Quantum groups and cylinder braiding

Tammo tom Dieck and Reinhard Häring-Oldenburg

Abstract. The purpose of this paper is to introduce a new structure into the representation theory of quantum groups. The structure is motivated by braid and knot theory. Representations of quantum groups associated to classical Lie algebras have an additional symmetry which cannot be seen in the classical limit. We first explain the general formalism of these symmetries (called cylinder forms) in the context of comodules. Basic ingredients are tensor representations of braid groups of type B derived from standard R-matrices associated to so-called four braid pairs. These are applied to the Faddeev-Reshetikhin-Takhtadjian construction of bialgebras from R-matrices. As a consequence one obtains four braid pairs on all representations of the quantum group. In the second part of the paper we study in detail the dual situation of modules over the quantum enveloping algebra U_{q}(sl_{2}). The main result here is the computation of the universal cylinder twist.

1. Cylinder forms

Let A = (A, m, e, μ, ε) be a bialgebra^{1} (over the commutative ring 𝔎) with multiplication m, unit e, comultiplication μ, and counit ε. Let r: A ⊗ A → 𝔎 be a linear form. We associate to left A-comodules M, N a 𝔎-linear map^{2}

z_{M,N}: M ⊗ N → N ⊗ M, x ⊗ y ↦ ∑ r(y^{1} ⊗ x^{1})y^{2} ⊗ x^{2},

T. tom Dieck and R. Häring-Oldenburg

where we have used the formal notation^{3} x ↦ ∑ x^{1} ⊗ x^{2} for a left A-comodule structure μ_{M}: M → A⊗M on M. (See [7, p. 186] formula (5.9) for our map z_{M,N} and also formula (5.8) for a categorical definition.) We call r a braid form on A, if the z_{M,N} yield a braiding on the tensor category A-COM of left A-comodules. We refer to [7, Def. VIII.5.1 on p. 184] for the properties of r which make it into a braid form and (A, r) into a cobraided bialgebra. (What we call braid form is called universal R-form in [7].)

1We prefer Latin-Greek duality; thus the comultiplication is not Δ.

2A notation like f: A → B, a ↦ f(a) is used in two different ways. Either the symbols a ↦ f(a) define f: A → B, or they specify a notation for an already defined morphism. For typographical reasons it often seems better not to obscure and interrupt formulas by inserting phrases like ‘defined by’.

3In the literature, notations of this type are attributed to Heyneman and Sweedler. Also the name sigma notation is used. We refer to this notation as the μ-convention. In this paper, we use superscripts for comodule actions and supscripts for comultiplications. The reader may notice that by proper use of the μ-convention summation indices are redundant. Strings of super- or supscripts have to be lexicographic sequences without gaps, and associativity amounts to the rule that any such string can be replaced by another one of the same length.

Let (C, μ, ε) be a coalgebra. Examples of our μ-convention for coalgebras are μ(a) = ∑ a_{1} ⊗ a_{2} and (μ ⊗ 1)μ(a) = ∑ a_{11} ⊗ a_{12} ⊗ a_{2}; if we set μ_{2}(a) = (μ ⊗ 1)μ, then we write μ_{2}(a) = ∑ a_{1} ⊗ a_{2} ⊗ a_{3}. The counit axiom reads in this notation ∑ ε(a_{1})a_{2} = a = ∑ ε(a_{2})a_{1}. The multiplication in the dual algebra C^{∗} is denoted by ∗ and called convolution: If f, g ∈ C^{∗} are 𝔎-linear forms on C, then the convolution product f ∗ g is the element of C^{∗} defined by a ↦ ∑ f(a_{1})g(a_{2}). The unit element of the algebra C^{∗} is ε. Therefore g is a (convolution) inverse of f, if f ∗ g = g ∗ f = ε. We apply this formalism to the coalgebras A and A ⊗ A. If f and g are linear forms on A, we denote their exterior tensor product by f ˆ⊗g; it is the linear form on A ⊗ A defined by a ⊗ b ↦ f(a)g(b). The twist on A ⊗ A is τ(a ⊗ b) = b ⊗ a.

Here is the main definition of this paper. Let (A, r) be a cobraided bialgebra with braid form r. A linear form f: A → 𝔎 is called a cylinder form for (A, r), if it is convolution invertible and satisfies

(1.1) f ∘ m = (f ˆ⊗ε) ∗ rτ ∗ (ε ˆ⊗f) ∗ r = rτ ∗ (ε ˆ⊗f) ∗ r ∗ (f ˆ⊗ε).

In terms of elements and the μ-convention, (1.1) assumes the following form:

(1.2) Proposition. For any two elements a, b ∈ A the identities

f(ab) = ∑ f(a_{1})r(b_{1} ⊗ a_{2})f(b_{2})r(a_{3} ⊗ b_{3}) = ∑ r(b_{1} ⊗ a_{1})f(b_{2})r(a_{2} ⊗ b_{3})f(a_{3})

hold.

Proof. Note that a four-fold convolution product is computed by the formula

(f_{1} ∗ f_{2} ∗ f_{3} ∗ f_{4})(x) = ∑ f_{1}(x_{1})f_{2}(x_{2})f_{3}(x_{3})f_{4}(x_{4}).

We apply this to the second term in (1.1). The value on a ⊗ b is then

(f(a_{1}) · ε(b_{1})) · r(b_{2} ⊗ a_{2}) · (ε(a_{3}) · f(b_{3})) · r(a_{4} ⊗ b_{4}).

By the counit axiom, we can replace ∑ ε(b_{1}) · b_{2} ⊗ b_{3} ⊗ b_{4} by ∑ b_{1} ⊗ b_{2} ⊗ b_{3} (an exercise in the μ-convention), and ∑ a_{1} ⊗ a_{2} ⊗ ε(a_{3}) · a_{4} can be replaced by ∑ a_{1} ⊗ a_{2} ⊗ a_{3}. This replacement yields the second expression in (1.2). The third expression is verified in a similar manner. The first value is obtained from the definition of f ∘ m. □

A cylinder form f (in fact any linear form) yields for each left A-comodule M a 𝔎-linear endomorphism

t_{M}: M → M, x ↦ ∑ f(x^{1})x^{2}.

If φ: M → N is a morphism of comodules, then φ ∘ t_{M} = t_{N} ∘ φ. Since t_{M} is in general not a morphism of comodules we express this fact by saying: The t_{M} constitute a weak endomorphism of the identity functor of A-COM. We call t_{M} the cylinder twist on M. The axiom (1.1) for a cylinder form has the following consequence.

(1.3) Proposition. The linear map t_{M} is invertible. For any two comodules M, N the identities

t_{M⊗N} = z_{N,M}(t_{N} ⊗ 1_{M})z_{M,N}(t_{M} ⊗ 1_{N}) = (t_{M} ⊗ 1_{N})z_{N,M}(t_{N} ⊗ 1_{M})z_{M,N}

hold.

Proof. Let g be a convolution inverse of f. Define the endomorphism s_{M}: M → M via x ↦ ∑ g(x^{1})x^{2}. Then

s_{M}t_{M}(x) = ∑ f(x^{1})g(x^{21})x^{22} = ∑ ε(x^{1})x^{2} = x,

by the definition of a convolution inverse and the counit axiom. Hence s_{M} is inverse to t_{M}.

