Uwe Franz

AND Michael Sch Ürmann

Monika Var Šo

BROWNIAN MOTION ON INVOLUTIVE BRAIDED SPACES

Keywords: 2010 Mathematics Subject Classification. Primary: 18D10 Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories, 60G51 Processes with independent increments, Lévy processes, Secondary: 16T05 Hopf algebras and their applications, 81S25 Quantum stochastic calculus Lévy process, Brownian motion, braided involutive bialgebra, braided space, symmetrization, quantum stochastic calculus

Quantum) stochastic processes with independent and stationary increments (i.e. Lévy processes) and in particular Brownian motions in braided monoidal categories are studied. The notion of increments is based on a bialgebra or Hopf algebra structure, as in [Sch93], and positivity is taken w.r.t. to an involution. We show that involutive bialgebras and Hopf algebras in the Yetter-Drinfeld categories of a quasi-or coquasi-triangular * -bialgebra admit a symmetrization (or bosonization) and that their Lévy processes are in one-to-one correspondence with a certain class of Lévy processes on their symmetrization. We classify Lévy processes with quadratic generators, i.e., Brownian motions, on several braided Hopf- * -algebras that are generated by their primitive elements (also called braided * -spaces), and on the braided SU (2)-quantum groups.

Introduction

The study of random variables and stochastic processes with values in algebraic structures has a long and rich history, see, e.g., the monograph [START_REF] Grenander | Probabilities on algebraic structures[END_REF]. In quantum probability [START_REF] Parthasarathy | An introduction to quantum stochastic calculus[END_REF], the commutative algebras of functions on the underlying probability space and the state space of a stochastic process are replaced by possibly noncommutative algebras, typically realized as algebras of operators on a Hilbert space. Quantum (or noncommutative) probability was initially motivated by the wish to have a common framework for classical probability and quantum physics, but it also opened the stage for many new interactions between probability and algebraic structures. In [START_REF] Accardi | Quantum independent increment processes on superalgebras[END_REF][START_REF] Schürmann | White noise on bialgebras[END_REF], Lévy processes on involutive bialgebras were introduced and studied as a common generalization of Lévy processes with values in groups and semigroups, and of factorizable representations of groups and Lie algebras.

In this paper we will study quantum stochastic processes and in particular Brownian motions in braided categories. In quantum probability, Brownian motion is defined as a Lévy process with quadratic generator, cf. [Sch93, Section 5.1]. This generalizes Brownian motions in Euclidean spaces, Lie groups, and Riemannian manifolds, which are fundamental examples of stochastic processes, and which have been the guiding examples in the development of the stochastic calculus and potential theory. The generator of the Markov semigroup of classical Brownian motion is a Laplace or Laplace-Beltrami operator and contains valuable information about the geometry of the underlying space.

In this paper we will develop the theory of Brownian motions and Lévy processes on braided involutive bialgebras, i.e., involutive bialgebras which are objects of a braided monoidal category, cf. [START_REF] Majid | Foundations of quantum group theory[END_REF][START_REF] Heckenberger | Hopf Algebras and Root Systems[END_REF]. Our main examples will be such algebras, also called braided spaces, which are generated by a finite number of primitive elements satisfying certain quadratic commutative relations coming from an R-matrix. This article builds on ideas and results from [START_REF] Majid | Quantum groups and quantum probability[END_REF][START_REF] Majid | Quantum random walks and time reversal[END_REF][START_REF] Ellinas | Random walk and diffusion on a smash line algebra[END_REF][START_REF] Franz | Diffusion on braided spaces[END_REF][START_REF] Franz | Lévy processes and Brownian motion on braided spaces[END_REF].

Our paper introduces several interesting new classes of quantum stochastic processes. In dimension one (i.e., with one self-adjoint primitive generator) there is only one R-matrix which leads to a braided involutive space in our setting, the 1 × 1 matrix R = (q) with q ∈ R\{0}. In this case the braiding can be defined via a group, and we obtain, e.g., the Azéma process, whose surprising properties have been studied in [START_REF] Émery | On the Azéma martingales[END_REF][START_REF] Parthasarathy | Azéma martingales and quantum stochastic calculus In: Probability, Statistics and Design of Experinwnts[END_REF][START_REF] Schürmann | The Azéma martingales as components of quantum independent increment processes[END_REF]. But in higher dimension there are many possibilities to choose an R-matrix with a compatible involution, and most of them cannot be obtained from groups. As an example, we will classify the Brownian motions (i.e., Lévy processes with quadratic generator) on the braided spaces associated to the sl 2 -and sl 3 -R-matrix in Section 6.

Let us outline the structure of this paper.

In Section 1, we give the basic definitions and several fundamental results needed to define and study braided Lévy processes.

In Section 2, we show how braided categories can be constructed from a Hopf algebra, a coquasi-triangular or a quasi-triangular bialgebra. Our particular setting has been choosen since it will be convenient when we consider Lévy processes on braided * -bialgebras in these categories. We consider three cases. In the first case we have a Hopf- * -algebra A and the objects of the category are Yetter-Drinfeld modules carrying an involution, which has to satisfy some compatibility conditions. In the second case A is a coquasi-triangular bialgebra equipped with an involution. We show that if the r-form satisfies a compatibility condition with respect to the involution, then we can build a braided category from the comodules. In the last case, A is a quasi-triangular bialgebra with an involution. If the R-matrix satisfies a certain compatibility condition w.r.t. the involution, then we can again build a braided category, this time from the modules.

In Section 3 we show that for every braided * -bialgebra B in either of the three types of category we can construct a symmetrization (or bosonization), i.e., we can embed it into a bigger * -bialgebra H as an algebra in a way that allows us to "lift" Lévy processes on B to Lévy processes on H. This generalizes the symmetrization in [START_REF] Schürmann | White noise on bialgebras[END_REF]Chapter 3], where the braiding was defined via actions and coactions of a group. For more general braidings this kind of construction was introduced by Majid, see [START_REF] Majid | Foundations of quantum group theory[END_REF]Section 9.4], but without the involution, which is crucial for defining and studying quantum stochastic processes. We show that the symmetrization reduces the construction and classification of Lévy processes on braided * -bialgebras to usual involutive bialgebras.

In the following section (Section 4), we explicitely construct a familiy of braided * -spaces and their symmetrization. The construction starts from a bi-invertible R-matrix of real type I and it yields a braided * -space in the category of comodules of the quasi-triangular * -bialgebra associated to the R-matrix by the Faddeev-Reshetikhin-Takhtajan construction (cf. [START_REF] Faddeev | Quantization of Lie groups and Lie algebras[END_REF]). Furthermore, these braided * -spaces always come with a canonical quadratic generator that gives rise to a standard Brownian motion on these spaces.

In Section 5, we prove the existence of these invariant, conditionally positive, quadratic functionals on our braided * -spaces and study the associated Brownian motions. These processes can be considered as multi-dimensional analogues of the Azéma martingale.

Section 6 contains the explicit quantum stochastic differential equations defining the standard Brownian motion on the braided line and the braided plane as well as the classification of all quadratic generators for the braided * -spaces associated to the 1 × 1-R-matrix (q), and the standard sl 2 -and sl 3 -R-matrices, and for the braided SU (2)-quantum groups.

1. Preliminaries 1.1. Lévy processes on braided Hopf- * -algebras. In this section we introduce braided Hopf- * -algebras, braided * -bialgebras, and Lévy processes on braided *bialgebras in order to recall some of their elementary theory, in particular the oneto-one correspondence between these processes and their generators. See [START_REF] Majid | Foundations of quantum group theory[END_REF][START_REF] Heckenberger | Hopf Algebras and Root Systems[END_REF] and the references therein for information on braided tensor categories and Hopf algebras, and [START_REF] Schürmann | White noise on bialgebras[END_REF][START_REF] Franz | Noncommutative mathematics for quantum systems[END_REF] for the theory of quantum Lévy processes.

A tensor category or monoidal category is a category C equipped with a bifunctor ⊗ : C × C → C satisfying certain conditions, see for example [START_REF] Maclane | Categories for the working mathematician[END_REF]. A braiding Ψ in a tensor category is a natural isomorphism between the functors ⊗ :

(A, B) → A ⊗ B and ⊗ • τ : (A, B) → B ⊗ A satisfying the so-called Hexagon Axioms. It is called a symmetry, if it is involutive, i.e. if Ψ B,A • Ψ A,B = id A⊗B for all objects A, B of C.
A braided tensor category or braided monoidal category is a pair (C, Ψ) consisting of a tensor category C and a braiding Ψ of C. It is called symmetric tensor category, if the braiding is a symmetry, cf. [START_REF] Maclane | Categories for the working mathematician[END_REF].

In this paper we always assume that the objects of our braided tensor categories (C, Ψ) are complex vector spaces and that the morphisms are linear maps. We call a linear map L :

V → W Ψ-invariant, if (id X ⊗ L) • Ψ V,X = Ψ W,X • (L ⊗ id X )
holds for any X. Note that due to the naturality of Ψ, morphisms of the category (C, Ψ) always have to satisfy both conditions (id

X ⊗ L) • Ψ X,V = Ψ X,W • (L ⊗ id X ) and (L ⊗ id X ) • Ψ X,V = Ψ X,W • (id X ⊗ L), i.e.
, they are Ψ-invariant and Ψ -1invariant. The notions of bialgebras and Hopf algebras can also be defined in a braided tensor category, this leads to braided bialgebras and braided Hopf algebras. The product, coproduct, unit, counit and antipode now have to be morphisms of the braided tensor category and satisfy similar axioms as in the usual case, cf. [START_REF] Majid | Foundations of quantum group theory[END_REF]. Bialgebras and Hopf algebras are special cases of braided bialgebras and braided Hopf algebras where the flip automorphism τ :

A ⊗ B → B ⊗ A, τ (u ⊗ v) = v ⊗ u is the braiding.
Since the axioms proposed by Majid in [START_REF] Majid | -structures on braided spaces[END_REF][START_REF] Majid | Foundations of quantum group theory[END_REF] do not guarantee that the braided coproduct is a * -algebra homomorphism, we follow the ones in [START_REF] Franz | Stochastic processes and operator calculus on quantum groups[END_REF].

We want the * -structure * B⊗B on B ⊗ B to be an anti-homomorphism and the canonical inclusions

B ι1 -→ B ⊗ B ι2 ← -B to be * -homomorphisms. Thus we get for a, b ∈ B * B⊗B (a ⊗ b) = ( * B⊗B • m B⊗B )(a ⊗ 1 ⊗ 1 ⊗ b) = (m B⊗B • ( * B⊗B ⊗ * B⊗B ) • τ B⊗B,B⊗B )(a ⊗ 1 ⊗ 1 ⊗ b) = ((m ⊗ m) • (id ⊗ Ψ ⊗ id))(1 ⊗ b * ⊗ a * ⊗ 1) = (m • (1 ⊗ id)) ⊗ (m • (id ⊗ 1)) • Ψ (b * ⊗ a * ) = (Ψ • ( * ⊗ * ) • τ )(a ⊗ b)
Hence we get the following

A such that ϕ is Ψ -1 -invariant or θ is Ψ-invariant, then the convolution ϕ ⋆ θ = (ϕ ⊗ θ) • ∆ is positive, cf. [FS99, Lemma 4.2.2].
We briefly recall the definition of Lévy processes on braided * -bialgebras, see also [START_REF] Franz | Stochastic processes and operator calculus on quantum groups[END_REF]Chapter 4]. A quantum probability space is a pair (A, Φ) consisting of a * -algebra and a state (i.e. a normalized positive linear functional) Φ on A. A quantum random variable j over a quantum probability space (A, Φ) on a *algebra B is a * -algebra homomorphism j : B → A. A quantum stochastic process is a family of quantum random variables over the same quantum probability space, indexed by some set, and defined on the same algebra. Two quantum stochastic processes (j t ) t∈I and (k t ) t∈I , indexed by the same set I, on the same * -algebra B over the quantum probability spaces (A j , Φ j ) and (A k , Φ k ) are called equivalent, if all their finite-dimensional distributions agree, i.e. if

Φ j j t1 (b 1 ) . . . j tn (b n ) = Φ k k t1 (b 1 ) . . . k tn (b n ) for all n ∈ N, t 1 , • • • , t n ∈ I, b 1 , • • • , b n ∈ B.
Definition 1.3. [FS99, Def. 4.2.1] Let (A, Φ) be a quantum probability space, B a * -algebra, and Ψ : B ⊗ B → B ⊗ B a linear map. An n-tuple (j 1 , . . . , j n ) of quantum random variables j i : B → A, i = 1, . . . , n, over (A, Φ) on B is called Ψ-independent or braided independent, if

(i) Φ j σ(1) (b 1 ) . . . j σ(n) (b n ) = Φ(j σ(1) (b 1 )) . . . Φ(j σ(n) (b n )) for all permutations σ ∈ S(n) and all b 1 , . . . , b n ∈ B, and (ii) m A • (j l ⊗ j k ) = m A • (j k ⊗ j l ) • Ψ for all 1 ≤ k < l ≤ n.
Definition 1.4. [FS99, Def. 4.3.1] Let B be a braided * -bialgebra. A quantum stochastic process (j st ) 0≤s≤t on B over some quantum probability space (A, Φ) is called a Lévy process, if the following conditions are satisfied.

(1) (Increment property)

j rs ⋆ j st = j rt for all 0 ≤ r ≤ s ≤ t, j tt = 1 • ε for all 0 ≤ t.,
where j rs ⋆ j st = m A • (j rs ⊗ j st ) • ∆ B denotes the convolution of j rs and j st . (2) (Independence of increments) The family (j st ) 0≤s≤t is Ψ-independent, i.e.

(j s1t1 , . . . , j sntn ) is Ψ-independent for all n ∈ N and all 0 ≤ Let (j st ) be a Lévy process on some * -bialgebra. The states on B defined by φ t = Φ • j 0t are called marginal distributions of (j st ) and uniquely determine a Lévy process (up to equivalence). The marginal distributions form a convolution semigroup. Using the fundamental theorem of coalgebras one can show that there exists a unique, conditionally positive (i.e. positive on the kernel of ε) hermitian, linear functional L : B → C with L(1) = 0, such that φ t = exp ⋆ tL. The functional L is called the generator of the process (j st ). Using Schoenberg correspondence, we see that the convolution semigroup generated by a conditionally positive, hermitian linear functional L with L(1) = 0 consists of states, i.e., normalized positive functionals. Due to Remark 1.2 and the braided version of the Schoenberg correspondence for Ψ-invariant functionals, see [START_REF] Gerhold | Additive deformations of braided Hopf algebras[END_REF], these results remain true for Lévy processes on braided * -bialgebras. This is summarised in the following proposition.

s 1 ≤ t 1 ≤ s 2 ≤ . . . ≤ t n . ( 3
Proposition 1.5. [FS99, Prop. 4.3.2] Let B be a * -bialgebra over C in a braided category (C, Ψ). Then there is a one-to-one correspondence between (equivalence classes of ) Lévy processes (j st ), the set of convolution semigroups (φ t ) t of Ψinvariant states on B , and the set of Ψ-invariant, hermitian, conditionally positive linear functionals L : B → C.

