Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 24, 2006, 13–42

\begin{matrix}c \\ ◯ \end{matrix} H. L. Huru

H. L. Huru

QUANTIZATIONS OF BRAIDED DERIVATIONS. 1. MONOIDAL CATEGORIES

(submitted by V. V. Lychagin)

Abstract. For monoidal categories we describe braidings and quantizations. We use them to find quantizations of braided symmetric algebras and modules, braided derivations, braided connections, curvatures and differential operators.

1 Introduction

We consider quantizations q, braidings σ and quantizations of braidings σ_{q} of monoidal categories. We mainly work with braidings that are symmetries.

We consider σ-symmetric algebras A, modules, co- and bialgebras and internal homomorphisms and find quantizations of these.

Internal homomorphisms of σ-symmetric modules has a braided Lie structure with respect to the braided commutator. Quantizations of the internal homomorphisms has the quantized braided Lie structure and can be realized within the original braided Lie structure by what we call dequantization.

We investigate braided derivations of σ-symmetric algebras and modules. The σ-bracket of two braided derivations is a braided derivation. We show that there is a braided Lie structure on the braided derivations.

A quantizations the braided derivations provides an isomorphism of the modules of braided derivations and quantized braided derivations. We also show that the quantizations of braided derivations has the braided Lie structure with respect to the quantizations of the braiding which can be realized within the original braided Lie structure by dequantization.

We define braided connections in modules and braided curvatures. We prove that the braided curvature is A-linear, skew σ-symmetric and is an A-module homomorphism.

We find quantizations of braided connections and braided curvatures. The quantization of the braided curvature is A-linear, skew σ_{q}-symmetric and an A-module homomorphism with respect to the quantized braiding.

Finally we consider braided differential operators. We show that there is a braided Lie structure on the braided differential operators. A quantization the braided differential operators provides an isomorphism of the original braided differential operators and quantized ones. The quantization of the braided Lie structure can be realized within the original one by dequantization.

This paper is the first in a trilogy.

We have found explicit descriptions of all quantizations and braidings in the monoidal category of modules graded by a finite commutative monoid, [7]. We have proved the same as for any monoidal category for this category, but the picture is somewhat more visible in this case. That is, we have a complete description for braided derivations of graded algebras and graded modules, braided connections, braided curvature, quantizations and so on. This is to be found in the second paper Quantizations of braided derivations. 2. Graded modules, [8].

In [7] we showed that the Fourier transform establishes an isomorphism between the categories of Ĝ-graded modules and G-modules where G is a finite abelian group and Ĝ is the dual of G. Using this we find a description of all quantizations and braiding also for the monoidal category of modules with action by a finite abelian group G. Again, we have a complete and explicit description for σ-derivations of algebras and modules, braided connections, curvature, differential operators and quantizations of these structures. This is to be found in the third paper Quantizations of braided derivations. 3. Modules with action by a group, [9].

There are many interesting applications of these results. One of the more interesting applications is quantizations of braided Lie algebras. In the paper [10], which is to be published, we show quantizations of semisimple Lie algebras by quantizations of derivations, for example an alternative quantization of 𝔰𝔩_{2} (ℂ).

Note that in all three papers we assume that the associativity constraint is trivial.

2 Quantizations and σ-commutativity

In this section we shall recall some results needed later. We have to define quantizations and σ-commutativity of algebras, modules, co- and bialgebras and internal homomorphisms. Most of this was done by V. V. Lychagin and many results are found in [17].

2.1 Quantizations

A quantization [17] of a monoidal category C is a natural isomorphism of the tensor bifunctor

q : ⊗ → ⊗,
q_{X,Y} : X ⊗ Y → X ⊗ Y,

X, Y ∈ Ob (C), which preserves the unit and associativity so that the following diagram

[figure: X ⊗ (Y ⊗ Z); α; (X ⊗ Y) ⊗ Z; X ⊗ (Y ⊗ Z); 1 ⊗ q_{Y,Z}; (X ⊗ Y) ⊗ Z; q_{X,Y} ⊗ 1; X ⊗ (Y ⊗ Z); q_{X,Y ⊗Z}; α; (X ⊗ Y) ⊗ Z; q_{X⊗Y,Z}]  (1)

commutes for all X, Y, Z ∈ Ob (C). We call this the coherence condition for quantizations.

The composition of two quantizations q_{1} and q_{2} is a quantization and the inverse q^{−1} of a quantization q is a quantization.

2.2 Braidings

A braiding [18] of a monoidal category C is a natural isomorphism

σ : ⊗ → ⊗ ∘ τ
σ = σ_{X,Y} : X ⊗ Y → Y ⊗ X,

X, Y ∈ Ob (C), which preserves the unit and associativity such that the following diagrams

[figure: X ⊗ (Y ⊗ Z); α; (X ⊗ Y) ⊗ Z; X ⊗ (Z ⊗ Y); 1 ⊗ σ; Z ⊗ (X ⊗ Y); σ; (X ⊗ Z) ⊗ Y; α; σ ⊗ 1; (Z ⊗ X) ⊗ Y; α; ,; X ⊗ (Y ⊗ Z); α; (X ⊗ Y) ⊗ Z; (Y ⊗ Z) ⊗ X; σ; (Y ⊗ X) ⊗ Z; σ ⊗ 1; Y ⊗ (Z ⊗ X); α^{(−1)}; 1 ⊗ σ; Y ⊗ (X ⊗ Z); α^{(−1)}]  (2)

commute. This is the coherence condition on braidings.

The braiding σ is a symmetry if

σ_{Y,X} ∘ σ_{Y,Z} = Id,  (3)

and a monoidal category equipped with such is called symmetric. We shall work only with symmetries.

When the associativity constraint is trivial, the coherence condition gives what we call the bihomomorphism conditions for any braiding σ

(σ_{X,Z} ⊗ 1) ∘ (1 ⊗ σ_{Y,Z}) = σ_{X⊗Y,Z},  (i)
(1 ⊗ σ_{X,Z}) ∘ (σ_{X,Y} ⊗ 1) = σ_{X,Y ⊗Z},  (ii)

X, Y, Z ∈ Obj (C).

The trivial braiding is the twist, τ : X ⊗ Y → Y ⊗ X.

Any braiding composed with the twist, τ ∘ σ, is a quantization since the coherence condition for quantizations then is satisfied.

Quantizations act on the set of braidings as follows

(σ_{q})_{X,Y} = q_{Y,X}^{−1} ∘ σ_{X,Y} ∘ q_{X,Y} ,

and σ_{q} is also a braiding.

2.3 Algebras

Let A be an algebra in a monoidal category C with multiplication

μ : A ⊗ A → A

and unit η : e → A.

Given a braiding σ we say that A is σ-commutative or σ-symmetric, [16], [12], if

μ = μ ∘ σ.

Note that when the associativity constraint is trivial, the bihomomorphism conditions for the σ-commutativity of algebras is

μ ∘ (μ ⊗ 1) ∘ (σ_{x,z} ⊗ 1) ∘ (1 ⊗ σ_{y,z}) = μ ∘ σ_{xy,z} ∘ (μ ⊗ 1) ,  (4)
μ ∘ (1 ⊗ μ) ∘ (1 ⊗ σ_{x,z}) ∘ (σ_{x,y} ⊗ 1) = μ ∘ σ_{x,yz} ∘ (1 ⊗ μ) ,  (5)

for x, y, z ∈ A.

Given a quantization q we define a quantization A_{q} of the algebra A as the same object A equipped with a new multiplication

μ_{q} = μ ∘ q_{A,A} : A ⊗ A → A.

A_{q} = (A, μ_{q}, η) is an algebra.

If an algebra A in a monoidal category is σ-commutative, then it’s quantization A_{q} is σ_{q}-commutative [17].

2.4 Modules

Let (A, μ) be an algebra in a monoidal category C. Let E be a left A-module with action

ν^{l} : A ⊗ E → E

in a monoidal category. If not stated otherwise, assume that all modules are left modules.

By a quantization E_{q} of the A-module E we mean the same object E equipped with a new action

ν_{q}^{l} = ν^{l} ∘ q_{A,E} : A ⊗ E → E.

E_{q} = (E, ν_{q}^{l}) is also a left A-module in C.