In order to verify the second equality, we insert the definitions and see that the second map is

x ⊗ y ↦ ∑ f(x^{1})r(y^{1} ⊗ x^{21})f(y^{21})r(y^{221} ⊗ x^{221})y^{222} ⊗ x^{222}

while the third map is

x ⊗ y ↦ ∑ r(y^{1} ⊗ x^{1})f(y^{21})r(y^{21} ⊗ x^{221})f(x^{221})y^{222} ⊗ x^{222}.

The coassociativity of the comodule structure yields a rewriting of the form

∑ y^{1} ⊗ y^{21} ⊗ y^{221} ⊗ y^{222} = ∑(y^{1})_{1} ⊗ (y^{1})_{2} ⊗ (y^{1})_{3} ⊗ y^{2}

and one has a similar formula for x. We now apply (1.2) in the case where (a, b) = (x^{1}, y^{1}).

By definition of the comodule structure of M ⊗N, the map t_{M⊗N} has the form x ⊗ y ↦ ∑ f(x^{1}y^{1})x^{2} ⊗ y^{2}. Again we use (1.2) in the case where (a, b) = (x^{1}, y^{1}) and obtain the first equality of (1.3). □

We also mention dual notions. Let A be a bialgebra with a universal R-matrix R ∈ A ⊗ A. An element v ∈ A is called a (universal) cylinder twist for (A, R), if it is invertible and satisfies

(1.4) μ(v) = (v ⊗ 1) · τR · (1 ⊗ v) · R = τR · (1 ⊗ v) · R · (v ⊗ 1).

The R-matrix R = ∑ a_{r} ⊗ b_{r} induces the braiding

z_{M,N}: M ⊗ N → N ⊗ M, x ⊗ y ↦ ∑ b_{r}y ⊗ a_{r}x.

Let t_{M}: M → M be the induced cylinder twist defined by x ↦ vx. Again the t_{M} form a weak endomorphism of the identity functor. If v is not central in A, then the t_{M} are not in general A-module morphisms. The relations (1.3) also holds in this context.

In practice one has to consider variants of this definition. The universal Rmatrix for the classical quantum groups A is not contained in the algebra A ⊗ A, but rather is an operator on suitable modules. The same phenomenon will occur for the cylinder twist. Will see an example of this situation in Section 8.

If a ribbon algebra is defined as in [7, p. 361], then the element θ^{−1}, loc. cit., is a cylinder twist in the sense above.

2. Tensor representations of braid groups

The braid group ZB_{n} associated to the Coxeter graph B_{n}

[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 (2.1).

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

We recall: The group ZB_{n} is the group of braids with n strings in the cylinder (C\0)×[0, 1] from {1, …, n}×0 to {1, …, n}×1. This topological interpretation is the reason for using the cylinder terminology. For the relation between the root system B_{n} and ZB_{n} see [2].

Let V be a 𝔎-module. Suppose X: V ⊗V → V ⊗V and F: V → V are 𝔎-linear automorphisms with the following properties:

(1) X is a Yang-Baxter operator, i. e., X satisfies the equation

(X ⊗ 1)(1 ⊗ X)(X ⊗ 1) = (1 ⊗ X)(X ⊗ 1)(1 ⊗ X)

on V ⊗ V ⊗ V.

(2) With Y = F ⊗ 1_{V}, the four braid relation Y XY X = XY XY is satisfied.

If (1) and (2) hold, we call (X, F) a four braid pair. For the construction of four braid pairs associated to standard R-matrices see [4]. For a geometric interpretation of (2) in terms of symmetric braids with 4 strings see [3].

Given a four braid pair (X, F), we obtain a tensor representation of ZB_{n} on the n-fold tensor power V^{⊗n} of V by the following assignment:

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

The X in X_{i} acts on factors i and i + 1.

These representations give raise to further operators if we apply them to special elements in the braid groups. We set

t(1) = t, t(j) = g_{j−1}g_{j−2} ⋯ g_{1}tg_{1}g_{2} ⋯ g_{j−1}, t_{n} = t(1)t(2) ⋯ t(n),
g(j) = g_{j}g_{j+1} ⋯ g_{j+n−1}, x_{m,n} = g(m)g(m − 1) ⋯ g(1).

The elements t(j) pairwise commute. We denote by T_{n}: V^{⊗n} → V^{⊗n} and by X_{m,n}: V^{⊗m} ⊗ V^{⊗n} → V^{⊗n} ⊗ V^{⊗m}, respectively, the operators induced by t_{n} and by x_{m,n}.

(2.3) Proposition. The following identities hold

T_{m+n} = X_{n,m}(T_{n} ⊗ 1)X_{m,n}(T_{m} ⊗ 1) = (T_{m} ⊗ 1)X_{n,m}(T_{n} ⊗ 1)X_{m,n}.

Proof. We use some facts about Coxeter groups [1, CH. IV, §1]. If we adjoin the relations t^{2} = 1 and g_{j}^{2} = 1 to (2.1) we obtain the Coxeter group CB_{n}. The element t_{n} is given as a product of n^{2} generators t, g_{j}. The uniquely determined element of CB_{n} has length n^{2} and is equal to t_{n}. The element x_{n,m}t_{n}x_{m,n}t_{m} of CB_{m+n} has length (m + n)^{2} and therefore equals t_{m+n} in CB_{m+n}. By a fundamental fact about braid groups [1, CH. IV, §1.5, Prop. 5], the corresponding elements in the braid group are equal. We now apply the tensor representation and obtain the first equality of (2.3). □

For later use we record:

(2.4) Proposition. The element t_{n} is contained in the center of ZB_{n}. □

3. Cylinder forms from four braid pairs

Let V be a free 𝔎-module with basis {v_{1}, …, v_{n}}. Associated to a Yang-Baxter operator X: V ⊗ V → V ⊗ V is a bialgebra A = A(V, X) with braid form r, obtained via the FRT-construction (see [7, VIII.6 for the construction of A and r). We show that a four braid pair (X, F) induces a canonical cylinder form on (A, r).

Recall that A is a quotient of a free algebra Ã. We use the model

Ã := ⨁_{n=0}^{∞} Hom(V^{⊗n}, V^{⊗n}).

The multiplication of Ã is given by the canonical identification E_{k} ⊗ E_{l} ≅ E_{k+l}, furnishedbyf ⊗ g ↦ f ⊗ g where E_{k} = Hom(V^{⊗k}, V^{⊗k}). The canonical basis of E, given by T_{i}^{j}: v_{k} ↦ δ_{i,k}v_{j} of E_{1}, induces the basis

T_{i}^{j} = T_{i_{1}}^{j_{1}} ⊗ ⋯ ⊗ T_{i_{k}}^{j_{k}}

of E_{k}, where, in multi-index notation, i = (i_{1}, …, i_{k}) and j = (j_{1}, …, j_{k}). The comultiplication of Ã is given by μ(T_{i}^{j}) = ∑_{k} T_{i}^{k} ⊗ T_{k}^{j} while the counit of Ã is given by ε(T_{i}^{j}) = δ_{i}^{j}.