1.2. Quasi-triangular and coquasi-triangular bialgebras. We will now recall some definitions (see e.g. [START_REF] Klimyk | Quantum groups and their representations[END_REF]) which allow us later on to "symmetrize" braided * -bialgebras. Let B be a bialgebra and

R ∈ B ⊗ B. For R = i x i ⊗ y i define R 12 , R 13 und R 23 ∈ B ⊗ B ⊗ B by R 12 := i x i ⊗ y i ⊗ 1, R 13 := i x i ⊗ 1 ⊗ y i , R 23 := i 1 ⊗ x i ⊗ y i . Definition 1.6. A quasi-triangular bialgebra B is a bialgebra B equipped with an invertible element R ∈ B ⊗ B, called universal R-Matrix, such that the equations ∆ op (a) = R • ∆(a) • R -1 , (∆ ⊗ id)(R) = R 13 R 23 , (id ⊗ ∆)(R) = R 13 R 12
(1) hold.

Note that the universal R-matrix of a quasi-triangular bialgebra satisifes the quantum Yang-Baxter equation (QYBE) A coquasi-triangular bialgebra is a bialgebra A equipped with a universal rform on A, i.e. a linear functional r : A ⊗ A → C that is invertible w.r.t. the convolution product (i.e. there exists another functional r :

R 12 R 13 R 23 = R 23 R 13 R 12 , (2) see, e.g.
A ⊗ A → C such that r ⋆ r = r ⋆ r = ε ⊗ ε) that satisfies m op = r ⋆ m ⋆ r, r 13 ⋆ r 23 = r • (m ⊗ id), r 13 ⋆ r 12 = r • (id ⊗ m), (3)
where r 12 , r 23 , r 13 : A ⊗ A ⊗ A → C are defined by r 12 := r ⊗ ε,

r 23 := ε ⊗ r und r 13 := (r ⊗ ε) • (id ⊗ τ ). Furthermore we have r • (1 ⊗ id) = ε = r • (id ⊗ 1) (4)
and, if A has an antipode S, then the antipode is invertible and satisfies

r • (S ⊗ id) = r = r • (id ⊗ S -1 ).
The inverse r satisfies similar conditions, i.e.

r12 ⋆ r13 ⋆ r23 = r23 ⋆ r13 ⋆ r12 , r23 ⋆ r13 = r • (m ⊗ id), r12 ⋆ r13 = r • (id ⊗ m), r • (1 ⊗ id) = r • (id ⊗ 1) = ε.
(5)

Construction of braided categories

We will now study the special case of tensor categories whose objects are Yetter-Drinfeld modules of some given bialgebra A. Our main goal is to construct such categories with objects that are equipped with an involution which is compatible with the braiding in the sense of Definition 1.1. Let A be a bialgebra. A C-vector space V is called (left) Yetter-Drinfeld module (over A), if it is both, a left Amodule with action α : A ⊗ V → V and a left A-comodule with left coaction γ : V → A ⊗ V , such that the (left) Yetter-Drinfeld equation

a (1) v (1) ⊗ a (2) .v (2) = (a (1) .v) (1) a (2) ⊗ (a (1) .v) (2)
or equivalently

(m ⊗ α)•(id ⊗ τ ⊗ id) • (∆ ⊗ γ) = (m ⊗ id) • (id ⊗ τ ) • (γ ⊗ id) • (α ⊗ id) • (id ⊗ τ ) • (∆ ⊗ id) (6)
is satisfied for all a ∈ A, v ∈ V , where a.b := α(a ⊗ b) and γ(v) = v (1) ⊗ v (2) . The category A A YD of (left) Yetter-Drinfeld-modules is well studied, see for example [Y90] or [START_REF] Montgomery | Hopf algebras and their actions on rings[END_REF]Ch. 10]. It is a braided tensor category with braiding Ψ = (α ⊗ id)

• (id ⊗ τ ) • (γ ⊗ id). For actions α V , α W on a bialgebra A a linear map Φ : V → W is called (left) module map, if Φ • α V = α W • (id ⊗ Φ) and (left) comodule map for coactions γ V , γ W , if γ W • Φ = (id ⊗ Φ) • γ V .

The category ( A

A YD * , Ψ) of Yetter-Drinfeld modules with an involution. Now we want to equip the objects in the category A A YD with an involution and construct a new category A A YD * . Lemma 2.1. Let A be a Hopf- * -algebra with antipode S and V ∈ Obj( A A YD) with action α and coaction γ. Suppose that there exists a * -structure on V , such that α and γ satisfy *

• α = α • ( * ⊗ * ) • (S ⊗ id), (7)
γ • * = ( * ⊗ * ) • γ. ( 8
)
Then * V ⊗V = (α ⊗ id) • (τ ⊗ id) • (id ⊗ γ) • ( * ⊗ * ) is an involution on V ⊗ V . Proof: Recall that the inverse of Ψ = (α ⊗ id) • (id ⊗ τ ) • (γ ⊗ id) is given by Ψ -1 = (id ⊗ α) • (τ ⊗ id) • (S -1 ⊗ id ⊗ id) • (γ ⊗ id) • τ, see [Y90, Theorem 7.2]. We have ( * ⊗ * ) • Ψ • ( * ⊗ * ) (7) and (8) = (α ⊗ id) • ( * ⊗ * ⊗ * ) • (S ⊗ τ ) • ( * ⊗ * ⊗ * ) • (γ ⊗ id) = τ • Ψ -1 • τ, since * • S • * = S -1 . This proves that * V ⊗V = Ψ • ( * ⊗ * ) • τ is its own inverse. □ Theorem 2.2.
Let A be a Hopf- * -algebra. Then we can define a braided category A A YD * as follows. The objects (V, α, γ, * ) are Yetter-Drinfeld modules equipped with an involution * , such that equations (7) and (8) are satisfied. The morphisms are the linear maps that are module and comodule maps. The tensor product of objects is given by

(V, α V , γ V , * V ) ⊗ (W, α W , γ W , * W ) := (V ⊗ W, α V ⊗W , γ V ⊗W , * V ⊗W ),
where

α V ⊗W := (α V ⊗ α W ) • (id ⊗ τ ⊗ id) • (∆ ⊗ id ⊗ id), γ V ⊗W := (m ⊗ id ⊗ id) • (id ⊗ τ ⊗ id) • (γ V ⊗ γ W ), * V ⊗W := Ψ W,V • ( * W ⊗ * V ) • τ V,W .
The braiding is again given by Ψ = (α ⊗ id)

• (id ⊗ τ ) • (γ ⊗ id).
Proof: To show that this is again a braided category it only remains to show that

(V ⊗ W, α V ⊗W , γ V ⊗W , * V ⊗W ) is again an object in the category A A YD * . First we show that * V ⊗W satisfies equation (7). We have * V ⊗W • α V ⊗W = Ψ • ( * ⊗ * ) • τ • (α ⊗ α) • (id ⊗ τ ⊗ id) • (∆ ⊗ id ⊗ id) (7) = Ψ • (α ⊗ α) • (id ⊗ τ ⊗ id) • (∆ ⊗ id ⊗ id) • (id ⊗ τ ) • ( * ⊗ * ⊗ * ) • (S ⊗ id ⊗ id) = (α ⊗ α) • (id ⊗ τ ⊗ id) • (∆ ⊗ id ⊗ id) • (id ⊗ Ψ) • ( * ⊗ * ⊗ * ) • (S ⊗ τ ) = α V ⊗W • ( * A ⊗ * V ⊗W ) • (S ⊗ id),
because Ψ is a module map, since it has to be a morphism in the category. Now we show that it also satisfies equation (8).

( * A ⊗ * V ⊗W ) • γ V ⊗W = (id ⊗ Ψ) • (m ⊗ id) • (τ ⊗ τ ) • (id ⊗ τ ⊗ id) • ( * ⊗ * ⊗ * ⊗ * ) • (γ ⊗ γ) = γ V ⊗W • * V ⊗W □
2.2. The category ( A C * , Ψ) of comodules over a coquasi-triangular *bialgebra. We start by considering the category A C of A-comodules over a coquasitriangular bialgebra A.

Lemma 2.3. Let A be a coquasi-triangular bialgebra. If γ is a coaction of A on V , then

α γ = (r ⊗ id) • (id ⊗ γ V ) (9)
defines an action of A on V and (V, α γ , γ) is a Yetter-Drinfeld module. Furthermore, if a linear map f : V → W between two comodules V, W is γ-invariant (i.e. a comodule map), then it is also α γ -invariant.

Proof: First we show that α γ is an action. It is

α γ • (m ⊗ id) = (r ⊗ id) • (m ⊗ id ⊗ id) • (id ⊗ id ⊗ γ) (5) = (r 23 ⋆ r13 ) ⊗ id • (id ⊗ id ⊗ γ) = (ε ⊗ r) ⊗ (r ⊗ ε) • (id ⊗ τ ) • ∆ A⊗A⊗A ⊗ id • (id ⊗ id ⊗ γ) = (r ⊗ id) • (id ⊗ γ) • id ⊗ (r ⊗ id) • (id ⊗ γ) = α γ • (id ⊗ α γ )
as well as

α γ • (1 ⊗ id) = (r • (1 ⊗ id)) ⊗ id • γ (4) = (ε ⊗ id) • γ = id.
Now we want to show that γ and α γ fulfill the Yetter-Drinfeld equation ( 6).

(m ⊗ α γ ) • (id ⊗ τ ⊗ id) • (∆ ⊗ γ) = (m ⊗ r ⊗ id) • (id ⊗ τ ⊗ id ⊗ id) • (∆ ⊗ ∆ ⊗ id) • (id ⊗ γ) = (m ⋆ r) ⊗ id • (id ⊗ γ) (3) = (r ⋆ m op ) ⊗ id • (id ⊗ γ) = (r ⊗ m ⊗ id) • (id ⊗ id ⊗ id ⊗ τ ) • (id ⊗ id ⊗ γ ⊗ id) • (id ⊗ γ ⊗ id) • (id ⊗ τ ) • (∆ ⊗ id) = (m ⊗ id) • (id ⊗ τ ) • (γ ⊗ id) • (α γ ⊗ id) • (id ⊗ τ ) • (∆ ⊗ id)
It remains to show that α γ is a module-map. We have

α W • (id ⊗ f ) = (r ⊗ id) • (id ⊗ id ⊗ f ) • (id ⊗ γ W ) □ Note that for the braiding Ψ = (α γ ⊗ id) • (id ⊗ τ ) • (γ ⊗ id) defined in this way between two A-comodules V and W , we have Ψ(v ⊗ w) = r(v (1) ⊗ w (1) )w (2) ⊗ v (2) , for v ∈ V , w ∈ W , and where γ V (v) = v (1) ⊗ v (2) , γ(w) = w (1) ⊗ w (2)
. Up to a conjugation by the flip and the use of r instead of r, this is the same as the definition of the braiding associated to a universal r-form in [Kas95, Equation (VIII.5.9)].

One can show as in [START_REF] Kassel | Quantum groups[END_REF] that the category of A-comodules becomes a braided category ( A C, Ψ) in this way. Unlike for the construction for Yetter-Drinfeld modules in Theorem 2.2, this does not require A to be a Hopf algebra, because the invertibility of Ψ follows from that of the universal r-form.

Now we want to extend the objects in the category A C by an involution * and define the category A C * .

A coquasi-triangular * -bialgebra (resp. Hopf- * -algebra) is a coquasi-triangular bialgebra (resp. Hopf-algebra) which is also a * -bialgebra (resp. Hopf- * -algebra) such that the universal r-form r fulfills the equation

¯• r = r • ( * ⊗ * ), (10)
where ¯denotes the complex conjugation on C. A universal r-form that satsifies equation ( 10) is called involutive r-form.

Theorem 2.4. Let A be a coquasi-triangular * -bialgebra. Then we can construct a braided category ( A C * , Ψ) as follows. The objects are triples (V, γ V , * V ) consisting of an A-comodule V with coaction γ and an involution * such that equation (8) is satsified. The morphisms between two objects are the comodule maps. The tensor product of objects is given by

(V, γ V , * V ) ⊗ (W, γ W , * W ) := (V ⊗ W, γ V ⊗W , * V ⊗W ),
where

γ V ⊗W := (m ⊗ id ⊗ id) • (id ⊗ τ ⊗ id) • (γ V ⊗ γ W ) * V ⊗W := Ψ W,V • ( * W ⊗ * V ) • τ V,W .
The braiding Ψ is given by

Ψ V,W = (α γ ⊗ id) • (id ⊗ τ ) • (γ ⊗ id) = τ • (r ⊗ id ⊗ id) • (id ⊗ τ ⊗ id) • (γ V ⊗ γ W ).
Proof: Let (V, γ V , * V ), (W, γ W , * W ) be two objects of ( A C * , Ψ), i.e., A-comodules with a coaction and an involution that satisfy (8). We have to check that * V ⊗W is an involution and that V ⊗ W is again an object of (

A C * , Ψ). For v ∈ V , w ∈ W with γ V (v) = v (1) ⊗ v (2) , γ(w) = w (1) ⊗ w (2) we have (v ⊗ w) * = Ψ(w * ⊗ v * ) = r (w (1) ) * ⊗ (v (1) ) * v (2) * ⊗ w (2) * = r w (1) ⊗ v (1) v (2) * ⊗ w (2) * and (v ⊗ w) * ) * = r v (1) ⊗ w (1) v (2) * ⊗ w (2) * * = r w (1) ⊗ v (1) Ψ w (2) ⊗ v (2) = r w (1) ⊗ v (1) r w (2) ⊗ v (2) v (3) ⊗ w (3) = v ⊗ w since r ⋆ r = ε ⊗ ε. Furthermore, γ V ⊗W • * V ⊗W (v ⊗ w) = γ V ⊗W r w (1) ⊗ v (1) v (2) * ⊗ w (2) * = r w (1) ⊗ v (1) w (2) v (2) * v (3) * ⊗ w (3) * = v (1) w (1) r w (2) ⊗ v (2) * v (3) * ⊗ w (3) * = v (1) w (1) * ⊗ r w (2) ⊗ v (2) v (3) * ⊗ w (3) * = γ V ⊗W (v ⊗ w) * ⊗ * V ⊗W , since r ⋆ m = m op ⋆ r. I.e., γ V ⊗W satisfies (8), and therefore (V ⊗ W, γ V ⊗W , * V ⊗W ) is an object of ( A C * , Ψ). □
Remark 2.5. In general, A does not have an antipode. Thus equation ( 7) is not satisfied and (V, α γ , γ, * ) is not an object in ( A A YD * , Ψ) and therefore ( A C * , Ψ) can not be interpreted asa subcategory of ( A A YD * , Ψ). But if A is a coquasitriangular Hopf- * -algebra, then the antipode S is automatically invertible and we have r = r • (S ⊗ id), which, combined with equation (10), implies that α γ satisfies equation (7). So if A is a coquasi-triangular Hopf- * -algebra, then because of Lemma 2.3, the category A C * can be viewed as a subcategory of A A YD * .

2.3. The category ( A C * , Ψ) of modules over a quasi-triangular * -bialgebra.

We start with the category A C of A-modules over a quasi-triangular bialgebra A, and then introduce the category A C * , if A has an involution that is compatible with the R-matrix, see below. Let A be a quasi-triangular bialgebra and define R :

C → A ⊗ A by R(c) = cR for all c ∈ C.
Lemma 2.6. If α is an action of A on V , then

γ α = (id ⊗ α) • (τ ⊗ id) • ( R ⊗ id)
defines a coaction of A on V and (V, α, γ α ) is a Yetter-Drinfeld module. Furthermore, if a linear map f : V → W between two modules V, W is α-invariant (i.e. a module map), then it is also γ α -invariant.