Let A be a σ-symmetric algebra and E be an A-A-bimodule. Given a braiding σ we say that E is σ-commutative if ν^{l} = ν^{r} ∘ σ_{A,E} and ν^{r} = ν^{l} ∘ σ_{E,A} [16], that is,

[figure: A ⊗ E; ν^{l}; X; E ⊗ A; σ; ν^{r}; X; =]  (6)

and

[figure: E ⊗ A; ν^{r}; X; A ⊗ E; σ; ν^{l}; X; =]  (7)

commutes. If σ is a symmetry then an A-module E is left σ-commutative if and only if it is right σ-commutative, that is (6) and (7) implies each other. This means that when σ is a symmetry we need only to consider left modules, not bimodules.

When the associativity constraint is trivial, the bihomomorphism conditions for any the σ-commutativity of modules is

M ∘ (M ⊗ 1) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ) = M ∘ σ ∘ (M ⊗ 1) ,  (8)
M ∘ (1 ⊗ M) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1) = M ∘ σ ∘ (1 ⊗ M) ,  (9)

where M is μ, ν^{l}or ν^{r}, depending on the triplet of A and E.

For any A-A-bimodule E the σ-symmetric part is

E_{σ} = {x ∈ E|ν^{l} (a ⊗ x) = ν^{r} ∘ σ_{A,E} (a ⊗ x) , ν^{r} (x ⊗ a) = ν^{l} ∘ σ_{E,A} (x ⊗ a) , ∀a ∈ A} .

The quantization of a right A-module E with the action

ν^{r} : E ⊗ A → E,

is done by giving E the new action

ν_{q}^{r} = ν^{r} ∘ q_{E,A} : E ⊗ A → E.

A quantization of a A-A-bimodule is the same object E equipped with two new actions E_{q} = (E, ν_{q}^{l}, ν_{q}^{r}). E_{q} is an A_{q} − A_{q}-bimodule in C.

E is σ-commutative, then E_{q} is σ_{q}-commutative;

ν^{l} ∘ q_{A,E} = ν_{q}^{l} = ν_{q}^{r} ∘ (σ_{q})_{A,E}

and similarly ν_{q}^{r} = ν_{q}^{l} ∘ (σ_{q})_{E,A}.

Let E_{σ} be the symmetric part of the A-A-bimodule E. Consider the quotient bimodule E/E_{σ} = E_{σ}^{(0)} and define E_{σ}^{(1)}as the preimage of (E/E_{σ})_{σ} with respect to the canonical projection E → E/E_{σ}. Proceeding, we get a filtration of E by bimodules E_{σ}^{(i)}, i = −1, 0, 1, …, E_{σ}^{(−1)} = 0. We call the bimodule

E_{σ}^{∗} = ∪E_{σ}^{(i)}

a differential approximation of the A-A-bimodule E.

2.5 Coalgebras

Let A be a coalgebra in C with comultiplication Δ : A → A⊗A and counit ε : A → e.

A is σ-cocommutative if

Δ = σ^{−1} ∘ Δ.

Define a quantization A_{q} of the coalgebra A as the same object A equipped with a new comultiplication defined by

Δ_{q} = q_{A,A}^{−1} ∘ Δ : A → A ⊗ A.

A_{q} = (A, Δ_{q}, ε) is a coalgebra.

2.6 Bialgebras

Let A be an algebra in C. Then the tensor square A ⊗ A can be considered as an algebra with multiplication

μ_{σ}^{⊗2} = (μ ⊗ μ) ∘ (1 ⊗ σ ⊗ 1) .

Let A be a coalgebra in C. Then the tensor square A ⊗ A can be considered as a coalgebra with comultiplication

Δ_{σ}^{⊗2} = (1 ⊗ σ ⊗ 1) ∘ Δ ⊗ Δ.

A σ-bialgebra (A, μ, Δ) in a monoidal category C is an algebra (A, μ) and a coalgebra (A, Δ) such that the diagonal

Δ : (A, μ) → (A ⊗ A, μ_{σ}^{⊗2})

and counit are algebra morphisms and the multiplication

μ : (A ⊗ A, Δ_{σ}^{⊗2}) → (A, Δ)

and unit are coalgebra morphisms.

The quantization A_{q} = (A, μ_{q}, Δ_{q}) is a bialgebra in C, [16].

2.7 Internal homomorphisms

For a closed monoidal category C and any two objects X and Y in C there is the internal homomorphism bifunctor hom and the internal homomorphism object hom (X, Y),

hom : X ⊗ Y → hom (X, Y)

together with the composition

μ^{h} : hom (Y, Z) ⊗ hom (X, Y) → hom (X, Z) ,
g ⊗ f ⟼ g ∘ f = g ∗ f.

The collection of internal homomorphisms is an ”algebra” with respect to ∗.

If C is equipped with a braiding σ, then the composition of internal homomorphisms is called σ-symmetric if

μ^{h} = μ^{h} ∘ σ.

The morphism

ev_{X,Y} : hom (X, Y) ⊗ X → Y

is called the evaluation map if any morphism f : X ⊗ Z → Y can be represented by the composition

f = ev_{X,Y} ∘ (f̂⊗ id_{X})

for a unique morphism f̂: Z → hom (X, Y). That is

Mor (Z, hom (X, Y)) ≅ Mor (Z ⊗ X, Y) .

The evaluation map also can be considered as a result of the multiplication μ^{h},

ev_{X,Y} (f ⊗ x) = f (x) = f ∗ x,

x ∈ X, f ∈ hom (X, Y).

Given a quantization q, define a quantization of the internal homomorphisms to be a quantization as an algebra and we equip the internal homomorphisms with a new multiplication. Namely a quantization is a natural isomorphism

Q_{q} : hom (X, Y) → hom (X, Y)

defined by

Q_{q} (f) ∗ x = f ∗_{q} x

for all f ∈ hom (X, Y) and x ∈ hom (1, X), where

f ∗_{q} g = μ^{h} (q_{hom(X,Y),hom(Y,Z)} (f ⊗ g)) = μ_{q}^{h} (f ⊗ g) .

For any quantization we have

Q_{q} (f) ∗ Q_{q} (f) = Q_{q} (f ∗_{q} g) ,  (10)
Q_{q} (x) = x.  (11)

3 Braided Lie structure of internal homomorphisms

We shall describe the module structure and braided Lie algebra structure of the internal homomorphisms. We describe the same for the quantizations of the internal homomorphisms.

3.1 Module structure of hom (E, E′)

Let σ be a braiding in a monoidal category C.

Let A be a σ-commutative algebra, let E and E′ be left A-modules and let hom (E, E′) be the set of internal homomorphisms.

The set hom (E, E′), is an A-A-bimodule with the left and right multiplications defined by

ν_{a}^{l} (f) (x) = ν^{l} (a ⊗ f) (x) = af (x) ,
ν_{a}^{r} (f) (x) = ν^{r} (f ⊗ a) (x) = f (ax) ,

a ∈ A, x ∈ E.

Proposition 1 Let σ be a symmetry and E and E′ be σ-commutative A-A-modules. Then hom (E, E′) is σ-commutative as a module, that is, the diagrams

[figure: A ⊗ hom(X, Y); ν^{l}; hom(X, Y); hom(X, Y) ⊗ A; σ; ν^{r}; hom(X, Y); =]  (12)

and

[figure: hom(X, Y) ⊗ A; ν^{r}; hom(X, Y); A ⊗ hom(X, Y); σ; ν^{l}; hom(X, Y); =]  (13)

commute.

Proof. Write the actions as

ev_{E,E′} ∘ (ν^{l} ⊗ 1) = v_{E′}^{l} ∘ (1 ⊗ ev_{E,E′}) : A ⊗ hom (E, E′) ⊗ E → E′,
ev_{E,E′} ∘ (ν^{r} ⊗ 1) = ev_{E,E′} ∘ (1 ⊗ v_{E}^{l}) : hom (E, E′) ⊗ A ⊗ E → E′.