In Section 2 we defined an operator T_{k} ∈ E_{k} from a given four braid pair (X, F). We express T_{k} in terms of our basis

T_{k}(v_{i}) = ∑_{j} F_{i}^{j}v_{j}

again using the multi-index notation v_{i} = v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{k}} when i = (i_{1}, …, i_{k}). We use these data in order to define a linear form

f̃: Ã → 𝔎, T_{i}^{j} ↦ F_{i}^{j}.

(3.1) Theorem. The linear form f̃ factors through the quotient map Ã → A and induces a cylinder form f for (A, r).

Proof. Suppose the operator X = X_{m,n}: V^{⊗m} ⊗ V^{⊗n} → V^{⊗n} ⊗ V^{⊗m} has the form X(v_{i} ⊗ v_{j}) = ∑_{ab} X_{ij}^{ab}v_{a} ⊗ v_{b}. We define a form r̃: Ã ⊗ Ã → 𝔎 by defining

r̃: E_{k} ⊗ E_{l} → 𝔎, T_{i}^{a} ⊗ T_{j}^{b} ↦ X_{ji}^{ab}.

The form r̃ factors through the quotient A ⊗ A and induces r.

Claim: The forms r̃ and f̃ satisfy (1.1) and (1.2). Proof of the Claim: In the proof we use the following summation convention: Summation occurs over an upper-lower index. We can then write μ_{2}(T_{i}^{c}) = T_{i}^{k} ⊗ T_{k}^{a} ⊗ T_{a}^{c} and μ_{2}(T_{j}^{d}) = T_{j}^{l} ⊗ T_{l}^{b} ⊗ T_{b}^{d}. The equality (1.2) amounts to

F_{ij}^{cd} = F_{i}^{k}X_{kj}^{la}F_{l}^{b}X_{ba}^{cd} = X_{ij}^{lk}F_{l}^{b}X_{bk}^{ad}F_{a}^{c}.

These equations are also a translation of (2.3) into matrix form. This completes the proof of the claim.

We have to show that f̃ maps the kernel I of the projection Ã → A to zero. But this is a consequence of (1.2), applied in the case b = 1, since one of the terms a_{1}, a_{2}, a_{3} is contained in I and r̃ is the zero map on I ⊗ Ã and Ã ⊗ I.

It remains to show that f is convolution invertible. The pair (X^{−1}, F^{−1}) is a four braid pair. Let r̄ and f̄ be the induced operators on Ã. Then f̃∗f̄ = ε = f̄∗f̃ on Ã, and (1.2) holds for (f̄, r̄) in place of (f, r). The Yang-Baxter operator X^{−1} defines the same quotient A of Ã as X. Hence the kernel ideal obtained from X^{−1} equals I; therefore f̄(I) = 0. □

We have the comodule structure map V → A⊗V defined via v_{i} ↦ ∑_{j} T_{i}^{j} ⊗v_{j}. One has a similar formula for V^{⊗k} using multi-index notation. By construction we have:

(3.2) Proposition. The cylinder form f induces on V^{⊗k} the cylinder twist t_{V^{⊗k}} = T_{k}. □

4. Tensor categories with cylinder braiding

We may summarize the results of the previous section from the following categorical perspective:

(1) B is a category;

(2) A is a subcategory with the same objects;

(3) The category A carries the structure of a tensor category with a braiding z_{M,N};

(4) B is a right tensor module category over A;

(5) For each object V an automorphism t_{V}: V → V in B is given. The t_{V} constitute a weak endomorphism of the identity functor of A.

(6) For each pair M, N of objects the identities (1.3) hold. The meaning of (4) is the following: There is given a functor ⊗: B × A → B and a natural associativity isomorphism a: A ⊗ (B ⊗ C) → (A ⊗ B) ⊗ C of functors B × A × A → B. The pentagon axiom of tensor category theory (which still makes sense in this context) is also assumed. The tensor product functor and the associativity a restrict to the given tensor product and associativity in the tensor category A. The unit object of A is a left and right unit for ⊗: B × A → B and the triangle axiom holds. An example of this type of module category arises from a tensor category B and a tensor subcategory A. (See [7, XI.2] for such notions.)

We considered the case where B was the category of A-comodules and 𝔎-linear maps and A the category of A-comodules and A-linear maps. (5) is induced by a cylinder form.

The prototype is given by the braid categories themselves. The objects are the natural numbers n ≥ 0. The morphisms in B from n to n are the elements in ZB_{n} with composition the group multiplication. There are no morphisms from m to n for m ≠ n. The morphisms in A from n to n are the elements of the Artin braid group ZA_{n−1}, the subgroup of ZB_{n} generated by g_{1}, …, g_{n−1}. The tensor product is given on objects as m ⊗ n = m + n and on morphisms as the following homomorphism ZB_{m} × ZB_{n} → ZB_{m+n}

t, g_{1}, …, g_{m−1} ∈ ZB_{m} ↦ t, g_{1}, …, g_{m−1} ∈ ZB_{m+n}

t, g_{1}, …, g_{n−1} ∈ ZB_{n} ↦ g_{m}g_{m−1} … g_{1}tg_{1}g_{2} … g_{m}, g_{m+1}, …, g_{m+n−1} ∈ ZB_{m+n}.

The braiding is given by the morphisms x_{m,n} of section 2 and the morphisms t_{n} are also specified in that section. By (2.4), the t_{n} constitute an endomorphism of the identity of B.

There is a natural quotient category of this braid category (when 𝔎-linearized), namely the Temperley-Lieb category of type B via the Kauffman functor (see [3]).

For an elaboration of the categorical viewpoint and applications to knot theory along the lines of [9] see [5] and [6].

5. The example SL_{q}(2)

We illustrate the theory with the quantum group associated to SL_{2}. For simplicity we work over the function field Q(q^{1/2}) = 𝔎.

Let V be a two-dimensional 𝔎-module with basis {v_{1}, v_{2}}. In terms of the basis {v_{1} ⊗ v_{1}, v_{1} ⊗ v_{2}, v_{2} ⊗ v_{1}, v_{2} ⊗ v_{2}} the matrix

(5.1) X = q^{−1/2} ( \begin{matrix}q \\ & q − q^{−1} & 1 \\ & 1 & 0 \\ & & & q \end{matrix} )

defines a Yang-Baxter operator. The FRT-construction associates to X the algebra A generated by a, b, c, and d (corresponding, respectively, to T_{1}^{1}, T_{1}^{2}, T_{2}^{1}, and T_{2}^{2} in the general setting) with relations

ab = qba  bd = qdb
ac = qca  cd = qdc
bc = cb
ad − da = (q − q^{−1})bc.

The matrix

(5.2) F = ( \begin{matrix}0 & β \\ α & θ \end{matrix} )

yields a four braid pair (X, F) for arbitrary parameters with invertible αβ. (See [4], also for an n-dimensional generalization.) The quantum plane P = 𝔎{x, y}/(xy − qyx) is a left A-comodule via the map μ_{P}: P → A ⊗ P given by

μ_{P} (x^{i}y^{j}) = ∑_{r=0}^{i} ∑_{s=0}^{j} q^{−s(i+j−r−s)−r(i−r)} \left[{i \atop r}\right] \left[{j \atop s}\right] a^{r}b^{i−r}c^{s}d^{j−s} ⊗ x^{r+s}y^{i+j−r−s}

where \left[{i \atop r}\right] is a q-binomial coefficient

\left[{i \atop r}\right] = \frac{[i]!}{[r]![i − r]!}, [i]! = [1][2] ⋯ [i], [i] = \frac{q^{i} − q^{−i}}{q − q^{−1}} .