Proof: First we show that γ α is a coaction. It is

(∆ ⊗ id) • γ α = (id ⊗ id ⊗ α) • (id ⊗ τ ⊗ id) • (τ ⊗ id ⊗ id) • (id ⊗ ∆ ⊗ id) • ( R ⊗ id) (1) = (id ⊗ id ⊗ α) • (id ⊗ τ ⊗ id) • (τ ⊗ id ⊗ id) • (m ⊗ id ⊗ id ⊗ id) • (id ⊗ id ⊗ τ ⊗ id) • (id ⊗ τ ⊗ id ⊗ id) • ( R ⊗ R ⊗ id) = (id ⊗ γ α ) • γ α ,
as well as

(ε ⊗ id) • γ α = α • (id ⊗ ε ⊗ id) • ( R ⊗ id) = α • (1 ⊗ id) = id.
Now we show that α and γ α satisify the Yetter-Drinfeld equation ( 6).

(m ⊗ α) • (id ⊗ τ ⊗ id) • (∆ ⊗ γ α ) = (m ⊗ α) • (id ⊗ τ ⊗ α) • (id ⊗ id ⊗ τ ⊗ id) • (∆ ⊗ R ⊗ id) = (id ⊗ α) • (m ⊗ id ⊗ α) • (id ⊗ τ ⊗ id ⊗ id) • (τ ⊗ τ ⊗ id) • ( R ⊗ ∆ ⊗ id) = (m ⊗ id) • (id ⊗ τ ) • (γ α ⊗ id) • (α ⊗ id) • (id ⊗ τ ) • (∆ ⊗ id).
That γ α is a comodule-map follows directly. □ Define a quasi-triangular * -bialgebra as a quasi-triangular bialgebra which is also a

* -bialgebra such that ( * ⊗ * )(R) = R -1 .
Theorem 2.7. Let A be a quasi-triangular Hopf- * -algebra with antipode S. Then we can construct a braided category ( A C * , Ψ) as follows. The objects are triples (V, α, * ) consisting of an A-module V with action α and an involution * such that equation (7) is satsified. The morphisms between two objects are the module maps. The tensor product of objects is given by

(V, α V , * V ) ⊗ (W, α W , * W ) := (V ⊗ W, α V ⊗W , * V ⊗W ),
where

α V ⊗W := (α V ⊗ α W ) • (id ⊗ τ ⊗ id) • (∆ ⊗ id ⊗ id), * V ⊗W := Ψ W,V • ( * W ⊗ * V ) • τ V,W .
The braiding Ψ is given by

Ψ V,W = (α ⊗ id) • (id ⊗ τ ) • (γ α ⊗ id) = (α ⊗ α) • (id ⊗ τ ⊗ id) • (τ ⊗ τ ) • ( R ⊗ id ⊗ id).
Proof: This follows directly from Lemma 2.6 and Theorem 2.2. □ Lemma 2.8. Let A be a quasi-triangular Hopf- * -algebra and let (V, α, * ) be an object in ( A C * , Ψ). Then γ α satisfies equation (8).

Proof:

Since (S ⊗ id) • R = R -1 , we have ( * ⊗ * ) • γ α = ( * ⊗ * ) • (id ⊗ α) ⊗ (τ ⊗ id) • ( R ⊗ id) (7) = (id ⊗ α) • ( * ⊗ * ⊗ * ) • (id ⊗ S ⊗ id) • (τ ⊗ id) • ( R ⊗ id) = (id ⊗ α) • (τ ⊗ id) • ( * ⊗ * ⊗ * ) • ( R -1 ⊗ id) = (id ⊗ α) • ( R ⊗ * ) = γ α • * □
Remark 2.9. It follows from Lemma 2.8 and Lemma 2.6 that (V, α, γ α , * ) ∈ Obj( A A YD) and hence ( A C * , Ψ) is a subcategory of ( A A YD * , Ψ).

Cocommutative bialgebras.

If the bialgebra A is cocommutative (i.e. τ •∆ = ∆), then 1⊗1 defines an R-matrix.

The corresponding braiding is simply the flip τ .

We show now that the construction in Theorem 2.2 includes as a special case the construction given in [START_REF] Schürmann | White noise on bialgebras[END_REF]. Schürmann's construction has as input the group algebra A = CΓ of some group Γ, and actions and coactions of A on the objects V that satisfy the compatibility condition

γ • α = (ad ⊗ α) • (id ⊗ τ ⊗ id) • (∆ ⊗ γ) (11) see [Sch93, page 14], where ad = m • (id ⊗ m) • (id ⊗ id ⊗ S) • (id ⊗ τ ) • (∆ ⊗ id) is the adjoint action of A on itself.
The following lemma shows that our construction is a generalisation of the one presented there.

Lemma 2.10. Let A be a cocommutative bialgebra. Let α and γ be an action and a coaction of A on some vector space V . If α and γ satisfy (11), then (V, α, γ) is a Yetter-Drinfeld module.

Proof: Substituting (11) into the right side of (6), we get after some simplifications

(m (4) ⊗ α) • (id ⊗ τ ⊗ τ ⊗ id) • (id ⊗ τ A⊗A,A • id) • (id A ⊗ S ⊗ id) • (∆ (4) ⊗ γ)
where

m (4) = m • (m ⊗ id) • (m ⊗ id ⊗ id), ∆ (4) = (∆ ⊗ id ⊗ id) • (∆ ⊗ id) • ∆.
Using the cocommutativity, we can produce a term of the form m • (S ⊗ id) • ∆, to which we can apply the antipode axiom. Using the unit and the counit axiom to clean up the resulting expression, we get the desired result. □ 3. Symmetrization of braided * -bialgebras and their Lévy processes 3.1. Symmetrizing braided * -bialgebras. Now we will present a construction that will allow us in Subsection 3.2 to associate with every Lévy process on a braided * -bialgebra B a Lévy process on a usual (i.e., symmetric or τ -braided) *bialgebra. The idea is to construct a bigger (symmetric) bialgebra H that contains the braided bialgebra B as a subalgebra and whose coproduct is related to that of B in a "nice way". For the case where the braiding is defined through the action and coaction of a group, this construction can be found in [Sch93, Chapter 3]. For the general case a similar construction, called bosonization, was introduced by Majid, see [Maj95b, Section 9.4] and the references indicated there. But the role of the involution is not studied there. In this section we study the left symmetrization of braided * -bialgebras. Note that the whole theory works analogously for categories consisting of right modules and comodules.

Theorem 3.1. Let A be a Hopf- * -algebra and let B be a braided * -bialgebra in

( A A YD * , Ψ). Then H = B ⊗ A (as a vector space) becomes a * -bialgebra with m H = (m B ⊗ m A ) • (id ⊗ α ⊗ id ⊗ id) • (id ⊗ id ⊗ τ ⊗ id) • (id ⊗ ∆ ⊗ id ⊗ id), 1 H = 1 B ⊗ 1 A , ∆ H = (id ⊗ m ⊗ id ⊗ id) • (id ⊗ id ⊗ τ ⊗ id) • (id ⊗ γ ⊗ id ⊗ id) • (∆ B ⊗ ∆ A ), ε H = ε B ⊗ ε A , * H = (α ⊗ id) • (τ ⊗ id) • (id ⊗ ∆) • ( * B ⊗ * A ).
The map id B ⊗ 1 A : B → B ⊗ A ∼ = H defines an embedding, i.e., an injective *algebra homomorphism. Furthermore we have the following commutative diagrams

B H B ⊗ B H ⊗ H id B ⊗ 1 A ∆ B id B ⊗ γ ⊗ 1 A ∆ H B H B ⊗ B H ⊗ H id B ⊗ ε A ∆ B id B ⊗ ε A ⊗ id B ⊗ ε A ∆ H Proof: For the proof that H is a bialgebra, see [Maj95b, Section 9.4].
To show that it is even a * -bialgebra we first have to verify that * H is its own inverse.

Using the facts that ∆ is a * -algebra homomorphism and * is self-inverse, as well as equation ( 7) and the relation

* • S • * = S -1 , we get * H • * H = (α ⊗ id) • (id ⊗ α ⊗ id) • (τ ⊗ id ⊗ id) • (id ⊗ τ ⊗ id) • (id ⊗ id ⊗ ∆) • (S -1 ⊗ id ⊗ id) • (τ ⊗ id) • (id ⊗ ∆).
Applying the module equation and the coassociativity, this expression transforms to

(α ⊗ id) • (m • (id ⊗ S -1 ) • τ • ∆) ⊗ id ⊗ id • (τ ⊗ id) • (id ⊗ ∆)
and using the antipode axiom to

(α ⊗ id) • ((1 • ε) ⊗ τ ) • (∆ ⊗ id) • τ = τ • τ = id ⊗ id,
which shows that * H is its own inverse. Next we want to show that * H is an algebra-antihomomorphism. After applying some basic transformations, as well as equation ( 7) and the bialgebra equation

∆ • m = (m ⊗ m) • (id ⊗ τ ⊗ id) • (∆ ⊗ ∆) twice, we get the expression * H • m H = (m ⊗ id) • (α ⊗ α ⊗ id) • (id ⊗ τ ⊗ τ ) • (m ⊗ m ⊗ id ⊗ id ⊗ id) • (id ⊗ τ ⊗ id ⊗ τ ⊗ id) • (∆ ⊗ ∆ ⊗ id ⊗ id ⊗ id) • (id ⊗ id ⊗ m ⊗ id ⊗ id) • (id ⊗ τ ⊗ id ⊗ id ⊗ id) • (∆ ⊗ ∆ ⊗ id ⊗ id) • τ H⊗H • (τ ⊗ τ ) • (id ⊗ α ⊗ id ⊗ id) • (id ⊗ S -1 ⊗ τ ⊗ id) • (id ⊗ ∆ ⊗ id ⊗ id) • ( * ⊗ * ⊗ * ⊗ * ).
Again after using some basic transformations and reordering, we get the expression

(m ⊗ m) • (id ⊗ α ⊗ id ⊗ id) • (id ⊗ id ⊗ τ ⊗ id) • (id ⊗ ∆ ⊗ id ⊗ id) • (α ⊗ id ⊗ α ⊗ id) • (id ⊗ τ ⊗ id ⊗ τ ) • (∆ ⊗ id ⊗ ∆ ⊗ id) • (τ ⊗ τ ) • (id ⊗ τ ⊗ id) • (τ ⊗ τ ) • (id ⊗ α ⊗ id ⊗ id) • (id ⊗ m ⊗ τ ⊗ id) • (id ⊗ S ⊗ id ⊗ S ⊗ id ⊗ id) • (id ⊗ ∆ ⊗ id ⊗ id ⊗ id) • (id ⊗ τ ⊗ id ⊗ id) • (id ⊗ ∆ ⊗ id ⊗ id) • (id ⊗ S -1 ⊗ id ⊗ id) • ( * ⊗ * ⊗ * ⊗ * ).
After applying the antipode axiom and some simplifications, this becomes

(m ⊗ m) • (id ⊗ α ⊗ id ⊗ id) • (id ⊗ id ⊗ τ ⊗ id) • (id ⊗ ∆ ⊗ id ⊗ id) • (α ⊗ id ⊗ α ⊗ id) • (τ ⊗ id ⊗ τ ⊗ id) • (id ⊗ ∆ ⊗ id ⊗ ∆) • ( * ⊗ * ⊗ * ⊗ * ) • (τ ⊗ τ ) • (id ⊗ τ ⊗ id) = m H • ( * H ⊗ * H ) • τ H⊗H .
Thus we have shown that * H is an algebra homomorphism. Now we want to show that ∆ is a * -algebra homomorphism. After using the coassociativity twice and some reordering, we get the expression

∆ H • * H = id ⊗ (m ⊗ id) • (id ⊗ τ ) • (γ ⊗ id) • (α ⊗ id) • (id ⊗ τ ) • (∆ ⊗ id) ⊗ id • (α ⊗ id) • (id ⊗ τ ) • (∆ ⊗ id) ⊗ id ⊗ id • (τ ⊗ id ⊗ id) • (id ⊗ τ ⊗ id) • (∆ ⊗ ∆) • ( * ⊗ * ).
Using the Yetter-Drinfeld equation as well as the equations ( 7) and (8), this transforms to

( * ⊗ * ⊗ * ⊗ * ) • id ⊗ (m ⊗ α) • (τ ⊗ id ⊗ id) • (id ⊗ τ ⊗ id) • (∆ ⊗ id ⊗ id) ⊗ id • (α ⊗ id) • (S -1 ⊗ τ ) • (∆ ⊗ id) • τ ⊗ id ⊗ id ⊗ id • (α ⊗ τ ⊗ id ⊗ id) • (S -1 ⊗ id ⊗ id ⊗ τ ⊗ id) • (τ ⊗ γ ⊗ id ⊗ id) • (id ⊗ γ ⊗ id ⊗ id) • (∆ ⊗ ∆).
After a few more basic transformations, this becomes

( * ⊗ * ⊗ * ⊗ * ) • ( (α • (S -1 ⊗ id) • τ ) ⊗ id ⊗ (α • (S -1 ⊗ id) • τ ) ⊗ id • id ⊗ (m ⊗ m) • (id ⊗ τ ⊗ id) • (∆ ⊗ ∆) ⊗ id ⊗ id • (id ⊗ id ⊗ τ ⊗ id ⊗ id) • (id ⊗ γ ⊗ id ⊗ ∆) • (∆ ⊗ ∆).
Using the bialgebra equation, we get

(α ⊗ id) • (τ ⊗ id) • (id ⊗ ∆) • ( * ⊗ * ) ⊗ (α ⊗ id) • (τ ⊗ id) • (id ⊗ ∆) • ( * ⊗ * ) • id ⊗ (m ⊗ id) • (id ⊗ τ ) • (γ ⊗ id) ⊗ id • (∆ ⊗ ∆)
and thus we have shown that ∆ is a * -algebra homomorphism. It remains to show that the inclusions are * -algebra homomorphisms and that both diagrams commute. This can easily be verified by straightforward calculations. □

Due to the connection between the categories we get the following corollaries for the categories ( A C * , Ψ) and ( A C * , Ψ).