Then

ν^{l} (a ⊗ f) (x) = ν_{E′}^{l} ∘ (1 ⊗ ev_{E,E′}) (a ⊗ f ⊗ x)
= ν_{E′}^{r} ∘ σ ∘ (1 ⊗ ev_{E,E′}) (a ⊗ f ⊗ x)
= ev_{E,E′} ∘ (1 ⊗ ν_{E}^{r}) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1) (a ⊗ f ⊗ x)
= ev_{E,E′} ∘ (1 ⊗ ν_{E}^{l}) ∘ (σ ⊗ 1) (a ⊗ f ⊗ x)
= ν^{r} ∘ σ_{A,hom(E,E′)} (a ⊗ f) (x) ,

MONOIDAL CATEGORIES 21

and

ν^{r} (f ⊗ a) (x) = ev_{E,E′} ∘ (1 ⊗ v_{E}^{l}) (f ⊗ a ⊗ x)
= ev_{E,E′} ∘ (1 ⊗ v_{E}^{r}) ∘ (1 ⊗ σ) (f ⊗ a ⊗ x)
= ν_{E′}^{r} ∘ (ev_{E,E′} ⊗ 1) ∘ (1 ⊗ σ) (f ⊗ a ⊗ x)
= ν_{E′}^{l} ∘ σ ∘ (ev_{E,E′} ⊗ 1) ∘ (1 ⊗ σ) (f ⊗ a ⊗ x)
= ν_{E′}^{l} ∘ (1 ⊗ ev_{E,E′}) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ) ∘ (1 ⊗ σ) (f ⊗ a ⊗ x)
= ν_{E′}^{l} ∘ (1 ⊗ ev_{E,E′}) ∘ (σ ⊗ 1) (f ⊗ a ⊗ x)
= ν^{l} ∘ σ (f ⊗ a) (x) ,

a ∈ A, f ∈ hom (E, E′), x ∈ E.

3.2 Braided commutators and Lie structure of hom (E, E)

Let σ be a braiding, A be an σ-symmetric algebra and E be a left σ-symmetric A-module. Consider hom (E, E). Definition 2 Define the σ-bracket or σ-commutator of hom (E, E)

c_{σ} : hom (E, E) ⊗ hom (E, E) → hom (E, E)

by

c_{σ} = [ , ]^{σ} = μ^{h} − μ^{h} ∘ σ.

Proposition 3 Let σ be a symmetry. The σ-bracket satisfies the conditions,

c_{σ} ∘ (ν^{l} ⊗ 1) (a ⊗ f ⊗ g) = (ν^{l} ∘ (1 ⊗ c_{σ}) − ν^{l} ∘ (ev_{E,E} ⊗ 1) ∘ σ) (a ⊗ f ⊗ g) ,
c_{σ} ∘ (1 ⊗ ν^{l}) (f ⊗ a ⊗ g) = (ν^{l} ∘ (1 ⊗ c_{σ}) ∘ (σ ⊗ 1) + ν^{l} ∘ (ev_{E,E} ⊗ 1)) (f ⊗ a ⊗ g) ,

a ∈ A, f, g ∈ hom (E, E). Proof.

c_{σ} ∘ (ν^{l} ⊗ 1) = μ^{h} ∘ (ν^{l} ⊗ 1) − μ^{h} ∘ σ ∘ (ν^{l} ⊗ 1)
= ν^{l} ∘ (1 ⊗ μ^{h}) − μ^{h} ∘ (1 ⊗ ν^{l}) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ)
= ν^{l} ∘ (1 ⊗ μ^{h}) − μ^{h} ∘ (ν^{r} ⊗ 1) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ)
−ν^{l} ∘ (1 ⊗ μ^{h}) ∘ (σ ⊗ 1) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ)
= ν^{l} ∘ (1 ⊗ μ^{h}) − μ^{h} ∘ σ ∘ (ν^{l} ⊗ 1) − ν^{l} ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ)
= ν^{l} ∘ (1 ⊗ c_{σ}) − ν^{l} ∘ (ev_{E,E} ⊗ 1) ∘ σ,

and

c_{σ} ∘ (1 ⊗ ν^{l}) = μ^{h} ∘ (1 ⊗ ν^{l}) − μ^{h} ∘ σ ∘ (1 ⊗ ν^{l})
= μ^{h} ∘ (ν^{r} ⊗ 1) + ν^{l} ∘ (1 ⊗ μ^{h}) ∘ (σ ⊗ 1)
−ν^{l} ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1)
= ν^{l} ∘ (ev_{E,E} ⊗ 1) + ν^{l} ∘ (1 ⊗ c_{σ}) ∘ (σ ⊗ 1) .

Proposition 4 The σ-bracket is σ-invariant,

μ^{h} ∘ σ ∘ (c_{σ} ⊗ 1) = μ^{h} ∘ (1 ⊗ c_{σ}) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ) .

Proof.

μ^{h} ∘ σ ∘ (c_{σ} ⊗ 1) = μ^{h} ∘ σ ∘ ((μ^{h} − μ^{h} ∘ σ) ⊗ 1)
= μ^{h} ∘ σ ∘ (μ^{h} ⊗ 1) − μ^{h} ∘ σ ∘ (μ^{h} ⊗ 1) ∘ (σ ⊗ 1) ,

and

μ^{h} ∘ (1 ⊗ c_{σ}) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ)
= μ^{h} ∘ (1 ⊗ (μ^{h} − μ^{h} ∘ σ)) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ)
= μ^{h} ∘ (1 ⊗ μ^{h}) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ) − μ^{h} ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ)
= μ^{h} ∘ (μ^{h} ⊗ 1) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ) − μ^{h} ∘ (μ^{h} ⊗ 1) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ)
= μ^{h} ∘ σ ∘ (μ^{h} ⊗ 1) − μ^{h} ∘ σ ∘ (μ^{h} ⊗ 1) ∘ (σ ⊗ 1) .

Proposition 5 Let σ be a symmetry. The σ-commutator satisfies skew σ-symmetricity, that is, the diagram

[figure: hom(X, X) ⊗ hom(X, X); c_{σ}; hom(X, X); hom(X, X) ⊗ hom(X, X); σ; −c_{σ}; hom(X, X); =]  (14)

commutes, equivalently

c_{σ} = −c_{σ} ∘ σ.

Proof.

c_{σ} ∘ σ (f ⊗ g) = μ^{h} ∘ σ (f ⊗ g) − μ^{h} ∘ σ ∘ σ (f ⊗ g)
= μ^{h} ∘ σ (f ⊗ g) − μ^{h} (f ⊗ g)
= −c_{σ} (f ⊗ g) ,

f, g ∈ hom (E, E).

Proposition 6 Let σ be a symmetry. Then hom (E, E) equipped with the σ-commutator satisfies the σ-Jacobi identity,

c_{σ} ∘ (1 ⊗ c_{σ}) = c_{σ} ∘ (c_{σ} ⊗ 1) + c_{σ} ∘ (1 ⊗ c_{σ}) ∘ (σ ⊗ 1) .

Proof. The Jacobi identity applied to three elements in hom (E, E) is satisfied when

c_{σ} ∘ (1 ⊗ c_{σ})
= μ^{h} ∘ (1 ⊗ μ^{h})  (a)
− μ^{h} ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ)  (b)
−μ^{h} ∘ σ ∘ (1 ⊗ μ^{h})  (c)
+μ^{h} ∘ σ ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ)  (d)

is equal to

c_{σ} ∘ (c_{σ} ⊗ 1) + c_{σ} ∘ (1 ⊗ c_{σ}) ∘ (σ ⊗ 1)
= μ^{h} ∘ (μ^{h} ⊗ 1)  (e)
−μ^{h} ∘ (μ^{h} ⊗ 1) ∘ (σ ⊗ 1)  (f)
−μ^{h} ∘ σ ∘ (μ^{h} ⊗ 1)  (g)
+μ^{h} ∘ σ ∘ (μ^{h} ⊗ 1) ∘ (σ ⊗ 1)  (h)
+μ^{h} ∘ (1 ⊗ μ^{h}) ∘ (σ ⊗ 1)  (i)
−μ^{h} ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1)  (j)
−μ^{h} ∘ σ ∘ (1 ⊗ μ^{h}) ∘ (σ ⊗ 1)  (k)
+μ^{h} ∘ σ_{g,hf} ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1) ,  (l)

which is the case since

(a) = (e) ;

μ^{h} ∘ (1 ⊗ μ^{h}) = μ^{h} ∘ (μ^{h} ⊗ 1) ,

(b) = (k) ;

μ^{h} ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ) = μ^{h} ∘ σ ∘ (1 ⊗ μ^{h}) ∘ (σ ⊗ 1)
= μ^{h} ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1) ∘ (σ ⊗ 1) ,

(c) = (j) ;

μ^{h} ∘ σ ∘ (1 ⊗ μ^{h}) = μ^{h} ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1) ,

(d) = (h) ;

μ^{h} ∘ σ ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ) = μ^{h} ∘ σ ∘ (μ^{h} ⊗ 1) ∘ (σ ⊗ 1)

μ^{h} ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ) = μ^{h} ∘ (μ^{h} ⊗ 1) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1) ,

(f) + (i) = 0;

μ^{h} ∘ (μ^{h} ⊗ 1) ∘ (σ ⊗ 1) = μ^{h} ∘ (1 ⊗ μ^{h}) ∘ (σ ⊗ 1) ,

and (g) + (l) = 0;

μ^{h} ∘ σ ∘ (μ^{h} ⊗ 1) = μ^{h} ∘ σ ∘ (1 ⊗ μ^{h}) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1) ,

since

μ^{h} ∘ σ ∘ (μ^{h} ⊗ 1) = μ^{h} ∘ (μ^{h} ⊗ 1) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ) ,

and

μ^{h} ∘σ ∘(1 ⊗ μ^{h})∘(1 ⊗ σ)∘(σ ⊗ 1) = μ^{h} ∘(1 ⊗ μ^{h})∘(1 ⊗ σ)∘(σ ⊗ 1)∘(1 ⊗ σ)∘(σ ⊗ 1) .

