temperley-Lieb algebra [9].

We now define the restricted BW-algebras of type B and the corresponding Brauer centralizer algebras (symmetric without free points in the terminology of Reich [27]). The algebra BW (B_{n}) has generators G_{0}, G_{1}, …, G_{n−1}, E_{1}, …, E_{n−1}. The relations between the G_{j}, E_{j} for 1 ≤ j ≤ n − 1 are as above for the graph A_{n−1}. The additional relations are:

(16)  G_{0}G_{1}G_{0}G_{1} = G_{1}G_{0}G_{1}G_{0}
(17)  G_{0}G_{i} = G_{i}G_{0},  i > 1
(18)  G_{0}E_{i} = E_{i}G_{0},  i > 1
(19) (G_{0} − q^{−1})(G_{0} + 1) = 0
(20)  G_{0}G_{1}G_{0}E_{1} = E_{1}.

Again, there is a version which does not use the E_{j}.

(7.2) Remark. The algebra BW (B_{n}) has dimension 2^{n} · (2n − 1)!!.

Proof. This is easily shown for the Brauer algebras, compare [27]. □

(7.3) Theorem. The irreducible representations of BW (B_{n}) are indexed by pairs of Young diagrams of size n − 2k. The Bratteli diagram is given by adding or omitting a box in the Young diagrams.

The proof of the last theorem will be given in a subsequent paper of the series. For a further study of these algebras see also [16].

Finally, we define unrestricted BW-algebras BW^{∗}(B_{n}) of type B_{n}. They are infinite dimensional algebras. In addition to the generators for BW (B_{n}), there is another generator E_{0}. The new additional relations are as follows: (23) replaces (19) above; and

(21)  E_{0}G_{j} = G_{j}E_{0},  j > 1
(22)  E_{0}^{2} = DE_{0}
(23) G_{0} − G_{0}^{−1} = (Q − Q^{−1})(E_{0} + 1)
(24)  E_{1}E_{0}E_{1} = F E_{1}.

Here D, F, and Q are new parameters. Relation (23) is analogous to (4). Occasionally, it is more convenient to use the relation G_{0} + G_{0}^{−1} = (Q + Q^{−1})(E_{0} − 1). This latter relation came up in section 3, with E, F in place of E_{0}, G_{0}.

The algebra BW (B_{n}) has the Hecke algebra H(B_{n}), with suitable parameters, as a quotient: Again set E_{s} = 0. Similarly for the algebra BW^{∗}(B_{n}). The algebra BW^{∗}(B_{n}) contains a Temperley-Lieb algebra of type B_{n}: The subalgebra generated by the E_{0}, …, E_{n−1} (compare [7, 9]).

The R-matrix identities of section 4 are used to construct tensor representations of these algebras and quantum traces; and the latter give Kauffman polynomials [23] of type B. The representation of BW (B_{n}) is on V^{⊗n}. Here we use the presentation by generators G_{0}, …, G_{n−1} alone. The definition is as in (1.5). It is only necessary to check the additional cubic relation and (20). The definition of quantum traces is modelled after [29], [28]. We defer the discussion to another paper, but see [8].

(7.4) Remark. There are similar definitions for BW-algebras associated to the affine root systems B̃_{n} and C̃_{n}. Remark (4.9) allows the construction of tensor representations for these extended algebras. ♡

We describe in detail the example BW (B_{2}) in order to motivate the definition. Consider the algebra with generators X, X^{−1}, Y, E and relations

(1) XX^{−1} = X^{−1}X = 1

(2) XY XY = Y XY X

(3) XE = EX = λE

(4) (Y − q^{−1})(Y + 1) = 0

(5) X − X^{−1} = (q − q^{−1})(E + 1)

(6) E^{2} = dE

(7) Y XY E = E Here λ and q are invertible parameters in the ground ring 𝔎 and d ∈ 𝔎. They have to satisfy (d + 1)(q − q^{−1}) = λ − λ^{−1}, as follows from (3), (5), and (6).

(7.5) Proposition. The following assertions are implied by the relations above: (1) (X − λ)(X − q)(X + q^{−1}) = 0.

(2) XY XY is contained in the center.

(3) EY XY = E.

(4) EY E = \frac{q+λ^{−1}}{q+1} E = : aE.

Proof. The first assertion is a consequence of (3) and (5). The second follows from (2) and (5). For the third, we compute

λY XY = EXY XY = XY XY E = Y XY XE = λY XY E = λE.

For the final identity we compute

EY E = (δ^{−1}(X − X^{−1}) − 1)Y E

T. tom Dieck  8. Braid groups of type B  23
= δ^{−1}(XY E − X^{−1}Y E) − Y E
= δ^{−1}(Y^{−1}E − X^{−1}(q^{−1}Y^{−1} + (q^{−1} − 1))E) − Y E
= δ^{−1}(Y^{−1}E − q^{−1}Y E − (q^{−1})λ^{−1}E) − Y E
= δ^{−1}((qY + (q − 1)E − q^{−1}Y E − (q^{−1} − 1)λ^{−1}E) − Y E
= δ^{−1}((q − 1) − (q^{−1} − 1)λ^{−1})E
= \frac{q + λ^{−1}}{q + 1} E.

We have used (4), (5), and (7) in the computation. □

The algebra is generated by

1, X, Y, XY, Y X, Y XY, XY X, XY XY, E, EY, Y E, Y EY.