Corollary 3.2. Let A be a coquasi-triangular * -bialgebra and B a braided *bialgebra in ( A C * , Ψ). Then H = B ⊗ A (as a vector space) becomes a * -bialgebra with 1 H , ∆ H and ε H as in Theorem 3.1 and

m H = (m B ⊗ m A ) • (id ⊗ α γ ⊗ id ⊗ id) • (id ⊗ id ⊗ τ ⊗ id) • (id ⊗ ∆ ⊗ id ⊗ id) = (m B ⊗ m A ) • (id ⊗ r ⊗ τ ⊗ id) • (id ⊗ id ⊗ τ ⊗ id ⊗ id) • (id ⊗ ∆ ⊗ γ ⊗ id), * H = (α γ ⊗ id) • (τ ⊗ id) • (id ⊗ ∆) • ( * B ⊗ * A ) = (r ⊗ id ⊗ id) • (τ ⊗ id ⊗ id) • (id ⊗ τ ⊗ id) • (γ ⊗ ∆) • ( * B ⊗ * A ).
Corollary 3.3. Let A be a quasi-triangular * -bialgebra and B a braided * -bialgebra in ( A C * , Ψ). Then H = B ⊗ A (as a vector space) becomes a * -bialgebra with m H , 1 H , ε H and * H as in Theorem 3.1 and

∆ H = (id ⊗ m ⊗ id ⊗ id) • (id ⊗ id ⊗ τ ⊗ id) • (id ⊗ γ α ⊗ id ⊗ id) • (∆ B ⊗ ∆ A ) = (id ⊗ m ⊗ id ⊗ id) • (id ⊗ id ⊗ τ ⊗ id) • (id ⊗ id ⊗ α ⊗ id ⊗ id) • (id ⊗ τ ⊗ id ⊗ id ⊗ id) • (id ⊗ R ⊗ id ⊗ id ⊗ id) • (∆ ⊗ ∆).

Symmetrizing braided Lévy processes.

The following proposition is important for symmetrizing Lévy processes, i.e. for constructing a Lévy processes on H, for a given Lévy processes on B. In Theorem 3.5 below we shall show that the process we construct on H allows to recover a process on B which is equivalent to the original process.

Proposition 3.4. The map F : B ′ → H ′ , φ → φ ⊗ ε A is a unital injective algebra homomorphism w.r.t. the convolution product. Furthermore, it maps positive (or hermitian, conditionally positive) Ψ-invariant functionals φ ∈ B ′ to positive (or hermitian, conditionally positive, resp.) functionals F (φ) ∈ H ′ .

Proof: The injectivity of F is clear, because F :

H ′ → B ′ ∼ = (B ⊗ 1) ′ defined by F (ψ) = ψ • (id ⊗ 1) is a left inverse of F . It is unital, since F (ε B ) = ε B ⊗ ε A = ε H .
Furthermore it preserves the convolution product, because

F (φ 1 ) ⋆ F (φ 2 ) = (F (φ 1 ) ⊗ F (φ 2 )) • ∆ = (φ 1 ⊗ ε ⊗ φ 2 ⊗ ε) • (id ⊗ m ⊗ id ⊗ id) • (id ⊗ id ⊗ τ ⊗ id) • (id ⊗ γ ⊗ id ⊗ id) • (∆ ⊗ ∆) = (φ 1 ⊗ ε ⊗ φ 2 ⊗ ε ⊗ ε) • (id ⊗ γ ⊗ id ⊗ id) • (∆ ⊗ ∆) = ((φ 1 ⊗ φ 2 ) • ∆) ⊗ ε = F ((φ 1 ⊗ φ 2 ) • ∆) = F (φ 1 ⋆ φ 2 ).
Assume now that φ is positive and

Ψ-invariant. Let c = k b k ⊗ a k ∈ B ⊗ A ∼ = H. We want to show that F (φ) is again positive. Because (b k ⊗ a k ) * = Ψ(a * k ⊗ b * k ), we have c * c = k b k ⊗ a k * k b k ⊗ a k = m H k,l (b k ⊗ a k ) * ⊗ (b l ⊗ a l ) = m H k,l Ψ(a * k ⊗ b * k ) ⊗ b l ⊗ a l .
Since m is a morphism and thus Ψ-invariant, it is

(m ⊗ m) • (id ⊗ Ψ ⊗ id) • (Ψ ⊗ id ⊗ id) = (id ⊗ m) • (Ψ ⊗ id) • (id ⊗ m ⊗ id)
and therefore we get

F (φ) (c * c) = (φ ⊗ ε) • m H k,l Ψ(a * k ⊗ b * k ) ⊗ b l ⊗ a l = (φ ⊗ ε • (m ⊗ m) • (id ⊗ Ψ ⊗ id) • (Ψ ⊗ id ⊗ id) k,l a * k ⊗ b * k ⊗ b l ⊗ a l = (φ ⊗ ε) • (id ⊗ m) • (Ψ ⊗ id) • (id ⊗ m ⊗ id) k,l a * k ⊗ b * k ⊗ b l ⊗ a l = ε • m • (Ψ A,C ⊗ id) • (id ⊗ φ ⊗ id) • (id ⊗ m ⊗ id) k,l a * k ⊗ b * k ⊗ b l ⊗ a l = (ε • m) k,l φ(b * k b l )a * k ⊗ a l = k,l φ(b * k b l )ε(a * k a l ).
This is positive, since it is the Schur product of two positive definite matrices. Conditional positivity can be shown similarly and hermitianity is a straightforward calculation. □ Theorem 3.5. Let B be a braided * -bialgebra in one of the categories ( A A YD * ), ( A C * , Ψ) or ( A C * , Ψ). Let (j st ) 0≤s≤t be a Lévy process on B with convolution semigroup (φ t ) t≥0 and let (j H st ) 0≤s≤t be a Lévy process on H ∼ = B⊗A with convolution semigroup (F (φ t )) t≥0 . Then (ȷ st ) 0≤s≤t with

ȷst := m • (j H 0s ⊗ j H st ) • (1 B ⊗ γ ⊗ 1 A )
defines a Lévy process on B. Furthermore (ȷ st ) 0≤s≤t is equivalent to (j st ) 0≤s≤t .

(b) = m • (j H 0s ⊗ j H st ) • (1 B ⊗ γ ⊗ 1 A )(b) = j H 0s (1 ⊗ b (1) ) • j H st (b (2) ⊗ 1) (12)
as well as

j H 0s (1 ⊗ a) = (j H 0r ⋆ j H rs )(1 ⊗ a) = m • (j H 0r ⊗ j H rs ) • ∆ H (1 ⊗ a) = j H 0r (1 ⊗ a (1) ) • j H rs (1 ⊗ a (2) ). ( 13
)
and

∆ H (b (2) ⊗ 1) = b (2) (1) ⊗ b (2) (1) (2) ⊗ b (2) (2) (2) ⊗ 1. ( 14
)
Let 0 ≤ r ≤ s ≤ t. Then we have

(ȷ rs ⋆ ȷst )(b) = m • (ȷ rs ⊗ ȷst ) • ∆(b) = ȷrs (b (1) ) • ȷst (b (2) ) (12) = j H 0r (1 ⊗ b (1) (1) ) • j H rs (b (2) (1) ⊗ 1) • j H 0s (1 ⊗ b (1) (2) ) • j H st (b (2) (2) ⊗ 1) (13) = j H 0r (1 ⊗ b (1) (1) ) • j H 0r (1 ⊗ b (1) (2) (1) ) • j H rs (b (2) (1)
⊗ 1)

• j H rs (1 ⊗ b (1) (2) (2) ) • j H st (b (2) (2) ⊗ 1) = j H 0r (1 ⊗ b (1) (1) • b (1) (2) (1) ) • j H rs (b (2) (1) ⊗ b (1) (2) (2) ) • j H st (b (2) (2) ⊗ 1) = m • (m ⊗ id) • (j H 0r ⊗ j H rs ⊗ j H st ) • (id ⊗ id ⊗ id ⊗ γ ⊗ id) • (id ⊗ m ⊗ id ⊗ id ⊗ id) • (id ⊗ id ⊗ τ ⊗ id ⊗ id) • (id ⊗ γ ⊗ γ ⊗ id) • (1 B ⊗ ∆ ⊗ 1 A )(b) = j H 0r (1 ⊗ b (1) ) • j H rs (b (2)
(1) ⊗ b

(2) (1) (2)

) • j H st (b (2) (2) (2)
⊗ 1)

(14) = j H 0r (1 ⊗ b (1) ) • m • (j H rs ⊗ j H st ) • ∆ H (b (2) ⊗ 1) = j H 0r (1 ⊗ b (1) ) • (j H rs ⋆ j H st )(b (2) ⊗ 1) = j H 0r (1 ⊗ b (1) ) • j H rt (b (2) ⊗ 1) = m • (j H 0r ⊗ j H rt ) • (1 B ⊗ γ ⊗ 1 A )(b) = ȷrt (b)
Thus we have shown that (ȷ st ) 0≤s≤t satisfies the increment property. Furthermore, because of the independence of increments, it follows

Φ • ȷst = (Φ • j H 0s ) ⊗ (Φ • j H st ) • (1 B ⊗ γ ⊗ 1 A ) = F (φ s ) ⊗ F (φ t-s ) • (1 B ⊗ γ ⊗ 1 A ) = (φ s ⊗ ε A ⊗ φ t-s ⊗ ε A ) • (1 B ⊗ γ ⊗ 1 A ) = φ t-s ,
i.e., the processes (j st ) 0≤s≤t and (ȷ st ) 0≤s≤t have the same marginal distributions. This implies the stationarity and the weak continuity of the increments of (ȷ st ) 0≤s≤t and completes the proof that (ȷ st ) 0≤s≤t is a Lévy process. Furthermore it establishes the equivalence of the two processes and completes the proof of the theorem. □ Lévy processes on symmetric * -bialgebras can be realized on Bose Fock spaces using quantum stochastic differential calculus [START_REF] Hudson | Quantum Ito's formula and stochastic evolutions[END_REF][START_REF] Parthasarathy | An introduction to quantum stochastic calculus[END_REF][START_REF] Meyer | Quantum probability for probabilists[END_REF]. The necessary input is a triple (ρ, η, L), where ρ is a * -representation of A, on some pre-Hilbert space P , η : A → P is a ρ-cocycle, i.e. Theorem 3.7. Let (j st ) 0≤s≤t be a Lévy process on a braided * -bialgebra B in ( A A YD * ), ( A C * , Ψ) or ( A C * , Ψ) with α-invariant generator L and triple (ρ, η, L). Furthermore let (j H st ) 0≤s≤t be the Lévy process on the symmetrization H ∼ = B ⊗ A from Theorem 3.5 with triple (ρ H , η H , L H ). Then we have -The pre-Hilbert spaces P and P H belonging to the triples are isometrically isomorphic, i.e. there exists a linear, bijective map Ť : P H → P such that ⟨a, b⟩

P H = ⟨ Ť (a), Ť (b)⟩ P . -Ť • η H (b ⊗ a) = ε A (a)η(b) and Ť • η H vanishes on 1 B ⊗ A.
-ρ H is determinded by

Ť ρ H (1 B ⊗ a)η H (b ′ 0 ⊗ 1 A ) = η(α(a ⊗ b ′ 0 )) Ť ρ H (b ⊗ 1 A )η H (b ′ 0 ⊗ 1 A ) = η(bb ′ 0 ), for a ∈ A, b ∈ B, b ′ 0 ∈ B 0 .
Proof: Using the construction above we know that the triple (ρ, η, L) belonging to (j st ) 0≤s≤t is defined on the pre-Hilbert-space P := B 0 /N 0 with inner product

⟨a + N 0 , b + N 0 ⟩ P := ⟨a, b⟩ B0 = L(a * b)
where

N 0 = {G ∈ B 0 |⟨G, G⟩ B0 = 0}. The representation ρ is given by ρ(a)(b + N 0 ) := (a • b + N 0 ) and the ρ-cocycle by η(a) = (a -ε(a) • 1 B ) + N 0 .
Analogously, the triple (ρ H , η H , L H ) of the process (j H st ) 0≤s≤t on the (left) symmetrization H of B, with L H = F (L), is defined on the pre-Hilbert space P H := H 0 /N H 0 with inner product

⟨a + N H 0 , b + N H 0 ⟩ P H := ⟨a, b⟩ H0 = L H (a * b), (15)
where

H 0 := ker ε H . The representation ρ H is defined by ρ H (a)(b + N H 0 ) := (a • b) + N H 0 and the ρ-cocycle η H is given by η H (b ⊗ a) = b ⊗ a -ε H (b ⊗ a) • 1 H + N H 0 . ( 16
)
First we want to show that the pre-Hilbert spaces P und P H are isometrically isomorphic, i.e. there exists a linear bijective map T : P H → P with ⟨a, b⟩

P H = ⟨ T (a), T (b)⟩ P for all a, b ∈ P H . Let b ⊗ a, d ⊗ c ∈ H 0 = ker(ε B ⊗ ε A ), b, d ∈ B and a, c ∈ A. We have (L ⊗ ε A )((b ⊗ a) * (d ⊗ c)) = ε A (a * c)L(b * d) since L H • m H • ( * H ⊗ id H ) = (L ⊗ ε) • (m ⊗ m) • (id ⊗ α ⊗ id ⊗ id) • (id ⊗ id ⊗ τ ⊗ id) • (id ⊗ ∆ ⊗ id ⊗ id) • (α ⊗ id) • (τ ⊗ id) • (id ⊗ ∆) • ( * ⊗ * ) ⊗ id ⊗ id = (L ⊗ ε ⊗ ε) • (m ⊗ id ⊗ id) • (α ⊗ α ⊗ id ⊗ id) • (τ ⊗ id ⊗ τ ⊗ id) • (id ⊗ ((∆ ⊗ id) • ∆) ⊗ id ⊗ id) • ( * ⊗ * ⊗ id ⊗ id) = (L ⊗ ε) • (m ⊗ id) • (α ⊗ α ⊗ id) • (id ⊗ τ ⊗ id ⊗ id) • (∆ ⊗ id ⊗ id ⊗ id) • (τ ⊗ id ⊗ id) • ( * ⊗ * ⊗ id ⊗ id) = (L ⊗ ε) • (α ⊗ id) • (id ⊗ m ⊗ id) • (τ ⊗ id ⊗ id) • ( * ⊗ * ⊗ id ⊗ id) = (ε ⊗ ε) • (id ⊗ id ⊗ m) • (id ⊗ τ ⊗ id) • (τ ⊗ τ ) • ( * ⊗ * ⊗ id ⊗ id) = (ε ⊗ L) • (m ⊗ m) • (id ⊗ τ ⊗ id) • (τ ⊗ τ ) • ( * ⊗ * ⊗ id ⊗ id)
and thus

L H (b ⊗ a) * (d ⊗ c) = L H • m H • ( * H ⊗ id H )(b ⊗ a ⊗ d ⊗ c) = (ε A ⊗ L) • (m ⊗ m) • (id ⊗ τ ⊗ id) • (τ ⊗ τ ) • ( * ⊗ * ⊗ id ⊗ id)(b ⊗ a ⊗ d ⊗ c) = ε A (a * c)L(b * d) From this it follows that ⟨b ⊗ a, d ⊗ c⟩ H0 = L H ((b ⊗ a) * (d ⊗ c)) = (L ⊗ ε A )((b ⊗ a) * (d ⊗ c)) = ε A (a * c)L(b * d) = ε A (a * c)⟨b, d⟩ B0 . ( 17
)
That (ρ, η, L) is the triple of the generator L means that

L(b * d) = ⟨η(b), η(d)⟩ P + ε(b * )L(d) + L(b * )ε(d).
Thus from equation ( 17) it follows that

ε A (a * c)L(b * d) = ε A (a * c)⟨η(b), η(d)⟩ P + ε A (a * )ε A (a)ε B (b * )L(d) + ε A (a * )ε A (a)L(b * )ε B (d).
It is

ker ε H = (ker ε B ⊗ (ker ε A ⊕ ⟨1 A ⟩)) ⊕ ((ker ε B ⊗ ⟨1 B )⟩) ⊗ ker ε A = (ker ε B ⊗ ker ε B ) ⊕ (ker ε B ⊗ ⟨1 A ⟩) ⊕ (⟨1 B ⟩ ⊗ ker ε A ).
and the second and the third addend vanish. Hence we have

⟨η(b), η(d)⟩ P = L(b * d) = ⟨b, d⟩ P (18)
and with equations ( 17) and (18) it follows that

⟨b ⊗ a, d ⊗ c⟩ H0 = ε A (a * c)L(b * c) = ε A (a * c)⟨η(b), η(d)⟩ P = ⟨ε A (a)η(b), ε A (c)η(d)⟩ P .