This is shown by using the identities (4), (5) and the fact that, σ satisfies the Yang-Baxter equation,

(1 ⊗ σ) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ) = (σ ⊗ 1) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1) .

Remark 7 In the proof of the propositions 5 and 6 we can assume that the internal homomorphisms are equipped with the σ-commutative composition μ^{h} = μ^{h} ∘σ instead of using the fact that σ is a symmetry.

Definition 8 A σ-Lie algebra is an algebra equipped with a skew σ-symmetric σcommutator that is σ-invariant and satisfies the σ-Jacobi identity.

This is the definition of braided Lie algebras as introduced by D. Gurevich, [4]. By the propositions 4, 5 and 6 we have proved the following.

Theorem 9 Let σ be a symmetry and an algebra A and a left A-module E be σsymmetric. Then hom (E, E) is a σ-Lie algebra.

3.3 Quantization of hom (E, E)

We will define the quantization of the internal homomorphisms and describe the structure on the set of all such, also within the original internal homomorphisms.

Definition 10 Given a quantization q and f ∈ hom (E, E), its quantization is defined by

f_{q} (x) = Q_{q} (f) (x) =^{def} ev_{E,E} ∘ q (f ⊗ x) ,  (15)

x ∈ E.

It is easy to see that Q_{q} (f) is an internal homomorphism of the quantized A_{q}- module E_{q}.

Let f and g be internal homomorphisms. The quantization of composition of internal homomorphisms is

f ∗_{q} g = μ_{q}^{h} (f ⊗ g) .  (16)

The collection of all Q_{q} (f), f ∈ hom (E, E), equipped with the quantization of the composition is denoted by hom^{q} (E_{q}, E_{q}).

By proposition 1 is hom^{q} (E_{q}, E_{q}) a σ_{q}-symmetric module if E is a σ-symmetric A-A-bimodule.

By theorem 9, if σ is a symmetry is hom^{q} (E_{q}, E_{q}) a σ_{q}-Lie algebra with respect to the σ_{q} − q-bracket (or simply σ_{q}-bracket when it is clear that the multiplication or composition is the quantized multiplication).

Let γ be a braiding and p any quantization. Define the γ − p-bracket on internal homomorphisms,

c_{γ}^{p} = [−, −]_{p}^{γ} = μ_{p}^{h} − μ_{p}^{h} ∘ γ.  (17)

Lemma 11 The σ_{q} − q-bracket satisfies

c_{σ_{q}}^{q} = c_{σ} ∘ q.

Proof.

c_{σ_{q}}^{q} = μ_{q}^{h} − μ_{q}^{h} ∘ σ_{q} = μ_{q}^{h} − μ^{h} ∘ q ∘ q^{−1} ∘ σ ∘ q = μ^{h} ∘ q − σ ∘ μ^{h} ∘ q = c_{σ} ∘ q.

The inverse of the quantization of f ∈ hom^{q} (E_{q}, E_{q}) is denoted by

Q_{q}^{−1} (f) = f_{c},

and is called the dequantization of f.

Lemma 12 The composition of internal homomorphisms satisfies

f_{c} ∗_{q} g_{c} = (f ∘ g)_{c} ,  (18)

f, g ∈ hom^{q} (E_{q}, E_{q}).

Proof.

f_{c} ∗_{q} g_{c} (x) = Q_{q}^{−1} (f) ∗_{q} Q_{q}^{−1} (g) (x)
= ev_{E,E} ∘ q^{−1} ∘ (1 ⊗ ev_{E,E}) ∘ (1 ⊗ q^{−1}) ∘ (q ⊗ 1) (f ⊗ g ⊗ x)
= ev_{E,E} ∘ q^{−1} ∘ (μ^{h} ⊗ 1) ∘ (q^{−1} ⊗ 1) ∘ (q ⊗ 1) (f ⊗ g ⊗ x)
= Q_{q}^{−1} (f ∘ g) (x)
= (f ∘ g)_{c} ,

Note,

a_{c} = a,

for a ∈ A_{q}.

The set of all Q_{q}^{−1} (f), f ∈ hom^{q} (E_{q}, E_{q}), is equipped with the bracket c_{σ_{q}}^{q} and the A-module structure

a ∗_{q} f_{c} = ν^{l} ∘ q (a ⊗ f_{c}) ,  (19)

a ∈ A.

The dequantization of a internal homomorphism operates on A in the classical manner, but satisfies somewhat different properties than the classical, as the following theorem states. (See also [14])

Theorem 13 Let σ be a symmetry. The σ_{q}-Lie algebra structure of hom^{q} (E_{q}, E_{q}) can be realized within the classical, hom (E, E), by dequantization as follows.

Let f, g ∈ hom^{q} (E_{q}, E_{q}), a ∈ A_{q}. Then linearity is

(f + g)_{c} = f_{c} + g_{c},  (i)

A-module structure,

(a ∘ f)_{c} = a ∗_{q} f_{c},  (ii)

and the braided commutator satisfies

Q_{q}^{−1} ∘ c_{σ_{q}} = c_{σ_{q}}^{q} ∘ (Q_{q}^{−1} ⊗ Q_{q}^{−1}) .  (iii)

Proof. (i):

Q_{q}^{−1} (f + g) = Q_{q}^{−1} (f) + Q_{q}^{−1} (g) .

(ii): By lemma 12,

a_{c} ∗_{q} f_{c} = a ∗_{q} f_{c} = (a ∘ f)_{c} .

(iii):

[Q_{q}^{−1} (f) , Q_{q}^{−1} (g)]_{q}^{σ_{q}} (x)
= ev_{E,E} ∘ q^{−1} ∘ (1 ⊗ ev_{E,E}) ∘ (1 ⊗ q^{−1}) ∘ (q ⊗ 1) (f ⊗ g ⊗ x)
−ev_{E,E} ∘ q^{−1} ∘ (1 ⊗ ev_{E,E}) ∘ (1 ⊗ q^{−1}) ∘ (q ⊗ 1) ∘ (σ_{q} ⊗ 1) (f ⊗ g ⊗ x)
= ev_{E,E} ∘ q^{−1} ∘ (μ^{h} ⊗ 1) ∘ (q^{−1} ⊗ 1) ∘ (q ⊗ 1) (f ⊗ g ⊗ x)
−ev_{E,E} ∘ q^{−1} ∘ (μ^{h} ⊗ 1) ∘ (q^{−1} ⊗ 1) ∘ (q ⊗ 1) ∘ (σ_{q} ⊗ 1) (f ⊗ g ⊗ x)
= Q_{q}^{−1} ([f, g]^{σ^{q}}) (x) ,

x ∈ E, f, g ∈ hom^{q} (E_{q}, E_{q}).

4 Braided derivations in monoidal categories

Let C be a monoidal category equipped with a braiding σ.

Let A be a σ-symmetric algebra in C with the multiplication μ, and E and E′ be A-A-modules.

Denote by hom_{σ} (E, E′) = (hom (E, E′))_{σ} the σ-symmetric part of the A − Abimodule hom (E, E′). The modules

(hom (E, E′))_{σ}^{(i)} = Diff_{i}^{σ} (E, E′)

are called the braided or σ-differential operators, see [16].

4.1 Braided derivations of algebras

With an additional condition we define σ-derivations of algebras.

Definition 14 Define σ-derivations or braided derivations of A with values in a A-A-bimodule E as

Der^{σ} (E) = {f ∈ Diff_{1}^{σ} (A, E) |f (1) = 0} .

An internal homomorphism f : A → E is a σ-derivation if and only if f : A → E_{σ} ⊂ E and f satisfies the braided or σ-Leibniz rule, [17],

ev_{A,E} ∘ (1 ⊗ μ) (f ⊗ a ⊗ b)  (20)
= (ν^{r} ∘ (ev_{A,E} ⊗ 1) + ν^{l} ∘ (1 ⊗ ev_{A,E}) ∘ (σ ⊗ 1)) (f ⊗ a ⊗ b) ,
ev_{A,E} ∘ (1 ⊗ μ) ∘ (σ ⊗ 1) (a ⊗ f ⊗ b)  (21)
= (ν^{l} ∘ (1 ⊗ ev_{A,E}) + ν^{r} ∘ (ev_{A,E} ⊗ 1) ∘ (σ ⊗ 1)) (a ⊗ f ⊗ b) .