(Compare with [7, IV], where different conventions are used.) The operator T_{2} = (F ⊗ 1)X(F ⊗ 1)X on V ⊗ V has the matrix (with δ = q − q^{−1})

( \begin{matrix}0 & 0 & 0 & β^{2} \\ 0 & αβδ & αβ & qβθ \\ 0 & αβ & 0 & βθ \\ α^{2} & qαθ & αθ & αβδ + qθ^{2} \end{matrix} ) = ( \begin{matrix}F_{11}^{11} & F_{12}^{11} & F_{21}^{11} & F_{22}^{11} \\ F_{11}^{12} & F_{12}^{12} & F_{21}^{12} & F_{22}^{12} \\ F_{11}^{21} & F_{12}^{21} & F_{21}^{21} & F_{22}^{21} \\ F_{11}^{22} & F_{12}^{22} & F_{21}^{22} & F_{22}^{22} \end{matrix} )

with respect to the basis {v_{1} ⊗v_{1}, v_{1} ⊗v_{2}, v_{2} ⊗v_{1}, v_{2} ⊗v_{2}}. This is also the matrix of values of the cylinder form f

f ( \begin{matrix}aa & ac & ca & cc \\ ab & ad & cb & cd \\ ba & bc & da & dc \\ bb & bd & db & dd \end{matrix} ) .

(The notation means: If we apply f to entries of the matrix we obtain the matrix for T_{2} diplayed above.) Let det_{q} = ad − qbc be the quantum determinant. It is a group-like central element of A. The quotient of A by the ideal generated by det_{q} is the Hopf algebra SL_{q}(2).

(5.3) Proposition. The form f has the value −q^{−1}αβ on det_{q}. If −q^{−1}αβ = 1, then f factors over SL_{q}(2).

Proof. The stated value of f(det_{q}) is computed from the data above. We use the fact that

r(x ⊗ det_{q}) = r(det_{q} ⊗ x) = ε(x).

(See [7, p. 195].) From (1.2) we obtain, for a ∈ A and b = det_{q}, that

f(ab) = ∑ f(a_{1})r(b_{1} ⊗ a_{2})f(b_{2})r(a_{3} ⊗ b_{3})
= ∑ f(a_{1})ε(a_{2})f(det_{q})ε(a_{3})
= f(a),

by using the assumption that f(det_{q}) = 1 together with the counit axiom. □

We consider the subspace W = V_{2} of the quantum plane generated by x^{2}, xy, and y^{2}. We have

μ_{P} (x^{2}) = b^{2} ⊗ y^{2} + (1 + q^{−2})ab ⊗ xy + a^{2} ⊗ x^{2}
μ_{P} (xy) = bd ⊗ y^{2} + (ad + q^{−1}bc) ⊗ xy + ac ⊗ x^{2}
μ_{P} (y^{2}) = d^{2} ⊗ y^{2} + (1 + q^{−2})cd ⊗ xy + c^{2} ⊗ x^{2}.

This yields the following matrix for t_{W} with respect to the basis {x^{2}, xy, y^{2}}:

( \begin{matrix}0 & 0 & β^{2} \\ 0 & qαβ & (q + q^{−1})βθ \\ α^{2} & qαθ & αβδ + qθ^{2} \end{matrix} ) .

In the Clebsch-Gordan decomposition V ⊗ V = V_{2} ⊕ V_{0} the subspace V_{0} (the trivial irreducible module) is spanned by u = v_{2} ⊗ v_{1} − q^{−1}v_{1} ⊗ v_{2}. This is the eigenvector of X with eigenvalue −q^{−3/2}. It is mapped by T_{2} to −q^{−1}αβu. If we require this to be the identity we must have αβ = −q. We already obtained this condition by considering the quantum determinant.

The matrix of t_{W} with respect to the basis \{w_{1} = x^{2}, w_{2} = \sqrt{1 + q^{−2}}xy, w_{3} = y^{2}} is

(5.4) F_{2} = ( \begin{matrix}0 & 0 & β^{2} \\ 0 & qαβ & \sqrt{1 + q^{2}}βθ \\ α^{2} & \sqrt{1 + q^{2}}αθ & αβδ + qθ^{2} \end{matrix} ) .

In case α = β this matrix is symmetric.

The R-matrix X on W ⊗ W with respect to the lexicographic basis consisting of elements w_{i} ⊗ w_{j} with w_{1} = x^{2}, w_{2} = \sqrt{1 + q^{−2}}xy, and w_{3} = y^{2} has the form

(5.5) X_{2} = \begin{aligned}q^{2} \\ & δ^{∗} & & 1 \\ & & μ & & λ & & q^{−2} \\ & 1 & & 0 \\ & & λ & & 1 \\ & & & & & δ^{∗} & & 1 \\ & & q^{−2} & & & & 0 \\ & & & & & 1 & & 0 \\ & & & & & & & & q^{2} \end{aligned}

It makes use of the identities δ^{∗} = q^{2} − q^{−2}, μ = δ^{∗}(1 − q^{−2}), and λ = q^{−1}δ^{∗}. By construction, (X_{2}, F_{2}) is a four braid pair.

One has the the problem of computing t_{W} on irreducible comodules W. We treat instead the more familiar dual situation of modules over the quantized universal enveloping algebra.

6. The cylinder braiding for U-modules

The construction of the cylinder form is the simplest method to produce a universal operator for the cylinder twist. In order to compute the cylinder twist explicitly we pass to the dual situation of the quantized universal enveloping algebra U. One can formally dualize comodules to modules and thus obtain a cylinder braiding for suitable classes of U-modules from the results of the previous sections. But we rather start from scratch.

We work with the Hopf algebra U = U_{q}(sl_{2}) as in [8]. As an algebra, it is the the associative algebra over the function field Q(q^{1/2}) = 𝔎 generated^{4} by K, K^{−1}, E, and F subject to the relations KK^{−1} = K^{−1}K = 1, KE = q^{2}EK, KF = q^{−2}F K, and EF − F E = (K − K^{−1})/(q − q^{−1}). Its coalgebra structure is defined by setting μ(K) = K⊗K, μ(E) = E⊗1+K⊗E, μ(F) = F ⊗K^{−1}+1⊗F, ε(K) = 1, and ε(E) = ε(F) = 0. A left U-module M is called integrable if the following condition holds:

(1) M = ⨁ M^{n} is the direct sum of weight spaces M^{n} on which K acts as multiplication by q^{n} for n ∈ ZZ.