Now we define the map

T : H 0 → P b ⊗ a → ε A (a)η(b).
Since T (b ⊗ a) = 0 for b ⊗ a ∈ N H 0 we can lift the map to P H and we get the isometric isomorphism

T : P H → P η H → ε A (a)η(b),
where the injectivity follows from ker T = {0} and the surjectivity of T from the surjectivity of η. Since we have

η(1 B ) = (1 B -ε B (1 B ) • 1 B ) + N H 0 = 0 + N H 0 (19) it follows that T • η H (b ⊗ a) (16) = T (b ⊗ a -ε H (b ⊗ a) • 1 H ) + N H 0 = T b ⊗ a + N H 0 -T ε B (b)ε A (a) • (1 B ⊗ 1 A ) + N H 0 = ε A (a)η(b) -ε B (b)ε A (a)ε A (1 A )η(1 B ) (19) = ε A (a)η(b).
From equation (19) also follows that

T • η H (1 B ⊗ a) = ε A (a)η(1 B ) -ε B (1 B )ε A (a)ε A (1 A )η(1 B ) = 0 for all a ∈ A and thus T • η H vanishes on 1 B ⊗ A. For the proof of the last part let b = b 0 + c b • 1 B ∈ ker ε B ⊕ C • 1 B = B und a = a 0 + c a • 1 A ∈ ker ε A ⊕ C • 1 A = A. It is ⟨b 0 ⊗ a 0 , b 0 ⊗ a 0 ⟩ H0 (15) = L H (b 0 ⊗ a 0 ) * (b 0 ⊗ a 0 ) = ε A (a 0 ) * ε A (a 0 )L (b 0 ) * b 0 = 0 and hence we get η H (b ⊗ a) = η H b 0 ⊗ a 0 + η H b 0 ⊗ c a • 1 A + η H c b • 1 B ⊗ a 0 + η H c b • 1 B + c a • 1 A = η H b 0 ⊗ c a • 1 A .

Therefore it remains to show the assertion for b

0 ⊗ c a • 1 A ∈ ker ε B ⊕ C • 1 B . Let b ∈ B, a ∈ A und b ′ 0 ∈ ker ε B . It is m H (b ⊗ a ⊗ d ⊗ c) = b • α(a (1) ⊗ d) ⊗ a (2) • c.
with the relations p,q R ik pq a p j a q l = p,q

a k q a i p R pq jl , k,j R ij kl b k q a p j = k,j a k l b i j R jp qk k ∆(a i j ) = a i k ⊗ a k j , ε(a i j ) = δ i j .
The proof that these relations define indeed a * -bialgebra is similar to the construction of the FRT-bialgebra in [START_REF] Faddeev | Quantization of Lie groups and Lie algebras[END_REF]. See also [START_REF] Malczak | Realisierung von"braided" Quanten-Lévy-Prozessen[END_REF].

Lemma 4.1. Let V be a finite dimensional vector space and R ∈ End(V ⊗ V ) a bi-invertible solution of the quantum Yang-Baxter equation (2). Then A * (R) is a coquasi-triangular * -bialgebra, with the unique involutive r-form

r : A * (R) ⊗ A * (R) → C such that r(a i j ⊗ a k l ) = R ik jl , r(a i j ⊗ b k l ) = R ki lj r(a i j ⊗ 1) = r(1 ⊗ a i j ) = δ i j . The inverse (w.r.t. the convolution) r : A * (R) ⊗ A * (R) → C is defined by r(a i j ⊗ a k l ) = (R -1 ) ik jl , r(a i j ⊗ b k l ) = R ki lj r(a i j ⊗ 1) = r(1 ⊗ a i j ) = δ i j .
Proof: The proof is similar to [KS97, Ch. 10.1.2, Theo. 7]. See [START_REF] Malczak | Realisierung von"braided" Quanten-Lévy-Prozessen[END_REF] for a more detailed treatment. □

The left symmetrization

H = V(R) ⊗ A * (R ′ ). Consider the FRT- * -bi- algebra A * (R ′ ) for R ′ := τ (R). From Lemma 4.1 it follows that A * (R ′ ) is coquasi- triangular with the r-form r ′ : A * (R ′ ) ⊗ A * (R ′ ) → C where r ′ (a i j ⊗ a k l ) = R ki lj , r ′ (a i j ⊗ b k l ) = R ik jl , r ′ (b i j ⊗ a k l ) = R ki lj , r ′ (b i j ⊗ b k l ) = (R -1 ) ik jl and the inverse r ′ : A * (R ′ ) ⊗ A * (R ′ ) → C with r ′ (a i j ⊗ a k l ) = (R -1 ) ki lj , r ′ (a i j ⊗ b k l ) = R ik jl , r ′ (b i j ⊗ a k l ) = R ki lj , r ′ (b i j ⊗ b k l ) = R ik jl .
We extend r ′ to general elements such that equations (3) are satisifed. Let V(R) be the free algebra generated by the elements x 1 , • • • , x n and their adjoints v

1 = (x 1 ) * , • • • , v n = (x n ) * . The map γ : V(R) → A * (R ′ ) ⊗ V(R) with γ(1) = 1 ⊗ 1, γ(x i ) = j b j i ⊗ x j , γ(v i ) = j a i j ⊗ v j defines a left coaction on V(R). Thus the left action α = (r ′ ⊗ id) • (id ⊗ γ) introduced in Lemma 2.3 is given by α(a i j ⊗ x k ) = l R il jk x l , α(a i j ⊗ v k ) = l (R -1 ) ki lj v l , α(b i j ⊗ x k ) = l R il jk x l , α(b i j ⊗ v k ) = l R ki lj v l
and the braiding Ψ is given by

Ψ(x i ⊗ x j ) = k,l R kl ij x l ⊗ x k , Ψ(x i ⊗ v j ) = k,l R jk li v l ⊗ x k , Ψ(v i ⊗ x j ) = k,l R il kj x l ⊗ v k , Ψ(v i ⊗ v j ) = k,l (R -1 ) ji lk v l ⊗ v k
on the generators and extended to arbitrary elements by

Ψ(1 ⊗ u) = u ⊗ 1, Ψ(u ⊗ 1) = 1 ⊗ u, Ψ • (id ⊗ m) = (m ⊗ id) • (id ⊗ Ψ) • (Ψ ⊗ id), Ψ • (m ⊗ id) = (id ⊗ m) • (Ψ ⊗ id) • (id ⊗ Ψ),
such that the multiplication is Ψ-invariant. V(R) becomes a bialgebra with comultiplication ∆ and counit ε given by

∆(x i ) = x i ⊗ 1 + 1 ⊗ x i , ∆(v i ) = v i ⊗ 1 + 1 ⊗ v i ε(1) = 1, ε(x i ) = ε(v i ) = 0
on the generators and extended such that it is a homomorphism from

V(R) to V(R) ⊗ V(R)
, where the latter is equipped with the multiplication (m ⊗ m)

• (id ⊗ Ψ ⊗ id). Hence V(R) is a braided * -bialgebra in the braided category ( A * (R ′ ) C * , Ψ)
and we can give the relations of the (left) symmetrization

H = V(R) ⊗ A(R ′ ) of V(R), cf. Corollary 3.2. H is the * -bialgebra generated by the elements x i ⊗ 1 A * (R ′ ) , v i ⊗ 1 A * (R ′ ) = (x i ) * ⊗ 1 A * (R ′ ) , 1 V(R) ⊗ a i j , 1 V * (R) ⊗ b j i = 1 V * (R) ⊗ (a i j ) * with the relations p,q R ik pq a p j a q l = p,q a k q a i p R pq jl , k,j R ij kl b k q a p j = k,j a k l b i j R jp qk a j k x i = p,l R jl pi x l a p k , b j k x i = p,l R pl ki x l b j p , a j k v i = p,l (R -1 ) ij lp v l a p k , b j k v i = p,l R ip lk v l b j p , ∆(a i j ) = k a i k ⊗ a k j , ∆(b i j ) = k b k j ⊗ b i k , ∆(x i ) = x i ⊗ 1 + j b j i ⊗ x j , ∆(v i ) = v i ⊗ 1 + j a i j ⊗ v j , ε(a i j ) = ε(b i j ) = δ i j , ε(x i ) = ε(v i ) = 0.

The right symmetrization H

R = A * (R) ⊗ V(R).
For the second construction we choose the coquasi-triangular * -bialgebra A * (R) with universal r-form r :

A * (R) ⊗ A * (R) → C from Lemma 4.1 given by r(a i j ⊗ a k l ) = R ik jl , r(a i j ⊗ b k l ) = R ki lj , r(b i j ⊗ a k l ) = R ik jl , r(b i j ⊗ b k l ) = (R -1 ) ki lj and inverse r : A * (R) ⊗ A * (R) → C r(a i j ⊗ a k l ) = (R -1 ) ik jl , r(a i j ⊗ b k l ) = R ki lj , r(b i j ⊗ a k l ) = R ik jl , r(b i j ⊗ b k l ) = R ki lj .
The map γ :

V(R) → V(R) ⊗ A * (R) with γ(1) = 1 ⊗ 1, γ(x i ) = j x j ⊗ a j i , γ(v i ) = j v j ⊗ b i j defines a right coaction on V(R). Thus α = (id ⊗ r) • (γ ⊗ id) gives us a right action α : V(R) ⊗ A * (R) → V(R)
as an analog of Lemma 2.3 which is given on the generators by

α(x i ⊗ a k j ) = R lk ij x l , α(x i ⊗ b k j ) = R kl ji x l , α(v i ⊗ a k j ) = R ik lj v l , α(v i ⊗ b k j ) = (R -1 ) ki jl v l .
Furthermore the braiding is defined by

Ψ(x i ⊗ x j ) = l R lk ij x k ⊗ x l , Ψ(x i ⊗ v j ) = l R jl ki v k ⊗ x l , Ψ(v i ⊗ x j ) = l R ik lj x k ⊗ v l , Ψ(v i ⊗ v j ) = k (R -1 ) ji kl v k ⊗ v l .
Therefore we can define a braided category ((C A * (R) ) * , Ψ) whose objects are right A * (R)-comodules in the same way as we constructed ( A C * , Ψ) in Subsection 2.2, and V(R) is a braided * -bialgebra in the category ((C A * (R) ) * , Ψ). Using a version of Corollary 3.2 for this category, we get the right symmetrization

H R = A * (R) ⊗ V(R). It is the * -bialgebra generated by the elements 1 A * (R) ⊗ x i , 1 A * (R) ⊗ v i , a i j ⊗ 1 V(R) , b i j ⊗ 1 V(R) with the relations p,q R ik pq a p j a q l = p,q a k q a i p R pq jl , k,j R ij kl b k q a p j = k,j a j l b i k R kp qj , x i a j k = p,l R lp ik a j p x l , x i b j k = p,l R jl pi x l b j p , v i a j k = p,l R ip lk v l a p k , v i b j k = p,l (R -1 ) ip lk v l b j p , ∆(a i j ) = k a i k ⊗ a k j , ∆(b i j ) = k b k j ⊗ b i k , ∆(x i ) = j x j ⊗ a i j + 1 ⊗ x i , ∆(v i ) = j v j ⊗ b j i + 1 ⊗ v j , ε(a i j ) = ε(b i j ) = δ i j , ε(x i ) = ε(v i ) = 0.

Realization of Quantum Lévy processes on braided * -bialgebras

In this section we will show that there always exists a Lévy process on the braided * -spaces constructed in the previous section that can be considered as a standard Brownian motion on these spaces. For the rest of this section R will denote a fixed bi-invertible R-matrix of real type I, and V(R) the associated free braided * -space from the previous section. For explicit calculations we will use the basis B consisting of the words in the generators x 1 , . . . , x n , v 1 , . . . , v n . Let L : V(R) → C be the functional defined by L(x i v j ) = δ j i on basis elements of the form x i v j , and zero on all other basis elements.

Proposition 5.2. The functional L is quadratic, Ψ-invariant, hermitian, and conditionally positive (i.e. positive on ker ε B ).

Proof: For the Ψ-invariance we have to show that (id ⊗ L)

• Ψ = Ψ • (L ⊗ id). It is (id ⊗ L) • Ψ(x i v j ⊗ x k ) = r,s,p,q R jq pk R rs iq x s ⊗ L(x r v p ) = s δ j i δ s k x s = Ψ • (L ⊗ id)(x i v j ⊗ x k ).
Similary we get

(id ⊗ L) • Ψ(x i v j ⊗ v k ) = r,s,p,q (R -1 ) kj pq R ps ri v r ⊗ L(x s v q ) = r δ k r δ j i v r = Ψ • (L ⊗ id)(x i v j ⊗ v k ).
A generator L is quadratic, if and only if the associated representation is of the form ρ = ε(•)id. This is the case, since we have ρ(x i ) = ρ(v i ) = 0 (see below). That L is hermitian and conditionally positive are straightforward calculations. □

We can now carry out the construction described in [Sch93, Chapter 2] to obtain quantum stochastic differential equations for the symmetrization of the process (j st ) 0≤s≤t associated to the generator L. Let

N := { b ∈ V(R) | L (u -ε(u)1) * (u -ε(u)1) = 0 }
and P := V(R)/N . P is a Hilbert space with the inner product induced by ⟨u, v⟩ = L u -ε(u)1) * (v -ε(v)1) . Furthermore (η(v i )) i∈{1,...,n} with η : V(R) → P b → b + N forms an orthonormal basis in P and we have

η(y) = v i + N , if y = v i 0, else.
Therefore it is

P ∼ = C n . Since η is a ρ-cocycle it is ρ(x i )η(v j ) = η(x i v j ) -η(x i )ε(v j ) = 0 ρ(v i )η(v j ) = η(v i v j ) -η(v i )ε(v j ) = 0 and thus ρ(x i ) = ρ(v i ) = 0. One verifies that L is α-invariant for the right action α = (id ⊗ r) • (γ ⊗ id) from Lemma 2.3, i.e. L(α(u ⊗ a)) = ε A (a)L(u)
for all a ∈ A(R), u ∈ V(R). Therefore we can use Theorem 3.7 to get the triple (ρ H R , η H R , L H R ) on the right symmetrization H R . The pre-Hilbert space is given by P

H R = (A(R) ⊗ V(R))/N H R with N H R := {a ⊗ b | L H R a ⊗ b -δ H R (a ⊗ b)1 H R * (a ⊗ b -δ H R (a ⊗ b)1 H R ) = 0}. The ρ-cocycle η H R : H R → P H R is given by η H R (1 ⊗ y) = η(1 ⊗ y) = v i + N , for y = v i 0, else. ( 20
)
and it follows for the * -representation ρ H R that we have

ρ H R (1 A ⊗ b)η H R (1 A ⊗ v i ) = v i + N , for b = 1 B (the empty word) 0, else (21)
and

ρ H R (a ⊗ 1 B )η H R (1 A ⊗ v i ) = R ki pj η H R (1 A ⊗ v p ), for a = a i j (R -1 ) ik jp η H R (1 A ⊗ v p ), for a = b i j . (22)
The generator L is given by

L H R (a ⊗ b) = δ ij δ kl , for a = a i j or b i j and b = x k v l 0, else . (23)
Using the theory of Schürmann [Sch93, Chapter 2] we get the following theorem.