If the braiding is a symmetry, then the two Leibniz rules implies each other.

From now on, assume that every braiding σ is a symmetry. We can then assume E is a left A-module.

Proposition 15 Der^{σ} (E) has a left A-module structure defined by

ν^{l} (a ⊗ ∂) (b) =^{def} ν_{E}^{l} ∘ (1 ⊗ ev_{A,E}) (a ⊗ ∂ ⊗ b) = a∂ (b) ,  (22)

∂ ∈ Der^{σ} (E), a, b ∈ A, ν_{E}^{l} is the action of A on E.

Proof. We need to show that ν_{a}^{l} (∂) is a σ-derivation, that is satisfies the σ-Leibniz rule, and using the σ-Leibniz rule for ∂,

a∂ (bc) = ( \begin{matrix}ν_{E}^{l} ∘ (1 ⊗ ν_{E}^{l}) ∘ (1 ⊗ ev_{A,A} ⊗ 1) \\ +ν_{E}^{l} ∘ (1 ⊗ ν_{E}^{l}) (1 ⊗ 1 ⊗ ev_{E,E}) ∘ (1 ⊗ σ ⊗ 1) \end{matrix} ) (a ⊗ ∂ ⊗ b ⊗ c) ,

a, b, c ∈ A, ∂ ∈ Der^{σ} (E), and

ν_{E}^{l} ∘ (1 ⊗ ν_{E}^{l}) ∘ (1 ⊗ ev_{A,A} ⊗ 1) + ν_{E}^{l} ∘ (1 ⊗ ν_{E}^{l}) (1 ⊗ 1 ⊗ ev_{E,E}) ∘ (1 ⊗ σ ⊗ 1)

= ν_{E}^{l} ∘ (ν_{E}^{l} ⊗ 1) ∘ (1 ⊗ ev_{A,A} ⊗ 1)
+ν_{E}^{l} ∘ ((μ ∘ σ) ⊗ 1) ∘ (1 ⊗ 1 ⊗ ev_{E,E}) ∘ (1 ⊗ σ ⊗ 1)
= ν_{E}^{l} ∘ (ev_{A,A} ⊗ 1) ∘ (ν^{l} ⊗ 1 ⊗ 1)
+ν_{E}^{l} ∘ (1 ⊗ ν_{E}^{l}) ∘ (1 ⊗ 1 ⊗ ev_{E,E}) ∘ (σ ⊗ 1 ⊗ 1) ∘ (1 ⊗ σ ⊗ 1)
= ν_{E}^{l} ∘ (ev_{A,A} ⊗ 1) ∘ (ν^{l} ⊗ 1 ⊗ 1)
+ν_{E}^{l} ∘ (1 ⊗ ev_{E,E}) ∘ (1 ⊗ ν^{l} ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1) ∘ (1 ⊗ σ ⊗ 1) ,

and the condition,

ν^{l} (a ⊗ ∂) (1) = a∂ (1) = 0,

is satisfied.

If we consider Diff_{1}^{σ} (A, E) and not Der^{σ} (E) then there is a right A-module structure.

Proposition 16 Diff_{1}^{σ} (A, E) has in addition to the left A-module structure defined by (22), a right A-module structure defined by

ν^{r} (∂ ⊗ a) (b) =^{def} ev_{A,E} ∘ (1 ⊗ μ) (∂ ⊗ a ⊗ b) = ∂ (ab) ,

∂ ∈ Diff_{1}^{σ} (A, E) , a, b ∈ A.

Proof. The σ-Leibniz rule for ∂a is,

∂a (bc) = ( \begin{matrix}ν_{E}^{r} ∘ (ev_{A,E} ⊗ 1) ∘ (ν^{r} ⊗ 1 ⊗ 1) \\ +ν_{E}^{l} ∘ (1 ⊗ ev_{A,E}) ∘ (σ_{∂a,b} ⊗ 1) ∘ (ν^{r} ⊗ 1 ⊗ 1) \end{matrix} ) (∂ ⊗ a ⊗ b ⊗ c) . (23)

Using the σ-Leibniz rule for ∂,

∂a (bc) = ev_{A,E} ∘ (1 ⊗ μ) ∘ (1 ⊗ 1 ⊗ μ) (∂ ⊗ a ⊗ b ⊗ c)
= ev_{A,E} ∘ (1 ⊗ μ) ∘ (1 ⊗ μ ⊗ 1) (∂ ⊗ a ⊗ b ⊗ c)
= ( \begin{matrix}ν_{E}^{r} ∘ (ev_{A,E} ⊗ 1) ∘ (1 ⊗ μ ⊗ 1) \\ +ν_{E}^{l} ∘ (1 ⊗ ev_{A,E}) ∘ (σ ⊗ 1) ∘ (1 ⊗ μ ⊗ 1) \end{matrix} ) (∂ ⊗ a ⊗ b ⊗ c) ,

a, b, c ∈ A, ∂ ∈ Diff_{1}^{σ} (A, E), we see that (23) is satisfied as

ν_{E}^{l} ∘ (1 ⊗ ev_{A,E}) ∘ (σ ⊗ 1) ∘ (1 ⊗ μ ⊗ 1)
= ν_{E}^{l} ∘ (1 ⊗ ev_{A,E}) ∘ (1 ⊗ ν^{l} ⊗ 1) ∘ (1 ⊗ σ ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1)
= ν_{E}^{l} ∘ (1 ⊗ ν^{l}) ∘ (1 ⊗ 1 ⊗ ev_{A,E}) ∘ (1 ⊗ σ ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1)
= ν_{E}^{l} ∘ (μ ⊗ 1) ∘ (1 ⊗ 1 ⊗ ev_{A,E}) ∘ (1 ⊗ σ ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1)
= ν_{E}^{l} ∘ (μ ⊗ 1) ∘ (σ ⊗ 1) ∘ (1 ⊗ 1 ⊗ ev_{A,E}) ∘ (1 ⊗ σ ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1)
= ν_{E}^{l} ∘ (1 ⊗ ev_{A,E}) ∘ (μ ⊗ 1 ⊗ 1) ∘ (σ ⊗ 1) ∘ (1 ⊗ σ ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1)
= ν_{E}^{l} ∘ (1 ⊗ ev_{A,E}) ∘ (σ ⊗ 1) ∘ (ν^{l} ⊗ 1 ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1)
= ν_{E}^{l} ∘ (1 ⊗ ev_{A,E}) ∘ (σ ⊗ 1) ∘ (ν^{r} ⊗ 1 ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1)
= ν_{E}^{l} ∘ (1 ⊗ ev_{A,E}) ∘ (σ ⊗ 1) ∘ (ν^{r} ⊗ 1 ⊗ 1) .

For the rest of the paper, consider the σ-derivations of a σ-symmetric algebra A with values in A, denoted by

Der^{σ} (A) .

Proposition 17 The σ-commutator of two σ-derivations of A is a σ-derivation of A,

c_{σ} : Der^{σ} (A) ⊗ Der^{σ} (A) → Der^{σ} (A) .