(2) E and F are locally nilpotent on M. Let U-INT denote the category of integrable U-modules and U-linear maps. (It would be sufficient to consider only finite dimensional such modules.) An integrable U-module M is semi-simple: It has a unique isotypic decomposition M = ⨁_{n≥0} M(n) with M(n) isomorphic to a direct sum of copies of the irreducible module V_{n}. The module V_{n} has a 𝔎-basis x_{0}, x_{1}, …, x_{n} with F (x_{i}) = [i + 1]x_{i+1}, E(x_{i}) = [n − i + 1]x_{i−1}, x_{−1} = 0, x_{n+1} = 0; moreover, x_{i} ∈ V_{n}^{n−2i}. The category of integrable U-modules is braided. The braiding is induced by the universal R-matrix R = κ ∘ Ψ with

4There is another use of the letter F. It has nothing to do with the 2 × 2-matrix F in (5.2).

(6.1) Ψ = ∑_{n≥0} q^{n(n−1)/2} \frac{(q − q^{−1})^{n}}{[n]!} F^{n} ⊗ E^{n}

and κ = q^{H⊗H/2}. Note that Ψ is a well-defined operator on integrable U-modules. (This operator is called Θ̄ in [8, section 4.1] and L′_{i} in [8, p. 46].) The operator κ acts on M^{m} ⊗ N^{n} as multiplication by q^{mn/2}. If we view H as the operator H: M^{m} → M^{m} given by x ↦ mx, then q^{H⊗H/2} is a suggestive notation for κ. The braiding z_{M,N}: M ⊗ N → N ⊗ M is τ ∘ R, i. e., the action of R followed by the interchange operator τ: x ⊗ y ↦ y ⊗ x.

A four braid pair (X, F) on the vector space V yields a tensor representation of ZB_{n} on V^{⊗n}. We start with the standard four braid pair determined by (5.1) and (5.2) on the two-dimensional U-module V = V_{1}. Let T_{n}: V^{⊗n} → V^{⊗n} be the associated cylinder twist as defined in Section 2. By the Clebsch-Gordan decomposition, V_{n} is contained in V^{⊗n} with multiplicity 1. Similarly, V_{m+n} ⊂ V_{m} ⊗ V_{n} with multiplicity one [7, VII.7].

(6.2) Lemma. There exists a projection operator e_{n}: V^{⊗n} → V^{⊗n} whose image, V_{n}, commutes with T_{n}.

Proof. Let H_{n} be the Hecke algebra over 𝔎 generated by x_{1}, …, x_{n−1} with braid relations x_{i}x_{j}x_{i} = x_{j}x_{i}x_{j} for |i − j| = 1 and x_{j}x_{i} = x_{i}x_{j} for |i − j| > 1 and quadratic relations (x_{i} +1)(x_{i} −q^{2}) = 0. Since X satisfies (X −q^{1/2})(X +q^{−3/2}) = 0, we obtain an action of H_{n} from the action of ZA_{n−1} ⊂ ZB_{n} on V^{⊗n} if we let x_{i} act as q^{3/2}g_{i}. Since T_{n} comes from a central element of ZB_{n} as noted in (2.4), the H_{n}-action commutes with T_{n}. It is well known that there exists an idempotent e_{n} ∈ H_{n} for which e_{n}V^{⊗n} = V_{n}. (This is quantized Schur-Weyl duality.) This fact implies the assertion of the Lemma. □

(6.3) Corollary. The subspace V_{n} ⊂ V^{⊗n} is T_{n}-stable. □

A similar proof shows that all summands in the isotypic decomposition of V^{⊗n} are T_{n}-stable.

We denote by τ_{n} the restriction of T_{n} to V_{n}; and we denote by τ_{m,n} = z_{n,m}(τ_{n}⊗ 1)z_{m,n}(τ_{m} ⊗ 1) the induced operator on V_{m} ⊗ V_{n} where z_{m,n} denotes the braiding on V_{m} ⊗ V_{n}.

(6.4) Lemma. The subspace V_{m+n} ⊂ V_{m} ⊗ V_{n} is τ_{m,n}-stable. The induced morphism equals τ_{m+n}.

Proof. Consider V_{m} ⊗ V_{n} ⊂ V^{⊗m} ⊗ V^{⊗n} = V^{⊗(m+n)}. The projection operator e_{m} ⊗ e_{n} is again obtained from the action of a certain element of the Hecke algebra H_{m+n}. Hence V_{m} ⊗ V_{n} is T_{m+n}-stable and the action on the subspace V_{m+n} is τ_{m+n}. We now use the equality (2.3)

T_{m+n} = X_{n,m}(T_{n} ⊗ 1)X_{m,n}(T_{m} ⊗ 1).

The essential fact is that X_{m,n} is the braiding on V^{⊗m} ⊗ V^{⊗n}. It induces, by naturality of the braiding, the braiding z_{m,n} on V_{m} ⊗ V_{n}. □

Let A(n) = (α_{i}^{j}(n)) be the matrix of τ_{n} with respect to x_{0}, …, x_{n}. In the next theorem we derive a recursive description of A(n). We need more notation to state it. Define inductively polynomials γ_{k} by γ_{−1} = 0, γ_{0} = 1 and, for k > 0,

(6.5) αγ_{k+1} = q^{k}θγ_{k} + βq^{k−1}δ[k]γ_{k−1}.

Here δ = q − q^{−1}, and γ_{k} = γ_{k}(θ, q, α, β) is a polynomial in θ with coefficients in ZZ[q, q^{−1}, α^{−1}, β]. Let D(n) denote the codiagonal matrix with α^{k}β^{n−k}q^{k(n−k)} in the k-th row and (n − k)-th column and zeros otherwise. (We enumerate rows and columns from 0 to n.) Let B(n) be the upper triangular matrix

(6.6) B(n) = ( \begin{matrix}γ_{0} & \left[{n \atop 1}\right]γ_{1} & \left[{n \atop 2}\right]γ_{2} & ⋯ & γ_{n} \\ & γ_{0} & \left[{n−1 \atop 1}\right]γ_{1} & ⋯ & γ_{n−1} \\ & & ⋯ & ⋯ & ⋯ \\ & & & γ_{0} & γ_{1} \\ & & & & γ_{0} \end{matrix} ) .

Thus the (n − k)-th row of B(n) is

0, …, 0, \left[{k \atop 0}\right]γ_{0}, \left[{k \atop 1}\right]γ_{1}, \left[{k \atop 2}\right]γ_{2}, …, \left[{k \atop k − 1}\right]γ_{k−1}, \left[{k \atop k}\right]γ_{k}.

(6.7) Theorem. The matrix A(n) is equal to the product D(n)B(n).

Proof. The proof is by induction on n. We first compute the matrix of τ_{n,1} on V_{n} ⊗ V_{1} and then restrict to V_{n+1}. In order to display the matrix of τ_{n,1} we use the basis

x_{0} ⊗ x_{0}, …, x_{n} ⊗ x_{0}, x_{0} ⊗ x_{1}, …, x_{n} ⊗ x_{1}.

The matrix of τ_{n,1} has the block form

( \begin{matrix}0 & βA(n) \\ αA(n) & A′(n) \end{matrix} ) .

The matrix A′(n) is obtained from A(n) in the following manner: Let α_{0}, …, α_{n} denote the columns of A(n) and β_{0}, …, β_{n} the columns of A′(n). We claim that

β_{i} = αq^{2i−n}θα_{i} + βq^{2i−n−1}δ[n − i + 1]α_{i−1} + αδ[i + 1]α_{i+1},

with α_{−1} = α_{n+1} = 0.