Theorem 5.3. Let j H st be the Lévy process in H R with triple (η H R , ρ H R , L H R ) given in equations (20) -(23). Then a realisation of the right symmetrization j H st on the Fock space Γ(L 2 (R + , C n )) is given by the unique solution of the quantum V(q) is conditionally positive if and only if the matrix

L := L(xv) L(xx) L(vv) L(vx)
is positive semi-definite. It is also hermitian if we have L(v) = L(x). Thus we get the following classification for the quadratic generators on V(q). Theorem 6.1. A quadratic functional L : V(q) → C is a generator of a Lévy process on V(q) if and only if

1. for q = 1: L is positive semi-definite and L(v) = L(x), 2. for q = -1: L is positive semi-definite and L(x) = L(v) = 0, 3. for q 2 ̸ = 1: L(xv), L(vx) ≥ 0 and L vanishes on all other basis elements.

The symmetrization of this braided * -space gives for A(q) the free commutative algebra with group-like generator a and its adjoint b. H is generated by a,

ρ H R (m) = q -1 , for m ∈ {a, b} 0, for m ∈ {x, v}.
Thus we get the stochastic differential equations dX = X dΛ(q -1 -1) + dA(1)

dV = V dΛ(q -1 -1) + dA * (1) dA = A dΛ(q -1 -1) dB = B dΛ(q -1 -1)
for the processes X(t) = j t (x), V (t) = j t (v), A(t) = j t (a), B(t) = j t (b). The solution of this system of quantum stochastic differential equations is the quantum Azéma martingale, see [START_REF] Parthasarathy | Azéma martingales and quantum stochastic calculus In: Probability, Statistics and Design of Experinwnts[END_REF][START_REF] Schürmann | The Azéma martingales as components of quantum independent increment processes[END_REF].

Proof: Let L be an arbitrary quadratic functional. L ist Ψ-invariant if and only if the following equations are satisifed

A ij δ l k = n1,n2,n3 t 2 A n3n1 R n3 l i n2 R n1 n2 j k , A ij δ k l = n1,n2,n3 t 2 A n3n1 R n2 n3 l i R k n1 n2 j , B j i δ l k = n1,n2,n3 B n1 n3 R n3 l i n2 R j n2 n1 k , B j i δ k l = n1,n2,n3 B n1 n3 R n2 n3 l i (R -1 ) k j n2 n1 , C i j δ l k = n1,n2,n3 C n3 n1 R i l n3 n2 R n1 n2 j k , C i j δ k l = n1,n2,n3 C n3 n1 (R -1 ) n2 i l n3 R k n1 n2 j , D ij δ l k = n1,n2,n3 t -2 D n3n1 R i l n3 n2 R j n2 n1 k , D ij δ k l = n1,n2,n3 t -2 D n3n1 (R -1 ) n2 i l n3 (R -1 ) k j n2 n1 , a i δ k j = n1 tR n1 k i j a n1 , a i δ j k = n1 tR j n1 k i a n1 , b i δ k j = n1 t -1 R i k n1 j b n1 b i δ j k = n1 t -1 (R -1 ) j i k n1 b n1
for all i, j, k, l = 1, . . . , n. These equations follow directly from the invariance condition. For the first equation, e.g., we apply Ψ • (L ⊗ id) = (id ⊗ L) • Ψ to x i x j ⊗ x k . Solving this system of linear equations (using e.g. a computer program for symbolic computation like Maple) one arrives at the results listed in the proposition. □

The functional L is conditionally positive semi-definite, if and only if the matrix

L := L x i v j x i x j v i v j v i x j = B A D C
is positive semi-definite. For L to be hermitian, we need to impose furthermore a i = L(x i ) = L(v i ) = b i , for i = 1, . . . , n. This leads to the following classification.

Theorem 6.3. Suppose q 2 ̸ = 1. a) If t 2 q 3 = 1, then all Ψ-invariant generators on V(tR 2 ) are of the form

L =     b 0 0 -qa 0 b a 0 0 ā q 2 c 0 -qā 0 0 c     ,
and L(x i ) = L(v i ) = 0 for i = 1, . . . , n, where b, c ≥ 0, bc ≥ q 2 |a| 2 , and bc ≥ q -2 |a| 2 . b) If t 2 q 3 ̸ = 1, then all Ψ-invariant generators on V(tR 2 ) are of the form

L =     b 0 0 0 0 b 0 0 0 0 q 2 c 0 0 0 0 c     ,
and L(x i ) = L(v i ) = 0 for i = 1, . . . , n, where b, c ≥ 0.

Proof:

a) Proposition 6.2 a) implies that a Ψ-invariant functional

L : V(tR 2 ) → C is of the form L =     b 0 0 -qa 0 b a 0 0 d q 2 c 0 -qd 0 0 c     , and L(x i ) = L(v i ) = 0, where a = L(x 2 x 1 ), b = L(x 1 v 1 ), c = L(v 2 x 2 )
and d = L(v 1 x 2 ). Such an invariant functional L is a generator, if and only if this matrix is positive semi-definite. This is the case if and only if the matrices b -qa -qd c and b a d q 2 c are positive semi-definite, which leads immediatly to to the conditions given in the theorem. b) Proposition 6.2 b) shows us that if t 2 q 3 ̸ = 1, then L is Ψ-invariant if and only if we also have a = d = 0. □ 6.3. The sl 3 -R-matrix. Let now

R 3 =               q 2 0 0 0 0 0 0 0 0 0 q 0 q 2 -1 0 0 0 0 0 0 0 q 0 0 0 q 2 -1 0 0 0 0 0 q 0 0 0 0 0 0 0 0 0 q 2 0 0 0 0 0 0 0 0 0 q 0 q 2 -1 0 0 0 0 0 0 0 q 0 0 0 0 0 0 0 0 0 q 0 0 0 0 0 0 0 0 0 q 2              
be the sl 3 -R-matrix. We get a similar classification for the generators on V(tR 3 ) as in the previous subsection, but there are no additional generators for the special case t 2 q 3 = 1 (as in Proposition 6.2 c)).

Theorem 6.4. Suppose q 2 ̸ = 1. Then all Ψ-invariant generators on V(tR 3 ) are of the form

L =         b 0 0 0 0 0 0 b 0 0 0 0 0 0 b 0 0 0 0 0 0 q 4 c 0 0 0 0 0 0 q 2 c 0 0 0 0 0 0 c         and L(x i ) = L(v i ) = 0 for i = 1, . . . , n
, where b, c ≥ 0.

Proof: We first determine all invariant quadratic functionals and then we check positivity, as in Proposition 6.2 and Theorem 6.3. □ 6.4. The braided quantum SU (2) groups SU q (2), q ∈ C\{0}. Finally, let us treat the braided Hopf- * -algebra underlying the braided compact quantum groups introduced and studied by Kasprzak, Meyer, Roy, and Woronowicz in [START_REF] Kasprzak | Braided quantum SU(2) groups[END_REF]. The braiding is not actually defined in [START_REF] Kasprzak | Braided quantum SU(2) groups[END_REF], instead the authors define a monoidal category of C * -algebras. Here we show how this example fits into the framework of braided *-bialgebras that we used in this paper.

Let CZ ∼ = Pol(T) be the group algebra of Z. We will write [z] for the canonical basis element of CZ associated to an integer z ∈ Z. Recall that CZ is a Hopf- *algebra with the following operations: for z, z 1 , z 2 ∈ Z, c ∈ C, we set

m CZ ([z 1 ] ⊗ [z 2 ]) = [z 1 + z 2 ] 1(c) = c[0] ∆([z]) = [z] ⊗ [z] ε([z]) = 1 S([z]) = [-z] = [z] * ,
and extend m CZ , 1, ∆, δ as algebra homomorphisms, S as an algebra anti-homomorphism, and * as a anti-linear algebra anti-homomorphism.

Let B be a CZ-graded * -bialgebra. In particular, we have deg(vw) = deg(v) + deg(w) and deg(v * ) = -deg(v). Let ζ = q q for 0 < |q| < 1.

We can use the grading to define an action of CZ. Proof: For the left hand side we get

(m CZ ⊗ θ) • (id ⊗ τ ⊗ id) • ( ∆ ⊗ ρ)([z] ⊗ v) = (m CZ ⊗ θ) • (id ⊗ τ ⊗ id)([z] ⊗ [z] ⊗ deg(v) ⊗ v) = (m CZ ⊗ θ)([z] ⊗ [deg(v)] ⊗ [z] ⊗ v) = ζ -z•deg(v) ([z + deg(v)]) ⊗ v,
and the right hand side can be reduced to the same expression, Proof:

(m CZ ⊗ id) • (id ⊗ τ ) • (ρ ⊗ id) • ( θ ⊗ id) • (id ⊗ τ ) • ( ∆ ⊗ id)([z] ⊗ v) = (m CZ ⊗ id) • (id ⊗ τ ) • (ρ ⊗ id) • ( θ ⊗ id) • (id ⊗ τ )([z] ⊗ [z] ⊗ v) = (m CZ ⊗ id) • (id ⊗ τ ) • (ρ ⊗ id) • ( θ ⊗ id)([z] ⊗ v ⊗ [z]) = (m CZ ⊗ id) • (id ⊗ τ ) • (ρ ⊗ id)(ζ -z•deg v • v ⊗ [z]) = (m CZ ⊗ id) • (id ⊗ τ )(ζ -z•deg v [deg(v)] ⊗ v ⊗ [z]) = (m CZ ⊗ id)(ζ -z•deg v • [deg(v)] ⊗ [z] ⊗ v) = ζ -
Equation 1: * • θ([z] ⊗ v) = (ζ -z deg(v) • v) * = ζ -z deg(v) • v = ζ -z deg(v) • v * = ζ z•deg(v) • v * = ζ -z•deg(v * ) • v * = θ([z] ⊗ v * ) = θ • ( * ⊗ * )([-z] ⊗ v) = θ • ( * ⊗ * )(S ⊗ id)([z] ⊗ v).
= α ⊗ α ⊗ α -qα ⊗ γ * ⊗ γ -qγ * ⊗ γ ⊗ α + qγ * ⊗ α * ⊗ γ = α ⊗ α ⊗ α -qγ * ⊗ γ ⊗ α -q(γ * ⊗ α * + α ⊗ γ * ) ⊗ γ = (α ⊗ α -qγ * ⊗ γ) ⊗ α -q(γ ⊗ α + α * ⊗ γ) * ⊗ γ = ∆(α) ⊗ α -q ∆(γ) * ⊗ γ = ( ∆ ⊗ id)(α ⊗ α -qγ * ⊗ γ) = ( ∆ ⊗ id) • ∆(α)
Similarly, we can check that (id ⊗ ∆) • ∆(γ) = ( ∆ ⊗ id) • ∆(γ). The counit property for α:

(id ⊗ ε) • ∆(α) = (id ⊗ ε)(α ⊗ α -qγ * ⊗ γ) = α ⊗ δ(α) -qγ * ⊗ ε(γ) = α = ε(α) ⊗ α -q ε(γ * ) ⊗ γ = (ε ⊗ id)(α ⊗ α -qγ * ⊗ γ) = (ε ⊗ id) • ∆(α)
The counit property for γ can be verified in the same way. □

It is also straightforward to check that ∆ preserves the grading.

Let I be the two-sided *-ideal in à generated by the relations Since I is a coideal in à and ∆ is coassociative, there exists an unique bialgebrastructure ∆, δ on quotient algebra A = Ã/ Ĩ. We have deg I = 0, so A inherits a grading, and we can lift the action and the coaction on A to an action ϑ : CZ⊗A → A and a coaction ρ : A → CZ ⊗ A on A. It follows that (A, ϑ, ρ) is again an object in CZ CZ YD * . With these structures, A = Pol(SU q (2)) is the braided Hopf-*-algebra of the braided compact quantum groups SU q (2), 0 < |q| < 1, defined and studied in [START_REF] Kasprzak | Braided quantum SU(2) groups[END_REF].

Schürmann and Skeide described all generators on Pol(SU q (2)) for q ∈ R\{0}, i.e., in the unbraided case, see [START_REF] Schürmann | Infinitesimal generators on the quantum group SUq(2) Infin[END_REF]. We extend the classification of the quadratic generators in [SS98, Corollary 3.3] to the braided SU (2) quantum groups. Proposition 6.11. Let H be a Hilbert space. For any vector v ∈ H and real number λ there exists a unique triple (ε, η, L) such that η(γ) = η(γ * ) = 0,

η(α) = -η(α * ) = v, L(γ) = L(γ * ) = 0, L(α) = iλ - ∥v∥ 2 2 , L(α * ) = -iλ - ∥v∥ 2 2 .
Two such triples determined by pairs (v, λ) and (v ′ , λ ′ ) have the same generator L if and only if ∥v∥ = ∥v ′ ∥ and λ = λ ′ . Furthermore, all quadratic generators on Pol(SU q (2)) arise in this way.

Proof: Similar to [START_REF] Schürmann | Infinitesimal generators on the quantum group SUq(2) Infin[END_REF]. □

It turns out that all quadratic generators on Pol(SU q (2)) are invariant and therefore define Brownian motions. In fact, like in [START_REF] Schürmann | Infinitesimal generators on the quantum group SUq(2) Infin[END_REF], ε, η, and L vanish on γ and γ * , which implies that the triple factorizes through the quotient map A → A/J γ ∼ = CZ ∼ = Pol(T), where J γ denotes the *-ideal generated by γ. This means that the Brownian motions on SU q (2) are induced from Brownian motions of the undeformed subgroup T ⊆ SU q (2).

  ) (Stationarity of increments) The distribution φ st = Φ • j st of j st depends only on the difference t -s, (4) (Weak continuity) j st converges to j ss ( = 1 • ε) in distribution for t ↘ s, i.e., we have lim t↘s Φ j st (b) = ε(b) for all b ∈ B.

Remark 3. 6 .