Proof. The σ-Leibniz rule is satisfied for the σ-commutator of two σ-derivations,

c_{σ} (∂_{1} ⊗ ∂_{2}) (ab) = μ^{h} (∂_{1} ⊗ ∂_{2}) (ab) − μ^{h} ∘ σ (∂_{1} ⊗ ∂_{2}) (ab)
= c_{σ} (∂_{1} ⊗ ∂_{2}) (a) b + μ ∘ (1 ⊗ ev_{A,A}) (σ (c_{σ} (∂_{1} ⊗ ∂_{2}) ⊗ a) ⊗ b) ,

a, b ∈ A, ∂_{1}, ∂_{2} ∈ Der^{σ} (A), as we see,

ev_{A,A} ∘ (1 ⊗ ev_{A,A}) ∘ (1 ⊗ 1 ⊗ μ) − ev_{A,A} ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1 ⊗ 1) ∘ (1 ⊗ 1 ⊗ μ) = ev_{A,A} ∘ (1 ⊗ μ) ∘ (1 ⊗ ev_{A,A} ⊗ 1) + ev_{A,A} ∘ (1 ⊗ μ) ∘ (1 ⊗ 1 ⊗ ev_{A,A}) ∘ (1 ⊗ σ ⊗ 1) −ev_{A,A} ∘ (1 ⊗ μ) ∘ (1 ⊗ ev_{A,A} ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1) −ev_{A,A} ∘ (1 ⊗ μ) ∘ (1 ⊗ 1 ⊗ ev_{A,A}) ∘ (1 ⊗ σ ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1) = μ ∘ (ev_{A,A} ⊗ 1) ∘ (1 ⊗ ev_{A,A} ⊗ 1) + μ ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1) ∘ (1 ⊗ ev_{A,A} ⊗ 1) +μ ∘ (ev_{A,A} ⊗ 1) ∘ (1 ⊗ 1 ⊗ ev_{A,A}) ∘ (1 ⊗ σ ⊗ 1) +μ ∘ (1 ⊗ ev_{A,A}) ∘ (1 ⊗ 1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1 ⊗ 1) ∘ (1 ⊗ σ ⊗ 1) −μ ∘ (ev_{A,A} ⊗ 1) ∘ (1 ⊗ ev_{A,A} ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1) −μ ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1) ∘ (1 ⊗ ev_{A,A} ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1) −μ ∘ (ev_{A,A} ⊗ 1) ∘ (1 ⊗ 1 ⊗ ev_{A,A}) ∘ (1 ⊗ σ ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1) −μ ∘ (1 ⊗ ev_{A,A}) ∘ (1 ⊗ 1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1 ⊗ 1) ∘ (1 ⊗ σ ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1) = μ ∘ (ev_{A,A} ⊗ 1) ∘ (c_{σ} ⊗ 1 ⊗ 1) + μ ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1) ∘ (μ^{h} ⊗ 1 ⊗ 1) −μ ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1) ∘ (μ^{h} ⊗ 1 ⊗ 1) ∘ (σ ⊗ 1 ⊗ 1) = μ ∘ (ev_{A,A} ⊗ 1) ∘ (c_{σ} ⊗ 1 ⊗ 1) + μ ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1) ∘ (c_{σ} ⊗ 1 ⊗ 1) .

Furthermore,

c_{σ} (∂_{1} ⊗ ∂_{2}) (1) = μ^{h} (∂_{1} ⊗ ∂_{2}) (1) − μ^{h} ∘ σ (∂_{1} ⊗ ∂_{2}) (1) = 0.

Corollary 18 Let the braiding σ be a symmetry. Then Der^{σ} (A) equipped with the σ-commutator is a σ-Lie algebra.

MONOIDAL CATEGORIES 29

4.2 Braided derivations of modules

Let A be a σ-symmetric algebra and let E be a σ-symmetric A-module. Definition 19 An operator ∂ : E → E is said to be a σ-derivation of E over ∂^{A} ∈ Der^{σ} (A) if ∂ satisfy the σ-Leibniz rule with respect to ∂^{A}

∂ (ax) = ∂^{A} (a) x + ν_{E}^{l} ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1) (∂ ⊗ a ⊗ x) ,  (24)

x ∈ E, a ∈ A and ν_{E}^{l} is the action of A on E.

The pair (∂, ∂_{A}) is called a σ-derivation of E over A. (We could also call this pair a σ-𝔇-module, [13].)

The morphism π : (∂, ∂_{A}) → ∂_{A} we call the projection from the σ-derivations of E over A to the σ-derivations of A.

The set of σ-derivations of E over A is denoted by Der^{(σ,A)} (E). Proposition 20 Der^{(σ,A)} (E) has a left A-module structure defined by

ν_{a}^{l} (∂) (x) =^{def} ν_{E}^{l} ∘ (1 ⊗ ev_{E,E}) (a ⊗ ∂ ⊗ x) = a∂ (x) ,

∂ ∈ Der^{(σ,A)} (E), a ∈ A, x ∈ E. Proof. Just repeat the proof of proposition 15.

If we consider Diff_{1}^{σ} (E, E) and not Der^{(σ,A)} (E) then there is a right A-module structure. Proposition 21 Diff_{1}^{σ} (E, E) has in addition a right A-module structure defined by

ν_{a}^{r} (∂) (x) =^{def} ev_{E,E} ∘ (1 ⊗ ν_{E}^{l}) (∂ ⊗ a ⊗ x) = ∂ (ax) ,

∂ ∈ Diff_{1}^{σ} (E, E), a ∈ A, x ∈ E. Proof. Just repeat the proof of proposition 16. Proposition 22 The σ-commutator of two (σ, A)-derivations of E is a (σ, A)-derivation of E,

c_{σ} : Der^{(σ,A)} (E) ⊗ Der^{(σ,A)} (E) → Der^{(σ,A)} (E) .

Proof. Simply repeat the proof of proposition 17. Corollary 23 Let the braiding σ be a symmetry. Then the (σ, A)-derivations of E equipped with the σ-commutator is a σ-Lie algebra.

We get the following sequence of σ-symmetric left A-modules and σ-Lie algebras

0 → hom (E, E) → Der^{(σ,A)} (E) →^{π} Der^{σ} (A) .

4.3 Quantization of braided derivations of algebras

Consider derivations of a σ-symmetric algebra A.

Definition 24 Given a quantization q and ∂ ∈ Der^{σ} (A) define the quantization of ∂ by

Q_{q} (∂) (a) = ev_{A,A} ∘ q (∂ ⊗ a) ,

a ∈ A.

Sometimes we use the notation ∂_{q} = Q_{q} (∂).

Q_{q} (∂) is an operator of the quantized algebra A_{q}. Denote by Der^{σ^{q}} (A_{q}) the set of all Q_{q} (∂), ∂ ∈ Der^{σ} (A), equipped with the quantized composition.

Theorem 25 Given a braiding σ, let σ_{q} be the quantization of σ. The operator

Q_{q} : (Der^{σ} (A) , c_{σ}) → (Der^{σ^{q}} (A_{q}) , c_{σ_{q}}^{q}) ,  (25)
∂ ∈ Der^{σ} (A) ⟼ Q_{q} (∂) ∈ Der^{σ^{q}} (A_{q}) ,

is an isomorphism of modules between the σ-derivations of A and the σ_{q}-derivations of A_{q}

Proof. We need to show that the σ_{q}-Leibniz rule is satisfied

Q_{q} (∂) (a ∗_{q} b) = Q_{q} (∂) (a) ∗_{q} b + μ_{A} ∘ q ∘ (1 ⊗ ev_{A,A}) ∘ (1 ⊗ q) ∘ (σ_{q} ⊗ 1) (∂ ⊗ a ⊗ b)

since

ev_{A,A} ∘ (1 ⊗ μ_{A}) ∘ q ∘ (1 ⊗ q) = (μ_{A} ∘ (ev_{A,A} ⊗ 1) + μ_{A} ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1)) ∘ q ∘ (1 ⊗ q) = (μ_{A} ∘ (ev_{A,A} ⊗ 1) + μ_{A} ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1)) ∘ q ∘ (q ⊗ 1) = (μ_{A} ∘ q ∘ (ev_{A,A} ⊗ 1) + μ_{A} ∘ (1 ⊗ ev_{A,A}) ∘ q ∘ (σ ⊗ 1)) ∘ (q ⊗ 1) = (μ_{A} ∘ q ∘ (ev_{A,A} ⊗ 1) + μ_{A} ∘ q ∘ (1 ⊗ ev_{A,A}) ∘ (1 ⊗ q) ∘ (q^{−1} ⊗ 1) ∘ (σ ⊗ 1)) ∘ (q ⊗ 1) = μ_{A} ∘ q ∘ (ev_{A,A} ⊗ 1) ∘ (q ⊗ 1) + μ_{A} ∘ q ∘ (1 ⊗ ev_{A,A}) ∘ (1 ⊗ q) ∘ (σ_{q} ⊗ 1)

where

q_{A⊗Der^{σ}(A),B} ∘ (σ_{Der^{σ}(A),A} ⊗ 1) = (σ_{Der^{σ}(A),A} ⊗ 1) ∘ q_{Der^{σ}(A)⊗A,B}

by naturality.

Note that the σ_{q}-Leibniz rule is satisfied for the σ_{q}-commutator of two σ_{q}-derivations which follows from proposition 22.

By proposition 1 is Der^{σ^{q}} (A_{q}) a σ_{q}-symmetric module and by theorem 9 a σ_{q}-Lie algebra with respect to the σ_{q} − q-bracket.

By theorem 13 the σ_{q}-Lie algebra structure of (Der^{σ^{q}} (A_{q}) , c_{σ_{q}}^{q}) can be realized within the classical one by dequantization.