Recall that τ_{n,1} = (τ_{n} ⊗ 1)z_{1,n}(τ_{1} ⊗ 1)z_{n,1}. In our case the universal R-matrix has the simple form

R = κ ∘ (1 + (q − q^{−1})F ⊗ E).

For the convenience of the reader we display the four steps in the calculation of τ_{n,1}, separately for x_{i} ⊗ x_{0} and x_{i} ⊗ x_{1}.

x_{i} ⊗ x_{0} ↦ q^{(n−2i)/2}x_{0} ⊗ x_{i}
↦ αq^{(n−2i)/2}x_{1} ⊗ x_{i}
↦ αx_{i} ⊗ x_{1}
↦ ∑_{j} αα_{i}^{j}x_{j} ⊗ x_{0}.

x_{i} ⊗ x_{1} ↦ q^{−(n−2i)/2}x_{1} ⊗ x_{i} + δ[i + 1]q^{(n−2i−2)/2}x_{0} ⊗ x_{i+1}
↦ q^{−(n−2i)/2}(βx_{0} + θx_{1}) ⊗ x_{i} + αδ[i + 1]q^{(n−2i−2)/2}x_{1} ⊗ x_{i+1}
↦ βx_{i} ⊗ x_{0} + βq^{−n+2i−1}δ[n − i + 1]x_{i−1} ⊗ x_{1}

+q^{2i−n}θx_{i} ⊗ x_{1} + αδ[i + 1]x_{i+1} ⊗ x_{1}

↦ ∑_{j} α_{i}^{j}x_{i} ⊗ x_{0} + ∑_{j} βq^{2i−n+1}δ[n − i + 1]α_{i−1}^{j}x_{j} ⊗ x_{1}

+ ∑_{j} q^{2i−n}θα_{i}^{j}x_{j} ⊗ x_{1} + ∑_{j} αδ[i + 1]α_{i+1}^{j}x_{j} ⊗ x_{1}.

This proves the claim about the matrix for τ_{n,1}.

We now use the following fact about the Clebsch-Gordan decomposition (it is easily verified in our case, but see e. g. [7, VII.7] for more general results): In the Clebsch-Gordan decomposition V_{n} ⊗ V_{1} = V_{n+1} ⊕ V_{n−1} a basis of V_{n+1} is given by

y_{j} = \frac{F^{j}}{[j]!}(x_{0} ⊗ x_{0}) = q^{−j}x_{j} ⊗ x_{0} + x_{j−1} ⊗ x_{1}.

We apply τ_{n,1} to the y_{j}. Since there are no overlaps between the coordinates of the y_{j}, we can directly write τ_{n,1}(y_{j}) as a linear combination of the y_{k}.

We assume inductively that A(n) has bottom-right triangular form, i. e., zero entries above the codiagonal, with codiagonal as specified by D(n). Then A′(n) has a nonzero line one step above the codiagonal and is bottom-right triangular otherwise. From the results so far we see that the columns of A(n + 1), enumerated from 0 to n + 1, are obtained inductively as follows: The 0-th row is (0, …, 0, β^{n+1}). Below this 0-th row the j-th column, for 0 ≤ j ≤ n + 1, has the form

(6.8) αq^{j}α_{j} + q^{2j−n−2}θα_{j−1} + βq^{2j−n−3}δ[n − j + 2]α_{j−2}.

From this recursive formula one derives immediately that the codiagonal of A(n) is given by D(n).

Finally, we prove by induction that A(n) is as claimed. The element in row k and column n − k + j equals

α^{k}β^{n−k}q^{k(n−k)}\left[{k \atop j}\right]γ_{j}.

For n = 1, we have defined τ_{1} as A(1). For the inductive step we use (6.8) in order to determine the element of A(n) in column n − k + j and row k + 1. The assertion is then equivalent to the following identity:

α^{k}β^{n−k}q^{k(n−k)} (α\left[{k \atop j}\right]γ_{j} + q^{n−2k+2j−2}θ\left[{k \atop j − 1}\right]γ_{j−1}

+βq^{n−2k+2j−3}δ[k − j + 2]\left[{k \atop j − 2}\right]γ_{j−2})

= α^{k+1}β^{n−k}q^{(n−k)(k+1)}\left[{k + 1 \atop j}\right]γ_{j}.

We cancel α-, β-, and q-factors, use the Pascal formula

(6.9) \left[{a + 1 \atop b}\right] = q^{b}\left[{a \atop b}\right] + q^{−a+b−1}\left[{a \atop b − 1}\right]

and the identity

δ[k − j + 2]\left[{k \atop j − 2}\right] = \left[{k \atop j − 1}\right][j − 1]

and see that the identity in question is equivalent to the recursion formula (6.5) defining the γ-polynomials. This completes the proof. □

We now formulate the main result of this section in a different way. First, we note that it was not essential to work with the function field 𝔎. In fact, 𝔎 could have been any commutative ring and q, α, and β could have been any suitable parameters in it. We think of θ as being an indeterminate.

Let L(α, β) be the operator on integrable U-modules which acts on V_{n} via

x_{j} ↦ α^{n−j}β^{j}q^{j(n−j)}x_{n−j}.

Let

(6.10) T (α, β) = ∑_{k=0}^{∞} γ_{k} \frac{E^{k}}{[k]!};

T (α, β) is well-defined as an operator on integrable U-modules. Then (6.7) can be expressed as follows:

(6.11) Theorem. The operator t(α, β) defined by setting t(α, β) = L(α, β) ∘ T (α, β) acts on V_{n} as τ_{n}. □

In Section 8 we give another derivation of this operator from the universal point of view.

One can develop a parallel theory by starting with the four braid pair (X^{−1}, F^{−1}). This leads to matrices which are top-left triangular, i. e., zero entries below the codiagonal. By computing the inverse of (5.1) and of (5.2) we see that, in the case (α, β) = (1, 1), we have to replace (q, θ) by (q^{−1}, −θ).

The following proposition may occasionally be useful. Introduce a new basis u_{0}, …, u_{n} in V_{n} by

x_{i} = q^{−i(n−i)/2}\sqrt{\left[{n \atop i}\right]}u_{i}.

Then a little computation shows:

(6.12) Proposition. Suppose α = β. With respect to the basis (u_{i}) the Rmatrix and the matrix for τ_{n} are symmetric. □

7. The γ-polynomials

For later use we derive some identities for the γ-polynomials of the previous section. A basic one, (7.1), comes from the compatibility of the cylinder twist with tensor products. Again we use δ = q − q^{−1}. We give two proofs of (7.1).

(7.1) Theorem. The γ-polynomials satisfy the product formula

γ_{m+n} = ∑_{k=0}^{min(m,n)} α^{−k}β^{k}q^{mn−k(k+1)/2}δ^{k}[k]!\left[{m \atop k}\right]\left[{n \atop k}\right]γ_{m−k}γ_{n−k}.

First proof of (7.1). The first proof is via representation theory. We have a unique U-submodule of V_{m} ⊗ V_{n} which is isomorphic to V_{m+n} (Clebsch-Gordan decomposition). We use the symbol V_{m+n} also for this module. The vector x_{m}⊗x_{n} is contained in this module and satisfies F (m_{m} ⊗ x_{n}) = 0. The latter property characterizes x_{m}⊗x_{n} inside V_{m+n} up to a scalar (lowest weight vector, F -primitive vector).