 6 This theorem generalizes [Sch93, Theorem 3.3.1]. Proof: We will use Sweedler's notation ∆ B (b) = b (1) ⊗ b (2) ∈ B ⊗ B and γ(b) = b (1) ⊗ b (2) ∈ B ⊗ A for the coproduct and coaction on an element b ∈ B. We have ȷst

  η(ab) = ρ(a)η(b) -η(a)ε(b) for a, b ∈ A, and L : A → C is a hermitian linear functional s.t. ⟨η(a * ), η(b)⟩ = ε(a)L(b) + L(ab) -L(a)ε(b) for a, b, ∈ A. Such triples are called Schürmann triples in [Mey93, Chapter VII] and [FS16]. Let us recall the GNS-type construction of the triple (ρ, η, L) from the functional L, see [Sch93, Section 2.3]. Let B be a * -algebra with a unital, hermitian character ε : B → C (i.e. ε(1) = 1, ε(b * ) = ε(b), and ε(ab) = ε(a)ε(b) for all a, b in B) and let L : B → C be a generator. We define an inner product ⟨•, •⟩ B0 : B 0 × B 0 → C on B 0 := ker ε by (a, b) → L(a * b). This inner product is positive semi-definite, since L is conditionally positive. We define the null space by N 0 := { b ∈ B 0 | ⟨b, b⟩ = 0 }. The quotient space P := B 0 /N 0 with inner product ⟨a + N 0 , b + N 0 ⟩ P := ⟨a, b⟩ B0 becomes a pre-Hilbert space. The (left) action α : B × B 0 → B 0 with (a, b) → a • b induces an action α on P , since α(N 0 ) ⊆ N 0 . Now we define ρ(a) ∈ L(P, P ) by ρ(a)(b + N 0 ) := α(a, b + N 0 ) for a ∈ B and b + N 0 ∈ P as well as η : B → P b → (b -ε(b) • 1 B ) + N 0 , where η(b) = b + N 0 , for all b ∈ B 0 0 + N 0 , for all b ∈ ⟨1 B ⟩. The equations η(a • b) = ρ(a)η(b) + η(a)ε(b) and ⟨η(a), η(b)⟩ = L(a * b) -ε(a * )L(b) -L(a * )ε(b) hold. Thus (ρ, η, L) is a surjective triple, i.e., a triple whose cocycle η is surjective. If we know the triple for a generator L on a braided * -bialgebra B, then the following proposition tells us how to extend it to a triple for L H = F (L).

  Definition 5.1. [Sch93, Section 5.1] Let B be a braided bialgebra. A linear functional ϕ : B → C is called quadratic (or Gaussian), if it satisfies ϕ(abc) = 0 for all a, b, c ∈ ker ε B . A Lévy process whose generator is quadratic is called Brownian motion.

  x and their adjoints b = a * and v = x * . The algebraic relations are ab = ba, xa = qax, bx = qxb and the coalgebraic relations are ∆(a) = a ⊗ a, ∆(x) = x ⊗ a + 1 ⊗ x. Let L now be the generator with L(xv) = 1 and L(u) = 0 on all other basis elements. The construction of the triple gives the pre-Hilbert space D H R = C, the ρ-cocycle η H R (m) = 0, for m ∈ {x, a, b} 1, for m = v and the representation

  CZ ⊗ B → B [z] ⊗ v → ζ -z deg(v) • v defines a left action and ρ : B → CZ ⊗ B v → [deg(v)] ⊗ v defines a left coaction.Lemma 6.6. θ and ρ satisfy the Yetter-Drinfeld-condition, i.e.(m CZ ⊗ θ) • (id ⊗ τ ⊗ id) • ( ∆ ⊗ ρ) = (m CZ ⊗ id) • (id ⊗ τ ) • (ρ ⊗ id) • ( θ ⊗ id) • (id ⊗ τ ) • ( ∆ ⊗ id)

  z•deg v • (deg(v) + z) ⊗ v □ Lemma 6.7. The equations * • θ = θ • ( * ⊗ * ) • (S ⊗ id) ρ • * = ( * ⊗ * ) • ρ hold.

  (v) = ρ(v * ) = [deg(v * )] ⊗ v * = [-deg(v)] ⊗ v * = [deg(v)] * ⊗ v * = ( * ⊗ * )([deg(v)] ⊗ v) = ( * ⊗ * ) • ρ(v).□Corollary 6.8. (B, θ, ρ) is an involutive Yetter-Drinfeld-module and thus an object in the category CZ CZ YD * .The category CZ CZ YD * is braided with the braiding Ψ given byΨ(v ⊗ w) = ( θ ⊗ id) • (id ⊗ τ ) • (ρ ⊗ id)(v ⊗ w) = ζ -deg(v) deg(w) • w ⊗ vThus the multiplication on the tensor product is given bym B⊗B (v 1 ⊗ v 2 ⊗ w 1 ⊗ w 2 ) = (m ⊗ m) • (id ⊗ Ψ ⊗ id)(v 1 ⊗ v 2 ⊗ w 1 ⊗ w 2 ) = ζ -deg(v2) deg(w1) v 1 w 1 ⊗ v 2 w 2and the involution is given by* B⊗B (v ⊗ w) = Ψ • ( * ⊗ * ) • τ (v ⊗ w) = ζ -deg(w * ) deg(v * ) v * ⊗ w * = ζ -deg(w) deg(v) v * ⊗ w *Let à := C⟨α, α * , γ, γ * ⟩ be equipped with a CZ-graduation determined bydeg(α) = deg(α * ) = 0, deg(γ) = 1, deg(γ * ) = -1. Lemma 6.9. ∆ : à → à ⊗ à given by ∆(α) = α ⊗ α -qγ * ⊗ γ ∆(γ) = γ ⊗ α + α * ⊗ γis coassociative and ε : à → C with ε(α) = 1 and ε(γ) = 0 satisfies the counit axiom.Proof: First we show (id ⊗ ∆) • ∆(α) = ( ∆ ⊗ id) • ∆(α). We have(id ⊗ ∆) • ∆(α) = (id ⊗ ∆)(α ⊗ α -qγ * ⊗ γ) = α ⊗ ∆(α) -qγ * ⊗ ∆(γ) = α ⊗ (α ⊗ α -qγ * ⊗ γ) -qγ * ⊗ (γ ⊗ α + α * ⊗ γ)

I 1 :

 1 = α * α + γ * γ -1 I 2 := αα * + |q| 2 γ * γ -1 I 3 := γγ * -γ * γ I 4 := αγ -qγαI 5 := αγ * -qγ * α and A := Ã/I the quotient algebra. Lemma 6.10. I is a coideal in Ã. Proof: One checks that ∆(I) ⊆ Ã ⊗ I + I ⊗ Ã ε(I) = {0} □

Acknowledgements

We thank René Schott for his suggestions and his participation in an earlier stage of this paper.

This work was supported by the DFG [project no. 397960675] and the DAAD, MESRI, and MEAE through the PROCOPE programme. U.F. was supported by the ANR Project No. ANR-19-CE40-0002. 1 * B⊗B := Ψ • ( * ⊗ * ) • τ is a self-inverse map turning the coproduct ∆ into a *algebra-homomorphism for * B⊗B .

Because of that we get

Since ρ H is an algebra homomorphism we have

and

This completes the proof of the theorem for triples constructed in the way described above. But since the pre-Hilbert spaces belonging to two surjective triples which come from the same generator L are isometrically isomorphic, this gives rise to an isometric isomorphism Ť :

, such that the assertions hold for arbitrary triples belonging to the same generator. □

A construction of braided * -spaces

In this section we will construct a large class of braided * -spaces and their symmetrizations.

Let R ∈ C n×n ⊗ C n×n be a universal R-matrix and thus R satisfies the quantum Yang-Baxter equation (2). We suppose that R is of real type I, i.e.

and that R is bi-invertible, i.e. that there exist matrices

Note that if R is of real type I and bi-invertible, than R -1 and R are also of real type I.

4.1.

The FRT- * -bialgebra. We denote by A * (R) the * -bialgebra generated by the elements {a i j } i,j=1,2,...,n and their adjoints {b k l } k,l=1,2,...,n , where b k l := (a l k ) * , stochastic differential equations

where

6. Examples 6.1. The one-dimensional R-matrix R = (q). Let us first consider the onedimensional R-matrix R 1 = (q). For q ∈ R, q ̸ = 0, this is a bi-invertible R-matrix of real-type I and defines therefore a braided * -space V(q). As an algebra V(q) is the free algebra generated by x and x * = v. We will use the words in x and v as a basis for V(q). The braiding is given by

To be a generator of a Brownian Motion a linear functional L : V(q) → C has to be Ψ-invariant, i.e. it has to satisfy

and a similar set of equations for v. Thus for q 2 ̸ = 1 a Ψ-invariant quadratic functional can have non-zero values only on xv and vx. A quadratic functional on 6.2. The sl 2 -R-matrix. Let R 2 be the R-Matrix of the standard two-dimensional quantum plane, i.e.

Then tR 2 with q, t ∈ R, q, t ̸ = 0 is bi-invertible and of real type I and we can therefore define a braided * -space V(tR 2 ) for it. As an algebra this is the free algebra generated by x 1 , x 2 and their adjoints x * 1 = v 1 , x * 2 = v 2 . We will use the words in these four elements as a basis of V(tR 2 ). It turns out that the Ψ-invariance restricts very much the possible generators. Proposition 6.2. Let L be a quadratic functional on V(tR 2 ), q, t ̸ = 0. Then L is characterized by

The functional L is Ψ-invariant if and only if for q 2 ̸ = 1 and a) t 2 q 3 = 1: qA 12 + A 21 = 0, B 1 1 = B 2 2 , C 1 1 = q 2 C 2 2 , D 12 + qD 21 = 0 and all other coefficients vanish b) t 2 q 3 ̸ = 1: B 1 1 = B 2 2 , C 1 1 = q 2 C 2 2 and all other coefficients vanish


Hsin Lee

Nicolas Puillandre

Yasunori Kano

Wei-Jen Chen

Sarah Samadi

Biodiversity and phylogeny of Cocculinidae (Gastropoda: Cocculinida) in 2 the Indo-West Pacific 3 4

Keywords: classification < Phylogenetics, deep-sea < Geography, species delineation < Taxonomy, Gastropoda < Taxa classification -deep sea -organic fall -species delimitation -37 sunken

The family Cocculinidae (Gastropoda: Cocculinida) consists of small, usually colorless benthic limpets living primarily at depth below 100 m and on decaying plant or animal remains. These habitats are difficult to sample and the knowledge about their species diversity, biogeography, ecology, and evolution is poor. To explore the species diversity of the Cocculinidae we examined 499 specimens collected from 196 sites, mainly explored during expeditions of the Tropical Deep-Sea Benthos program in the Indo-West Pacific (IWP). To propose species-hypotheses, we used an integrated approach to taxonomy in which we combined DNA-based methods , with morphological, geographical, and ecological considerations. To classify the species-hypotheses into genera, we used a combination of one mitochondrial and two nuclear genes fragments to reconstruct a phylogenetic tree. We then used six morphological characters to diagnose the identified genera. Our results revealed an exceptionally high diversity of the IWP Cocculinidae, with 51 specieshypotheses that were mostly not assigned to available species names. We also discovered a previously unknown type of copulatory structure in the group. At higher taxonomic level, we identified ten main clades within the family. While six of them match to existing genera, four others should be regarded as new genera awaiting formal

363

Protoconch. A reticulate sculpture is dominant in the protoconchs of the Cocculinidae 364 (Fig. 5A). However, in some cocculinids the protoconch shows concentric lines in the first 365 half and smooth in the last half (Fig. 5B).

366

Radula. The rachidian tooth of the cocculinoid radula is characterized by overhanging 367 cusps, which may be present or obsolete [START_REF] Strong | Phylogeny of the Cocculinoidea 1006 (Mollusca, Gastropoda)[END_REF]. Then, we distinguished the 368 radula based on two features, the shape of the rachidian and the number of cusps on the 369 rachidian. We defined six distinct types of the cocculinid rachidian tooth: (a) "obsolete", no 370 rachidian observed (Fig. 5C); (b) "acuspate-flat", tip flat without a cusp (Fig. 5D); (c)

371 "unicuspidid narrow", tip narrow with a pointed cusp (Fig. 5E; (d) "unicuspid broad", tip 372 broad and pointed (Fig. 5F); (e) "multicuspidid narrow", tip narrow with more than one cusp 373 (Fig. 5G); and (f) "multicuspid broad", tip broad with more than one cusp (Fig. 5H). 382 letters, corresponding to eight morphogroups as defined with the four primary morphological 383 characters (shell shape, teleoconch sculpture, conditions of copulatory and epipodial tentacles;

384 Tables 1,S3), followed by a number, corresponding to PSHs (see below), namely MA1-385 MA12, MB1-MB10, MC1-MC19, MD1-MD8, ME1-ME3, MF1-MF9, MG1-MG18 and 386 MH1-MH2.

387

We aligned 415 bp of the sequences from the 147 samples successfully sequenced 388 (including one outgroup) in the 28S "master" dataset. The inferred ML tree allowed us to 389 reveal seven genetically distinct clades or groups (Fig. S2). 396 Finally, the inferred tree shows eight groups (Fig. S2). (Marshall, 642 [1985(Marshall, 642 [ ] 1986;;Haszprunar, 1987). All specimens attributed to MG, corresponding to nine 643 distinct species (Fig. 6A), displayed both conditions and were thus considered as the members 644 of Coccopigya (Tables 1,2).

645

Specimens of MG2-MG4 and MG15 were identified respectively as Coccopigya 646 punctoradiata (Kuroda & Habe, 1949) and Coccopigya okutanii Hasegawa, 1997 based on 647 their morphology and collection sites (Japan). The specimens of MG11-MG13, gathered into 648 a single SSH, were attributed to Coccopigya hispida based on a published cox1 sequence from 649 a paratype of the species (AY296823, voucher NMNZ M075188) (Fig. 6A).

651 Coccocrater

652

The use of the right cephalic tentacle as a copulatory organ is considered plesiomorphic 653 within the Cocculinoidea and shared by the species of Coccocrater, Coccopigya, Fedikovella, 654 Macleaniella and Teuthirostria (Table 2; [START_REF] Strong | Phylogeny of the Cocculinoidea 1006 (Mollusca, Gastropoda)[END_REF]. This condition was observed in 655 morphogroups MA, MD, MG and MH ( (Moskalev, 1976;[START_REF] Marshall | Recent and Tertiary cocculinidae and pseudococculinidae (Mollusca: 915 Gastropoda) from New Zealand and New South Wales[END_REF]Marshall, [ ] 1986;;McLean & Harasewych, 1995;Leal & Harasewych, 676 1999). The right cephalic tentacle of F. caymanensis is supposedly used as a copulatory organ 677 but not hypertrophied or modified [START_REF] Leal | Macleaniella moskalevi is the monotypic 730 species of the genus, known exclusively from the lower abyssal and hadal zones of the Puerto 731 Rico Trench (5179-8595 m). It is easily recognized by its large internal septum of the shell as 732 a unique character within Cocculinidae[END_REF]. The rachidian, first and 678 second lateral teeth are all multicuspidid [START_REF] Leal | Macleaniella moskalevi is the monotypic 730 species of the genus, known exclusively from the lower abyssal and hadal zones of the Puerto 731 Rico Trench (5179-8595 m). It is easily recognized by its large internal septum of the shell as 732 a unique character within Cocculinidae[END_REF]. These conditions were 679 found in our specimens of the clade MD (Tables 1,2; Figs 10A-D, 11M, N).

680

An additional MD specimen from the Lesser Antilles at 500-550 m deep (MNHN-IM-681 2013-60187; Table S1) was grouped with the IWP specimens in the 28S analysis (Fig. S2),

682 supporting the identity of this clade as Fedikovella from a geographic perspective. This clade 683 also included species from off Faro, Portugal (M-unknown1, AORI_YK1662) and from off 684 Costa Rica, east Pacific (M-unknown2, SMNH-108733) (Fig. 8). "Cocculina" pacifica 685 described from off Japan should likewise be attributed to Fedikovella based on its shell, 686 radula and copulatory organ (Kuroda & Habe, 1949: pl. 3, fig. 5;Hasegawa, 1997: fig. 5).

