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 Uq(sl2). The main result here is the computation of the universal cylinder twist. 1. Cylinder forms Let A = (A,m, e, μ, e) be a bialgebra (over the commutative ring K) with multiplication m, unit e, comultiplication μ, and counit e. Let r: A ⊗ A → K be a linear form. We associate to left A-comodules M,N a K-linear map zM,N :M ⊗N → N ⊗M, x⊗ y 7→ ∑ r(y ⊗ x)y ⊗ x, where we have used the formal notation x 7→ ∑x1 ⊗ x for a left A-comodule structure μM :M → A⊗M onM . (See [7, p. 186] formula (5.9) for our map zM,N and also formula (5.8) for a categorical definition.) We call r a braid form on A, if the zM,N yield a braiding on the tensor category A-COM of left A-comodules. We prefer Latin-Greek duality; thus the comultiplication is not ∆. A notation like f : A → B, a 7→ f(a) is used in two different ways. Either the symbols a 7→ 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’. In 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 superor 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. T. tom Dieck and R. Haring-Oldenburg 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].) Let (C, μ, e) be a coalgebra. Examples of our μ-convention for coalgebras are μ(a) = ∑ a1 ⊗ a2 and (μ⊗ 1)μ(a) = ∑ a11 ⊗ a12 ⊗ a2; if we set μ2(a) = (μ⊗ 1)μ, then we write μ2(a) = ∑ a1 ⊗ a2 ⊗ a3. The counit axiom reads in this notation ∑ e(a1)a2 = a = ∑ e(a2)a1. 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 7→ ∑ f(a1)g(a2). The unit element of the algebra C∗ is e. Therefore g is a (convolution) inverse of f , if f ∗ g = g ∗ f = e. 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⊗g; it is the linear form on A⊗ A defined by a⊗ b 7→ 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⊗e) ∗ rτ ∗ (e⊗f) ∗ r = rτ ∗ (e⊗f) ∗ r ∗ (f⊗e). 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(a1)r(b1 ⊗ a2)f(b2)r(a3 ⊗ b3) = ∑ r(b1 ⊗ a1)f(b2)r(a2 ⊗ b3)f(a3) hold. Proof. Note that a four-fold convolution product is computed by the formula (f1 ∗ f2 ∗ f3 ∗ f4)(x) = ∑ f1(x1)f2(x2)f3(x3)f4(x4). We apply this to the second term in (1.1). The value on a⊗ b is then (f(a1) · e(b1)) · r(b2 ⊗ a2) · (e(a3) · f(b3)) · r(a4 ⊗ b4). By the counit axiom, we can replace ∑ e(b1) · b2 ⊗ b3 ⊗ b4 by ∑ b1 ⊗ b2 ⊗ b3 (an exercise in the μ-convention), and ∑ a1 ⊗ a2 ⊗ e(a3) · a4 can be replaced by ∑ a1⊗ a2⊗ a3. 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. 2 A cylinder form f (in fact any linear form) yields for each left A-comodule M a K-linear endomorphism tM :M →M, x 7→ ∑ f(x)x. If φ: M → N is a morphism of comodules, then φ ◦ tM = tN ◦ φ. Since tM is in general not a morphism of comodules we express this fact by saying: The tM