MONOIDAL CATEGORIES 31

4.4 Evaluations and commutators

For both σ- and σ_{q}-derivations, evaluating a derivation of some element corresponds to taking the braided bracket of the derivation and that element. Proposition 26 Let A be σ-commutative algebra, ∂ ∈ Der^{σ} (A) and a ∈ A. Then

∂ (a) = c_{σ} (∂ ⊗ a) .

Let

∂_{q} = Q_{q} (∂_{c}) ∈ Der^{σ^{q}} (A_{q})

and a ∈ A_{q}. Then

∂_{q} (a) = c_{σ_{q}}^{q} (∂_{q} ⊗ a) .

Proof. Let ∂ ∈ Der^{σ} (A) and a ∈ A, b ∈ A. The σ-Leibniz rule is

∂ (ab) = ∂ (a) b + μ ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1) (∂ ⊗ a ⊗ b)

and since

∂ (ab) = ev_{A,A} ∘ (ν^{r} ⊗ 1) (∂ ⊗ a ⊗ b) ,

when we consider ∂ simply as an internal homomorphism, and

μ ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1) (∂ ⊗ a ⊗ b) = ev_{A,A} ∘ (ν^{l} ⊗ 1) ∘ (σ ⊗ 1) (∂ ⊗ a ⊗ b) , clearly, by rearranging the Leibniz rule we get

∂ (a) = c_{σ} (∂ ⊗ a) .

Let ∂ ∈ Der^{σ} (A). By the σ_{q}-Leibniz rule we have ev_{A,A} ∘ (1 ⊗ μ) (Q_{q} (∂) ⊗ a ⊗ b)

= (μ ∘ (ev_{A,A} ⊗ 1) + μ ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1)) (∂ ⊗ a ⊗ b) .

Since

ev_{A,A} ∘ (1 ⊗ μ) (Q_{q} (∂) ⊗ a ⊗ b) = ev_{A,A} ∘ (ν^{r} ⊗ 1) (Q_{q} (∂) ⊗ a ⊗ b)

and

μ ∘ (1 ⊗ ev_{A,A}) ∘ (σ ⊗ 1) = ev_{A,A} ∘ (ν^{l} ⊗ 1) ∘ (σ ⊗ 1) ,

clearly, by rearranging, the evaluation of Q_{q} (∂) on a ∈ A_{q} is

ev_{A,A} (Q_{q} (∂) ⊗ a) = (ν^{r} − ν^{l} ∘ σ) (Q_{q} (∂) ⊗ a)
= c_{σ_{q}}^{q} (Q_{q} (∂) ⊗ a) .

4.5 Quantization of braided derivations of modules

Consider derivations of a σ-symmetric algebra A and a σ-symmetric A-module E. Let (∂, ∂_{A}) ∈ Der^{(σ,A)} (E).

Definition 27 Given a quantization q, define the quantization of (∂, ∂_{A}) by

Q_{q} (∂) (x) = ev_{A,A} ∘ q (∂ ⊗ x) ,

x ∈ E. If x ∈ A, then this is the quantization of ∂_{A} defined in section 4.3.

Q_{q} (∂) is an operator of the quantized module E_{q}. Denote by Der^{(σ_{q},A_{q})} (E_{q}) the set of all Q_{q} (∂), ∂ ∈ Der^{(σ,A)} (E), equipped with the quantized composition.

Theorem 28 Given a braiding σ, let σ_{q} be the quantization of σ. The operator

Q_{q} : (Der^{(σ,A)} (E) , c_{σ}) → (Der^{(σ_{q},A_{q})} (E_{q}) , c_{σ_{q}}^{q}) ,  (26)
∂ ∈ Der^{(σ,A)} (E) ⟼ Q_{q} (∂) ∈ Der^{(σ_{q},A_{q})} (E_{q}) ,

is an isomorphism of modules between the σ-derivations of E over A and the σ_{q}- derivations of E_{q} over A_{q}.

By proposition 1 is Der^{(σ_{q},A_{q})} (E_{q}) a σ_{q}-symmetric module and by theorem 9 a σ_{q}-Lie algebra with respect to the σ_{q} − q-bracket.

By theorem 13 the σ-Lie algebra of the structure of (Der^{(σ_{q},A_{q})} (E_{q}) , c_{σ_{q}}^{q}) can be realized within the classical one by dequantization.

5 Braided connections and curvatures

Let σ be a symmetry, A be a σ-symmetric algebra and E a σ-commutative A-module.

Definition 29 A σ-connection in E is a A-module homomorphism ∇

∇ : Der^{σ} (A) → Der^{(σ,A)} (E) ,

such that

π ∘ ∇ = Id.

Definition 30 A σ-connection ∇ is flat if it is a σ-Lie algebra homomorphism, that is,

∇ ∘ c_{σ} = c_{σ} ∘ (∇ ⊗ ∇) .

Definition 31 In general, define the σ-curvature of ∇ to be

K_{∇} : Der^{σ} (A) ⊗ Der^{σ} (A) → hom (E, E) ,
K_{∇} = c_{σ} ∘ (∇ ⊗ ∇) − ∇ ∘ c_{σ}.

MONOIDAL CATEGORIES 33

Theorem 32 The σ-curvature K_{∇} is a σ-homomorphism of E, that is,

ev_{E,E} ∘ (K_{∇} ⊗ ν^{l}) = ν^{l} ∘ (1 ⊗ ev_{E,E}) ∘ (σ ⊗ 1) ∘ (K_{∇} ⊗ 1 ⊗ 1) ,  (i)

which maps Der^{σ} (A) ⊗ Der^{σ} (A) ⊗ A ⊗ E to E, and it is skew σ-symmetric,

K_{∇} = −K_{∇} ∘ σ.  (ii)

Furthermore K_{∇} satisfies

K_{∇} ∘ (ν^{l} ⊗ 1) = ν^{l} ∘ K_{∇},  (iii)
K_{∇} ∘ (1 ⊗ ν^{l}) = ν^{l} ∘ K_{∇} ∘ (σ ⊗ 1) .  (iv)

Proof. (i): Note that

ev_{E,E} ∘ (K_{∇} ⊗ ν^{l}) = ev_{E,E} ∘ (1 ⊗ ν^{l}) ∘ (K_{∇} ⊗ 1 ⊗ 1) .

Then

ev_{E,E} ∘ (K_{∇} ⊗ ν^{l}) = ν^{l} ∘ (1 ⊗ ev_{E,E}) ∘ (σ ⊗ 1) ∘ (K_{∇} ⊗ 1 ⊗ 1)

if and only if

ev_{E,E} ∘ (1 ⊗ ν^{l}) = ν^{l} ∘ (1 ⊗ ev_{E,E}) ∘ (σ ⊗ 1) : hom (E, E) ⊗ A ⊗ E → E. By proposition 1 and by the symmetry of σ, v_{E′}^{l} ∘(1 ⊗ ev_{E,E′})∘(σ_{hom(E,E′),A} ⊗ 1) = ev_{E,E′} ∘(1 ⊗ v_{E}^{l}) : hom (E, E′)⊗A⊗E → E′ for σ-commutative A-modules E and E′.

(ii):

−K_{∇} ∘ σ = −c_{σ} ∘ (∇ ⊗ ∇) ∘ σ + ∇ ∘ c_{σ} ∘ σ
= −c_{σ} ∘ σ ∘ (∇ ⊗ ∇) + ∇ ∘ c_{σ} ∘ σ
= c_{σ} ∘ (∇ ⊗ ∇) − ∇ ∘ c_{σ}
= K_{∇},

by proposition 5.

(iii):

K_{∇} ∘ (ν^{l} ⊗ 1) = c_{σ} ∘ (∇ ⊗ ∇) ∘ (ν^{l} ⊗ 1) − ∇ ∘ c_{σ} ∘ (ν^{l} ⊗ 1)
= c_{σ} ∘ (ν^{l} ⊗ 1) ∘ (1 ⊗ ∇ ⊗ ∇) − ∇ ∘ c_{σ} ∘ (ν^{l} ⊗ 1)
= ν^{l} ∘ (1 ⊗ c_{σ}) ∘ (1 ⊗ ∇ ⊗ ∇) − ν^{l} ∘ (ev_{A,A} ⊗ 1) ∘ σ ∘ (1 ⊗ ∇ ⊗ ∇)
−∇ ∘ ν^{l} ∘ (1 ⊗ c_{σ}) + ∇ ∘ ν^{l} ∘ (ev_{A,A} ⊗ 1) ∘ σ
= ν^{l} ∘ (1 ⊗ c_{σ}) ∘ (1 ⊗ ∇ ⊗ ∇) − ν^{l} ∘ (1 ⊗ ∇) ∘ (1 ⊗ c_{σ})
= ν^{l} ∘ K_{∇},

if and only if

ν^{l} ∘ (ev_{A,A} ⊗ 1) ∘ σ ∘ (1 ⊗ ∇ ⊗ ∇) = ∇ ∘ ν^{l} ∘ (ev_{A,A} ⊗ 1) ∘ σ,

which clearly is satisfied.