It is easily checked from the relations that left and right multiplication by X, Y, E always leads to linear combinations of the displayed elements. If we take the quotient by the relation E = 0, then we obtain a Hecke algebra of type B_{2}. The kernel of the quotient to the Hecke algebra is the twosided ideal spanned by E, Y E, EY, Y EY. It decomposes into the left ideals spanned by E, EY and Y E, Y EY. The matrices of X, Y, and E in the basis Y, EY are

X = ( \begin{matrix}λ & q − 1 \\ 0 & q \end{matrix} ) , Y = ( \begin{matrix}0 & q^{−1} \\ 1 & q^{−1} − 1 \end{matrix} ) , E = ( \begin{matrix}d & a \\ 0 & 0 \end{matrix} ) .

One can verify directly that these matrices satisfy the relations of the algebra and thus define a two-dimensional representation. From the Hecke algebra, we obtain 4 one-dimensional representations and another two-dimensional one. Altogether, we see that the algebra has dimension 12 and the elements above form a basis.

The corresponding Brauer algebra of symmetric Brauer graphs (without free points) is obtained by using the relations with λ = 1 and q = 1.

Because of the inhomogeneous nature of the relation Y XY E = E, the algebra is sensitive to the quadratic relation for Y. If one uses the relation Y − Y^{−1} = Q−Q^{−1}, then the algebra is at most 9-dimensional, and 8-dimensional if λ ≠ Q^{±2}.

8. 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

(8.1) \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}

We also need another presentations of this group.

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

(8.2) \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}.

(8.3) 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

(8.4)  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 (8.4) 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 (8.4)

(8.5) gh = kg.

We use this to show that the relations 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

(8.6) \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}

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 (8.7) ZÃ_{n−1} → G_{n} → ZZ;

the generator 1 ∈ ZZ acts through s on ZÃ_{n−1}. There is a similar semi-direct product where ZZ is replaced by ZZ/nk. 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 (8.6) for the g_{j} together with

(8.8)  sg_{i} = g_{i−1}s,  i mod n.

(8.9) 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 (8.6) and (8.8). This is obvious for (8.6). The relation (e, 1)(x, 0)(e, 1)^{−1} = (s(x), 0) is used to show compatibility with (8.8).

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 (8.4). It is inverse to ψ. □

(8.10) 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

(8.11) ZÃ_{n−1} → ZB_{n} → ZZ.

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

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

The homomorphism ZB_{n} → ZZ in (8.11) 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 (8.11). The arguments which lead to (8.11) 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}.

(8.13) 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

(8.14) N → Q′_{n} → S_{n}

T. tom Dieck  8. Braid groups of type B  27

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 (8.15) 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 (8.13) the subgroup Q′_{n} corresponds to the subgroup

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

(8.16) 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 (8.21). In the proof of (8.16) we use the following: (8.17) 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 (8.5) yields

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

By commutation 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 (8.13), (8.16) between W^{∞}B_{n} and W A_{n−1} has a counterpart for the braid groups. The homomorphism

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

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

(8.18) K_{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.

(8.19) 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. □

(8.20) 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 imitating (8.20):

(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:

(8.21) 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

(8.22) F_{n} → Γ_{n} → ZA_{n−1},

in which g_{j} ∈ ZA_{n−1} acts on F_{n} through δ_{j}. By (8.19) 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 (8.20). □

Proof of (8.16). The elements t_{j} of (8.17) and the elements y_{j} coincide in W^{∞}B_{n}, since g_{j} = g_{j}^{−1} in this group. Lemma (8.19) 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 (8.20) 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 (8.6) and the same quadratic relations. The algebra HB_{n} has generators t, g_{1}, …, g_{n−1}, the braid relations (8.1), 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 t, 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

(8.23) 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}.

(8.24) Proposition. The algebra (8.23) is canonically isomorphic to H^{∞}B_{n}.

Proof. We use the isomorphism (8.3) to redefine the algebra H^{∞}B_{n} by generators c, g_{1}, …, g_{n−1} relations (8.2) 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 (8.12)). 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. □

(8.25) 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}. □

9. Braids of type B

We use a theorem of Brieskorn [4] 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

(9.1) (ZZ/2)^{n} → W B_{n} → S_{n}.

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

T. tom Dieck  9. Braids of type B  31

(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 [4] shows:

(9.2) 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 (9.2) 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.

(9.3) 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 the S_{n}-action. The map (z_{j}) ↦ (z_{j}^{2}) yields an S_{n}-equivariant homeomorphism Y′ → Y. □

By (9.2) and (9.3), 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:

(9.4) 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 [7]. 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:

(9.5) 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].

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 [24]. 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^{∗}.

(9.6) 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 (8.11), 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.

(9.7) 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 (8.11) 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. It is geometrically obvious that the same configuration of crossings can be realized by a descending braid.

(9.8) 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.

(9.9) 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. □

10. Categories of bridges

This section introduces some general terminology for certain graphical categories and algebras.

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). The notation horizontal and vertical is evident in this context. The horizontal arcs with endpoints in P

T. tom Dieck 10. Categories of bridges 35

are called the lower part of the bridge, the horizontal arcs with endpoints in Q the upper part.

(10.1) 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-equivariant. 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, gσ(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:

(10.2) 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.

(3) This follows since G acts cellularly on |σ|′ and the orbit space has a finite number of cells.

Suppose σ is a G-equivariant (P, Q)-bridge and τ a G-equivariant (Q, R)- bridge. Consider the G-space |τ| ∪_{Q} |σ|. The components of this space can be of 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:

(10.3) 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:

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

(10.5) 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 (10.5) by τ ∧σ. 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.

(10.6) 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 two 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 [27] 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, finite orbit set and effictive action on orbits.

T. tom Dieck 11. References 37

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.

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