687 Fedikovella thus occupies a wide range, both geographically (worldwide) and bathymetrically 688 (bathyal to hadal zones). We should stress that [START_REF] Strong | Phylogeny of the Cocculinoidea 1006 (Mollusca, Gastropoda)[END_REF] [START_REF] Watson | Report on the Scaphopoda and Gastropoda collected by H.M.S. 1035 Challenger during the years 1873-1876[END_REF]Schepman, 1908;Thiele, 1925;[START_REF] Marshall | Recent and Tertiary cocculinidae and pseudococculinidae (Mollusca: 915 Gastropoda) from New Zealand and New South Wales[END_REF]Hasegawa, 1997;[START_REF] Lesicki | Checklist of gastropod species referred to the order Cocculiniformia 903 Haszprunar, 1987 (Gastropoda: Cocculinoidea et Lepetelloidea) with some remarks on 904 their food preferences[END_REF]Zhang & Zhang, 2018)

  ), the University of Tokyo.

379

  The final alignment of the cox1 dataset contained 657-bp long sequences from 498 sampled 380 cocculinids and 19 GenBank entries, plus two bathysciadiids for rooting. The inferred 381 phylogenetic tree is presented in Figure S1. The terminal taxon labels are composed of two

Figure 2 .

 2 Figure 1. Localities of cocculinid specimens collected during our expeditions (red) and those in literature records (yellow;[START_REF] Watson | Report on the Scaphopoda and Gastropoda collected by H.M.S. 1035 Challenger during the years 1873-1876[END_REF] Schepman, 1908; Thiele, 1925;[START_REF] Marshall | Recent and Tertiary cocculinidae and pseudococculinidae (Mollusca: 915 Gastropoda) from New Zealand and New South Wales[END_REF] Hasegawa, 1997;[START_REF] Lesicki | Checklist of gastropod species referred to the order Cocculiniformia 903 Haszprunar, 1987 (Gastropoda: Cocculinoidea et Lepetelloidea) with some remarks on 904 their food preferences[END_REF] Zhang & Zhang, 2018) in Indo-West Pacific. The square indicates the 11 regions identified. AFR: Mozambique Channel; MAD: Madagascar; TW: Taiwan plus South China Sea; ECS: East China Sea; PHI: seas surrounding the Philippines; JP: seas surrounding Japan; BIS: Bismarck Sea; SOL: Solomon Sea; COR: Coral Sea; Van: seas surrounding Vanuatu; NC: seas surrounding New Caledonia. 254x190mm (300 x 300 DPI)

Figure 3 .

 3 Figure 3. Character used in morphological examination. Teleoconch structure: (A) radially ribbed (MNHN-IM-2013-42733, MC10), (B) radially ribbed with pits and hairs (MNHN-IM-2013-42896, MG5), (C) plainly raised radially ribbed (MNHN-IM-2013-62346), (D) clathrate (MNHN-IM-2013-62341, MD5), (E) concentric (MNHN-IM-2009-11985, MH1), or (F) smooth (MNHN-IM-2013-42654, ME1).

Figure 4 .

 4 Figure 4. Character used in morphological examination. Copulatory organ: (A) associated with or branched from right cephalic tentacle, (B) originated from foot, (C) originated from oral lappet, or (D) originated from mantle margin. c.o.: copulatory organ, r.c.t.: right cephalic tentacle, o. l.: oral lappet.

Figure 5 .

 5 Figure 5. Characters used in morphological examination. Protoconch: (A) reticulate (MNHN-IM-2013-42752, MC2) or (B) concentric (MNHN-IM-2013-62343, MD8). Radula: (C) "obsolete," (C. japonica) (D) "acuspateflat," (MNHN-IM-2013-42608, MC9) (E) "unicuspid narrow," (MNHN-IM-2013-42715, MB1) (F) "unicuspid broad," (MNHN-IM-2013-42803, MC14) (G) "multicuspid narrow," (MNHN-IM-2013-42570, MD3) or (H) "multicuspid broad" (MNHN-IM-2013-62344, MA5).

Figure 6 .Figure 6 .Figure 6 .

 666 Figure 6. Maximum-likelihood tree based on cox1 dataset of cocculinids with results of species delimitation analyses. Numbers at tree nodes represent bootstrap values in percentage (values < 70% not shown). Names of primary species hypotheses (PSHs) are followed by number of sequences determined (cox1 and 28S). Boxes for PSHs are highlighted in gray when differently delimited in ABGD and PTP analyses (numerals in boxes represent numbers of cox1 sequences; asterisk and plus for MA9-MA12 denote same PSH detected by ABGD and PTP, respectively). 28S sequence monophyly and identity are also shown as empty boxes (NA: no data); cross marks (X) indicate non-monophyly or genotype shared with other PSHs. The "shell morph." denotes distinguishable forms of the shell with different numerals in each morphogroup (MA-MH); shell morph. C1 for MC1-MC3, for example, means that the three PSHs are identical in conchological characteristics. Known bathymetric and geographic distributions and habitat are also provided for each PSH (bathymetric distribution -sampled depth range shown in lines, dotted lines represent the estimated distributed range; geographic distribution -AFR: Mozambique channel, MAD: north Madagascar, TW: Taiwan and South China Sea, ECS: East China Sea, PHI: Philippines, JP: Japan, BIS: Bismarck Sea, SOL: Solomon Sea, COR: Coral Sea, VAN: Vanuatu, NC: New Caledonia, OTH: others, ME: Mediterranean, NZ: New Zealand, RS: Red Sea, AT: Atlantic, EP: East Pacific, WS: Weddell Sea; habitat -w: wood, l: leaf, b: bone, c: coral, e: egg case, vent*: hydrothermal vent area). Final, secondary species hypotheses (SSHs) are shown with generic and species names if available; those found from more than one area (northwestern Pacific, southwestern Pacific, and Indian Ocean) are highlighted in gray. 285x199mm (300 x 300 DPI)

Figure 7 .Figure 8 .Figure 9 .Figure 10 .

 78910 Figure 7. Nonmetric multidimensional scaling (NMDS) plot based on Jaccard index for cocculinid species composition among 157 stations in Indo-West Pacific. 254x190mm (300 x 300 DPI)

  Phylogenetic trees inferred from the 397 28S subdatasets are shown in Figure S3A-H. The ABGD and bPTP analyses delimited 89 and 92 PSHs, respectively, 78 of which were 402 recovered in both analyses. Figure 6 summarizes results from ABGD, bPTP and other criteria 403 used for species delimitation. Also shown in this figure are the numbers of shell morphotypes, 404 which were classified on the basis of global shell morphology, within each morphogroup Five more species have been described from the IWP under the genus Cocculina: C.628 pacificaKuroda & Habe, 1949, C. tosaensis Kuroda & Habe, 1949, C. surugaensis 629 Hasegawa, 1997 and C. tenuitesta Hasegawa, 1997, all on sunken wood off Japanese coasts, 630 and C. delphinicula Zhang & Zhang, 2018 from a dolphin skull at a depth of 300-400 m in 631 the East China Sea(Zhang & Zhang, 2018). Although none of them were found in our 632 material, we believe that the first species should be moved from the genus to either

	494 study are new to science as discussed below. 625 might have resulted from worn teeth or different angles in SEM shots and examination of
	626 more radulae is needed to confirm our species identification.
	627
	398
	399	SPECIES DELIMITATION ANALYSES
	400
	F o r R F o r R F o r R F o r R F o r R F o r R F o r R 401 410 geographic populations of single species (i.e. MG2-MG4; MG6-MG10; MD4 and MD5; 411 MB9 and MB10; Fig. 6A, B). Figure 6 further illustrates how we integrated the different lines 412 of evidence to make our decisions. Note that in all cases, our final decision was conservative, 413 i.e. if there was no clear evidence to choose between one single species or two different e v i e w e v i e w e v i e w e v i e w e v i e w e v i e w e v i e w
	414 species, we decided to choose the single species option. Examples include MC13, MC14 and O O O O O O O 415 MC15 that lacked 28S data (Fig. 6C). By taking all lines of evidence into account, we inferred 416 60 SSHs (including one without cox1 data but with 28S, morphological and geographic data). n l n l n l n l n l n l n l
	417 Of these, we recorded 51 from the 187 stations sampled in the IWP. Morphological characters y y y y y y y
	418 of each SSH are shown in Table S2.
	419
	420	REGIONAL SPECIES DIVERSITY
	421
	422 We included the 187 IWP stations in our NMDS analysis. The stress value reached close to 0
	423 (around 0.0007) after 200 NMDS runs, suggesting the presence of outlier stations. This result
	424 was not surprising since some stations contained a single, unique species. For a better
	425 recognition of underlying patterns, we removed 25 dissimilar stations that showed
	426 NMDS1/NMDS2 over 20 or below -20. The re-analysis with 162 stations resulted in the final
	427 stress value of 0.022. NMDS plots showed two main groups. One contained most stations in

405 (MA-MH). We considered PSHs as separate species when they were reciprocally 406 monophyletic in a 28S subtree. Some pairs or triplets of

 MD2 and 407 MD3; MA6 and MA7; MA11 and MA12; MC16 and MC17; MF3 and MF4; Fig. 6)

shared 408 the same 28S sequence, and we conservatively considered such PSHs as a single species. 409 Similarly, if two allopatric PSHs were gathered in a same 28S clade we considered them as 633 Fedikovella ("C." pacifica; see below) or Lepetellidae ("C." tosaensis; see Hasegawa, 1997: 634 65). The sequences we gathered from GenBank similarly illustrated the difficulty of the 635 morphological identification at both genus and species levels. Examples include published 636 sequences attributed to Coccopigya punctoradiata (Table S1) that were clustered with an 637 entirely different species (MC5) in a potential new genus (nGen1). 638 639 Coccopigya 640 Coccopigya is the most easily recognized genus in Cocculinidae. It is distinguished from 641 other cocculinids by having periostracal spines and by lacking epipodial tentacles

Table 1 ;

 1 Macleaniella and Teuthirostria do not match any of our specimens (see section Macleaniella 659 and Teuthirostria). We thus consider morphogroup MA as Coccocrater.660The occurrence of Coccocrater in the IWP region, including the type species C. radiatus 661 (with a type locality off western Sumatra), suggests a generic identity for one or both of the 662 two MA lineages. As we cannot exclude that MA is monophyletic (see above) and the 663 original description of C. radiatus by[START_REF] Thiele | Die beschalten Gastropoden der deutschen Tiefsee-Expedition[END_REF] was too simple to be linked to any of our 664 SSHs, both lineages are assigned here to Coccocrater with a possibility of establishing a new 665 genus in future pending more data on their relationships and the type specimen of C. radiatus.

	658 666
	667 Fedikovella
	F
	o r
	R
	e v i e w
	O
	n l
	y

Fig. 6). Of these, MD and MG were 656 attributed to Fedikovella and Coccopigya, respectively, and MH to an undescribed genus with 657 ecological uniqueness (see below). The morphological and ecological features of 668 The genus Fedikovella contains two described species. The type species Fedikovella 669 caymanensis Moskalev, 1976 was described from a hadal depth (6800 m) in the Cayman 670 Trough. Fedikovella beanii (Dall, 1882) was described from the north-western part of the 671 Atlantic (lectotype USNM 333751 from 613 m deep, off Martha's Vineyard Island, 672 Massachusetts) and has been collected from the Lesser Antilles and Martinique at around 400 673 m to 1000 m deep (McLean & Harasewych, 1995). The concentric sculpture of the 674 protoconch and reticulate ribs of the teleoconch are diagnostic characters of the genus 675

, Copley JT, Dahlgren TG, Horton T, Kemp KM, Rogers AD, Glover AG.

  The data underlying this article are available in the article and its online supplementary 762 material, and in the GenBank Nucleotide Database at https://www.ncbi.nlm.nih.gov/genbank/ 763 and can be accessed with the GenBank accession numbers OL800712-OL801212 and

	F F F 761 764 OL956554-OL956700. 757 oceans and habitats. 758 759 760 765 766 F F F F F F F F F 793 Amon DJF 827 Coykendall DK, Johnson SB, Karl SA, Lutz RA, Vrijenhoek RC. 2011. Genetic diversity DATA AVAILABILITY 1027 Turner RD. 1977. Wood, mollusks, and deep-sea food chains. American Malacological 1085
	o r o r o r o r o r o r o r o r o r o r o r o r o r
	R R R R R R R R R R R R R
	e v i e w e v i e w e v i e w e v i e w e v i e w e v i e w e v i e w e v i e w e v i e w e v i e w e v i e w e v i e w e v i e w
	O O O O O O O O O O O O O
	n l n l n l n l n l n l n l n l n l n l n l n l n l
	y y y y y y y y y y y y y

found Fedikovella + 689 Teuthirostria as sister to the rest of the cocculinids, but this sister group relationship

Table 1 .

 1 Eight morphogroups of Cocculinidae and their habitats.

	1086

Table 2 .

 2 Diagnoses of cocculinid genera (with unique conditions shown in bold).

	Genus	Protoconch	Teleoconch	Copulatory organ	Epipodial	Shell shape	Habitat	References
		sculpture	sculpture		tentacles
	Cocculina	reticulate	radial, smooth	on oral lappet or foot	present	round	wood, dolphin skull	Haszprunar, 1987;
			or concentric					McLean & Harasewych, 1995;
								Zhang & Zhang, 2018
	Coccopigya	reticulate	spinose	modified right cephalic	absent	round	wood, leaf, bone	Marshall, [1985] 1986;
				tentacle				McLean & Harasewych, 1995;
								see main text for habitat
	Coccocrater	reticulate	radial or plainly	modified right cephalic	present	round	wood	Haszprunar, 1987;
			raised radial	tentacle				McLean & Harasewych, 1995
								Strong et al., 2003
	Paracocculina reticulate	radial	on foot or oral lappet	present	round	wood	Haszprunar, 1987;
								Strong et al., 2003
	Fedikovella	concentric	reticulate	right cephalic tentacle;	present	round	wood	Moskalev, 1976;
				unmodified				McLean & Harasewych, 1995;
								Leal & Harasewych, 1999
	Teuthirostria	smooth	reticulate	modified right cephalic	present	round	cephalopod beak	Moskalev, 1976;
				tentacle				Leal & Harasewych, 1999;
								Strong et al., 2003
	Macleaniella	smooth	radial (with an	modified right cephalic	present	round	wood	Leal & Harasewych, 1999
			inner septum)	tentacle
	nGen1	reticulate	radial, smooth	on oral lappet	present	round	wood	This study
			or concentric
	nGen2	reticulate	smooth	on mantle margin	present	round	wood	This study
	nGen3	reticulate	plainly raised	on oral lappet	present	spindle or	wood	This study
			radial			narrow round
	nGen4	reticulate	concentric or	modified right cephalic	present	round	chondrichthyan egg	This study
			smooth	tentacle			case, deep-sea coral

Figure 1. Localities of cocculinid specimens collected during our expeditions (red) and those in literature records (yellow;

geographic regions defined above, we used non-metric multidimensional scaling (NMDS)

Zoological Journal of the Linnean Society

ACKNOWLEDGEMENTS 767 768 We thank all PIs and participants of the "Tropical Deep-Sea Benthos" expeditions AURORA