We consider

τ_{m,n} = (τ_{m} ⊗ 1)z_{n,m}(τ_{n} ⊗ 1)z_{m,n}

on V_{m+n} ⊂ V_{m}⊗V_{n} where it equals τ_{m+n}. We first express this equality formally in terms of matrices and then evaluate the formal equation by a small computation. We already have intruced the matrices for τ_{m} in Section 6

τ_{m}(x_{j}) = ∑_{k} α_{j}^{k}(m)x_{k}.

We write

z_{n,m}(x_{j} ⊗ x_{m}) = ∑_{u,v} r_{jm}^{uv}x_{u} ⊗ x_{v}.

From the form of the action of E and F on the modules V_{t} and from the form of the universal R-matrix we see that the sum is over (u, v) with u+v = j+m. Since F x_{m} = 0 for x_{m} ∈ V_{m}, we also observe directly z_{m,n}(x_{m} ⊗ x_{n}) = q^{mn/2}x_{n} ⊗ x_{m}. The two expressions for τ_{m+n}, applied to x_{m} ⊗ x_{n} ∈ V_{m+n} ⊂ V_{m} ⊗ V_{n}, now yield the formal identity

α_{m+n}^{m+n}(m + n) = ∑_{k≥0} q^{mn/2}α_{n}^{n−k}(n)r_{n−k,m}^{m−k,n}α_{m−n}^{m}(n).

By (6.7) we have

α_{m+n}^{m+n}(m + n) = α^{m+n}γ_{m+n},
α_{n}^{n−k}(n) = α^{n−k}β^{k}q^{(n−k)k}γ_{n−k}, and
α_{m−k}^{m}(m) = α^{m}\left[{m \atop k}\right]γ_{m−k}.

This already yields a relation of type (7.1). It remains to compute the coefficient r_{n−k,m}^{m−k,n}. For this purpose we use the definition z_{n,m} = τ ∘ κ ∘ Ψ of the braiding, the action of E and F on vectors x_{j}, and the explicit form (6.1) of the operator Ψ. Put together, this yields

z_{n,m}(x_{j} ⊗ x_{m}) = ∑_{k≥0} v^{•(k)}δ^{k}[j + 1] ⋯ [j + k]x_{m−k} ⊗ x_{j+k}

with

•(k) = k(k − 1)/2 + (n − 2j − 2k)(2k − m)/2.

We now have enough data to rewrite the formal identity above and give it the form (7.1). □

The dependence of γ_{k} on the parameters α and β is not essential. Define, inductively, polynomials γ′_{k} in θ over ZZ[q, q^{−1}] by setting γ′_{−1} = 0, γ′_{0} = 1 and, for k ≥ 0,

γ′_{k+1} = q^{k}θγ′_{k} + q^{k−1}δ[k]γ′_{k−1},

i. e., by setting γ′_{k}(θ, q) = γ_{k}(θ, q, 1, 1). A simple rewriting of the recursion formula then yields the identity

(7.2) γ_{k}(θ, q, α, β) = γ′_{k} (\frac{θ}{\sqrt{αβ}} , q) (\frac{β}{α})^{k/2} .

Note that γ′_{k} contains only powers θ^{l} with l ≡ k mod 2.

Normalize the γ′ to obtain monic polynomials β_{k}(θ) = q^{−k(k−1)/2}γ′_{k}(θ). The new polynomials are determined by the recursion relation

(7.3) β_{−1} = 0, β_{0} = 1, and β_{k+1} = θβ_{k} + (1 − q^{−2k})β_{k−1} for k ≥ 0.

In order to find an explicit expression for the β_{k}, we introduce a new variable ρ via the quadratic relation θ = ρ − ρ^{−1}. We then consider the recursion formally over the ring ZZ[q, q^{−1}, ρ, ρ^{−1}]. Let us set

B_{n}(ρ) = ∑_{j=0}^{n}(−1)^{j}q^{−j(n−j)}\left[{n \atop j}\right]ρ^{n−2j}.

(7.4) Proposition. The polynomials β satisfy the identity

β_{k}(ρ − ρ^{−1}) = B_{k}(ρ).

Proof. We verify the recursion (7.3) with θ replaced by ρ−ρ^{−1} and β_{k} replaced by B_{k}. We use the definition of the B_{k} in the right hand side of (7.3). Then the coefficient of ρ^{k+1−2j}, for 1 ≤ j ≤ k, turns out to be

(−1)^{j}q^{−j(k−j} (\left[{k \atop j}\right] + q^{k−2j+1}\left[{k \atop j − 1}\right] − q^{−j}δ[k]\left[{k − 1 \atop j − 1}\right]) .

We use the identity

[k]\left[{k − 1 \atop j − 1}\right] = [k − j + 1]\left[{k \atop j − 1}\right]

and arrive at

(−1)^{j}q^{−j(k−j} (\left[{k \atop j}\right] + q^{−k−1}\left[{k \atop j − 1}\right]) .

The Pascal formula (6.9) now shows that this is the coefficient of ρ^{k+1−2j} in B_{k+1}. It is easy to check that the coefficients of ρ^{±(k+1)} on both sides coincide. □

We can write ρ^{k}+(−1)^{k}ρ^{−k} as an integral polynomial P_{k} in θ where θ = ρ−ρ^{−1}. That polynomial satisfies the recursion relation

θP_{k} = P_{k+1} − P_{k−1}.

It is possible to write P_{k} in terms of Tschebischev- or Jacobi-polynomials. The last proposition says that

β_{n}(θ) = ∑_{j=0}^{[n/2]}(−1)^{j}q^{−j(n−j)}\left[{n \atop j}\right]P_{n−2j}(θ).

The product formula (7.1) was a consequence of representation theory. In view of the applications to be made in Section 8 it is desirable to have a proof which uses only the recursive definition of the γ-polynomials. We now give such a proof. By (7.2), it suffices to consider the case α = β = 1.

Second proof of (7.1). We write

C_{k}^{m,n} = q^{mn−k(k+1)/2}δ^{k}[k]!\left[{m \atop k}\right]\left[{n \atop k}\right],

and want to show that

γ_{m+n} = ∑_{k=0}^{min(m,n)} C_{k}^{m,n}γ_{m−k}γ_{n−k}.

Denote the right hand side by γ(m, n). Then γ(m, n) = γ(n, m). We will use the recursion (6.5) and the Pascal formula (6.9), with q replaced by q^{−1}, to show γ(m + 1, n) = γ(m, n + 1). Since γ(m + n, 0) = γ_{m+n} the proof will then be complete. Set γ_{k} = 0 for k < 0. We can then sum just over k ≥ 0. The C-coefficients satisfy the following Pascal type relation

(7.5) C_{k}^{m+1,n} = q^{n−k}C_{k}^{m,n} + δq^{n−k+1}q^{m−k}[n − k + 1]C_{k−1}^{m,n}.

The verification that this is so uses the Pascal formula for \left[{m+1 \atop k}\right] and a little rewriting. Next we apply this relation in the sum γ(m + 1, n) and obtain (with an index shift k → k + 1 in the second summand) the identity

γ(m + 1, n) = ∑_{k} q^{n−k}C_{k}^{m,n}γ_{m−k+1}γ_{n−k}

+ ∑_{k} (δ[n − k]q^{n−k−1}γ_{n−k−1}) q^{m−k}C_{k}^{m,n}γ_{m−k}.

In the second sum apply the recursion to the factor in parentheses to obtain the identity

γ(m+1, n) = ∑_{k} C_{k}^{m,n}(q^{n−k}γ_{m−k+1}γ_{n−k} +q^{m−k}γ_{m−k}γ_{n−k+1} −q^{n+m−2k}θγ_{m−k}γ_{n−k}).

Since γ(m, n) = γ(n, m), we obtain γ(m, n+1) upon interchanging m and n in the foregoing identity: That interchanges the first two summands in the parentheses and leaves the third fixed. □

8. The universal cylinder twist

In this section we work with operators on integrable U-modules. These are 𝔎linear weak endomorphisms of the category U-INT. Left multiplication by x ∈ U is such an operator; it will be denoted by x or by l_{x}. If t is an operator, then μ(t) is the operator on U-INT × U-INT which is given by the action of t on tensor products of modules. If τ denotes the twist operator, then we define τ(t) = τ ∘t∘τ. We have the compatibilities μ(l_{x}) = l_{μ(x)} and τμ(l_{x}) = l_{τμ(x)}. The operators μ(t) and τ(t) are again weak endomorphisms of the categories involved.

Typical examples of such operators which are not themselves elements of U are the universal R-matrix R and its factors κ and Ψ, (See (6.1).) as are the operators L = T′_{i,1} and L^{#} = T′′_{i,1} of Lusztig [8, p. 42].

Since R acts by U-linear maps each operator t satisfies the standard relation

(8.1) R ∘ μ(t) = τμ(t) ∘ R

of a braiding.

An operator t is called a universal cylinder twist on U-INT if it is invertible and satisfies the analogue of (1.4), namely,

(8.2) μ(t) = τR(1 ⊗ t)R(t ⊗ 1) and

(8.3) τR(1 ⊗ t)R(t ⊗ 1) = (t ⊗ 1)τR(1 ⊗ t)R.

We denote by t_{V} the action of t on the module V. Then (1.3) holds if we use R to define the braiding. Recall the operator t(α, β) defined at the end of Section 6. Here is the main result, proved following (8.6).

(8.4) Theorem. Suppose αβ = −q. Then t(α, β) is a universal cylinder twist.

We treat the case (α, β) = (−q, 1) in detail and reduce the general case formally to this one. We skip the notation α, β and work with t = LT. Note that L is Lusztig’s operator referred to above. We collect a few properties of L in the next lemma.

(8.5) Lemma. The operator L satisfies the following identities:

(1) LEL^{−1} = −KF, LF L^{−1} = −EK^{−1}, LKL^{−1} = K^{−1}.

(2) μ(L) = (L ⊗ L)Ψ = τR(L ⊗ L)κ^{−1}.

(3) κ(L ⊗ 1) = (L ⊗ 1)κ^{−1}, κ(1 ⊗ L) = (1 ⊗ L)κ^{−1}.

(4) (L ⊗ L)Ψ(L ⊗ L)^{−1} = κ ∘ τΨ ∘ κ^{−1}.

Proof. For (1), in the case L^{#}, see [8, Proposition 5.2.4.]. A simple computation from the definitions yields (3) and (4). For the first equality in (2) see [8, Proposition 5.3.4]; the second one follows by using (3) and (4). □

In the universal case one of the axioms for a cylinder twist is redundant, namely:

(8.6) Proposition. If the operator t satisfies (8.2), then it also satisfies (8.3).

Proof. Apply τ to (8.2) and use (8.1). □

Proof of theorem (8.4). The operator L is invertible. The operator T is invertible since its constant term is 1. Thus it remains to verify (8.2). We show that (8.2) is equivalent to

(8.7) μ(T) = κ(1 ⊗ T)κ^{−1} ∘ (L^{−1} ⊗ 1)Ψ(L ⊗ 1) ∘ (T ⊗ 1),

given the relations of Lemma (8.5). Given (8.2), we have

μ(T) = μ(L^{−1})τ(R)(1 ⊗ LT)κΨ(LT ⊗ 1).

We use (8.5.2) for μ(L^{−1}), cancel τ(R) and its inverse, and then use (8.5.3); (8.7) drops out. In like manner, (8.2) follows from (8.1).

In order to prove (8.7), one verifies the following identities from the definitions:

κ(1 ⊗ T)κ^{−1} = ∑_{k=0}^{∞} \frac{γ_{k}}{[k]!}(K^{k} ⊗ E^{k}) and

(L^{−1} ⊗ 1)Ψ(L ⊗ 1) = ∑_{k=0}^{∞}(−1)^{k}q^{−k(k−1)/2} \frac{δ^{k}}{[k]!}K^{k}E^{k} ⊗ E^{k}.

Using this information, we compute the coefficient of K^{r}E^{s} ⊗ E^{r} on the right hand side of (8.7) to be

∑_{n=0}^{min(r,s)}(−1)^{n}q^{−n(n−1)/2} \frac{δ^{n}}{[n]![s − n]![r − n]!}γ_{s−n}γ_{r−n}.

The coefficient of the same element in μ(T) is, by the q-binomial formula, equal to

q^{−rs} \frac{1}{[s]![r]!}γ_{r+s}.

Equality of these coefficients is exactly the product formula (7.1) in the case where (α, β) = (−q, 1). This finishes the proof of the theorem in this special case.

A similar proof works in the general case. Specifically, a formal reduction to the special case uses the following observation. Write α = q^{ζ}. Then, formally, L(α, β) = K^{ζ}L in case αβ = −q. This fact is used to deduce similar properties for L_{#} = L(α, β) from lemma (8.5), in particular

L_{#}^{−1}F L_{#} = α^{−1}βqKE.

The final identity leads to (7.1) in the general case. □

We point out that the main identity in the construction of the universal twist involves only the Borel subalgebra of U generated by E and K. Of course, there is a similar theory based on F and K and another braiding. The constructions of section 6 show that the universal twist is determined by its action on the 2-dimensional module V_{1}. Hence our main theorem gives all possible universal cylinder twists associated to the given braided category U-Int.

References

1. Bourbaki, N.: Groupes et algèbres de Lie. Chapitres 4,5 et 6. Paris, Hermann 1968.

2. Brieskorn, E.: Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe. Inv. math. 12, 57 – 61 (1971).

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

4. tom Dieck, T.: On tensor representations of knot algebras. To appear.

5. tom Dieck, T., and R. Häring-Oldenburg: Tensor categories with cylinder braiding. In preparation.

6. Häring-Oldenburg, R.: Dissertation Göttingen 1996/97.

7. Kassel, C.: Quantum groups. Berlin, Springer 1995.

8. Lusztig, G.: Introduction to quantum groups. Basel, Birkhäuser 1993.

9. Turaev, V. G.: Quantum invariants of knots and 3-manifolds. Berlin, W. de Gruyter 1994.

Version of November 24, 2000