(iv):

K_{∇} ∘ (1 ⊗ ν^{l}) = c_{σ} ∘ (∇ ⊗ ∇) ∘ (1 ⊗ ν^{l}) − ∇ ∘ c_{σ} ∘ (1 ⊗ ν^{l})
= c_{σ} ∘ (1 ⊗ ν^{l}) ∘ (∇ ⊗ 1 ⊗ ∇) − ∇ ∘ c_{σ} ∘ (1 ⊗ ν^{l})
= ν^{l} ∘ (1 ⊗ c_{σ}) ∘ (σ ⊗ 1) ∘ (∇ ⊗ 1 ⊗ ∇) + ν^{l} ∘ (ev_{A,A} ⊗ 1) ∘ (∇ ⊗ 1 ⊗ ∇)
−∇ ∘ ν^{l} ∘ (1 ⊗ c_{σ}) ∘ (σ ⊗ 1) − ∇ ∘ ν^{l} ∘ (ev_{A,A} ⊗ 1)
= ν^{l} ∘ (1 ⊗ c_{σ}) ∘ (1 ⊗ ∇ ⊗ ∇) ∘ (σ ⊗ 1) − ν^{l} ∘ (1 ⊗ ∇) ∘ (1 ⊗ c_{σ}) ∘ (σ ⊗ 1)
= ν^{l} ∘ K_{∇} ∘ (σ ⊗ 1) .

Note that

σ ∘ (∇ ⊗ 1) = (1 ⊗ ∇) ∘ σ

by the naturality of braidings.

5.1 Quantization of braided connections and curvatures

Let ∇ be a σ-connection in E. Then the quantization of ∇,

Q_{q} (∇) = ∇_{q} : Der^{σ^{q}} (A_{q}) → Der^{(σ_{q},A_{q})} (E_{q}) .

is defined by

∇_{q} =^{def} Q_{q} ∘ ∇ ∘ Q_{q}^{−1},  (27)

that is, the following diagram commutes

[figure: Der^{(σ,A)}(E); ∇; Der^{σ}(A); Der^{(σ_{q},A_{q})}(E_{q}); Q_{q}; ∇_{q}; Der^{σ^{q}}(A_{q}); Q_{q}].

Proposition 33 The quantization ∇_{q} is a σ_{q}-connection in E_{q}.

Proof. Clearly,

π_{q} ∘ ∇_{q} = Id,

as

π_{q} = Q_{q} ∘ π ∘ Q_{q}^{−1}.

Let ∇_{q} be a σ_{q}-connection in E_{q}. Then the σ_{q} − q-curvature of ∇_{q} (or simply σ_{q}-curvature when it is clear that the multiplication or composition is the quantized multiplication),

K_{∇_{q}}^{q} : Der^{σ^{q}} (A_{q}) ·_{σ_{q}} Der^{σ^{q}} (A_{q}) → End_{A_{q}} (E_{q}) ,

is defined by

K_{∇_{q}}^{q} = c_{σ_{q}}^{q} ∘ (∇_{q} ⊗ ∇_{q}) − ∇_{q} ∘ c_{σ_{q}}^{q}.

Let us state the properties of the σ_{q} − q-curvature.

Theorem 34 The σ_{q} − q-curvature is a σ_{q}-homomorphism, that is,

ev_{E,E} ∘ (K_{∇_{q}}^{q} ⊗ ν_{q}^{l}) = ν_{q}^{l} ∘ (1 ⊗ ev_{E,E}) ∘ (σ_{q} ⊗ 1) ∘ (K_{∇_{q}}^{q} ⊗ 1 ⊗ 1) ,  (i)

which maps Der^{σ^{q}} (A_{q}) ⊗ Der^{σ^{q}} (A_{q}) ⊗ A_{q} ⊗ E_{q} to E_{q}, and is skew σ_{q}-symmetric,

K_{∇_{q}}^{q} = −K_{∇_{q}}^{q} ∘ σ_{q}.  (ii)

Furthermore K_{∇_{q}}^{q} satisfies

K_{∇_{q}}^{q} ∘ (ν_{q}^{l} ⊗ 1) = ν_{q}^{l} ∘ (1 ⊗ K_{∇_{q}}^{q}) ,  (iii)
K_{∇_{q}}^{q} ∘ (1 ⊗ ν_{q}^{l}) = ν_{q}^{l} ∘ K_{∇_{q}}^{q} ∘ (σ_{q} ⊗ 1) ,  (iv)

∀a ∈ A.

Proof. (i): See the proof of (i) of theorem 32.

(ii):

K_{∇_{q}}^{q} = c_{σ_{q}}^{q} ∘ (∇_{q} ⊗ ∇_{q}) − ∇_{q} ∘ c_{σ_{q}}^{q}
= −c_{σ_{q}}^{q} ∘ σ_{q} ∘ (∇_{q} ⊗ ∇_{q}) + ∇_{q} ∘ c_{σ_{q}}^{q} ∘ σ_{q}
= −K_{∇_{q}} ∘ σ_{q}.

(iii):

K_{∇_{q}}^{q} ∘ (ν_{q}^{l} ⊗ 1)
= c_{σ_{q}}^{q} ∘ (∇_{q} ⊗ ∇_{q}) ∘ (ν_{q}^{l} ⊗ 1) − ∇_{q} ∘ c_{σ_{q}}^{q} ∘ (ν_{q}^{l} ⊗ 1)
= c_{σ_{q}}^{q} ∘ (ν_{q}^{l} ⊗ 1) ∘ (1 ⊗ ∇_{q} ⊗ ∇_{q}) − ∇_{q} ∘ c_{σ_{q}}^{q} ∘ (ν_{q}^{l} ⊗ 1)
= ν_{q}^{l} ∘ (1 ⊗ c_{σ_{q}}^{q}) ∘ (1 ⊗ ∇_{q} ⊗ ∇_{q}) − ν_{q}^{l} ∘ (ev_{A,A} ⊗ 1) ∘ σ_{q} ∘ (1 ⊗ ∇_{q} ⊗ ∇_{q})
−∇_{q} ∘ ν_{q}^{l} ∘ (1 ⊗ c_{σ_{q}}^{q}) + ∇_{q} ∘ ν_{q}^{l} ∘ (ev_{A,A} ⊗ 1) ∘ σ_{q}
= ν_{q}^{l} ∘ (1 ⊗ c_{σ_{q}}^{q}) ∘ (1 ⊗ ∇_{q} ⊗ ∇_{q}) − ν_{q}^{l} ∘ (1 ⊗ ∇_{q}) ∘ (1 ⊗ c_{σ_{q}}^{q})
= ν_{q}^{l} ∘ (1 ⊗ K_{∇_{q}}^{q})

if and only if,

ν_{q}^{l} ∘ (ev_{A,A} ⊗ 1) ∘ σ_{q} ∘ (1 ⊗ ∇_{q} ⊗ ∇_{q}) = ∇_{q} ∘ ν_{q}^{l} ∘ (ev_{A,A} ⊗ 1) ∘ σ_{q},

which clearly is satisfied. (iv):

K_{∇_{q}}^{q} ∘ (1 ⊗ ν_{q}^{l}) = c_{σ_{q}}^{q} ∘ (∇_{q} ⊗ ∇_{q}) ∘ (1 ⊗ ν_{q}^{l}) − ∇_{q} ∘ c_{σ_{q}}^{q} ∘ (1 ⊗ ν_{q}^{l})
= ν_{q}^{l} ∘ (1 ⊗ c_{σ_{q}}^{q}) ∘ (σ_{q} ⊗ 1) ∘ (1 ⊗ ∇_{q} ⊗ ∇_{q}) + ν_{q}^{l} ∘ (ev_{A,A} ⊗ 1) ∘ (1 ⊗ ∇_{q} ⊗ ∇_{q})
−∇_{q} ∘ ν_{q}^{l} ∘ (1 ⊗ c_{σ_{q}}^{q}) ∘ (σ_{q} ⊗ 1) − ∇_{q} ∘ ν_{q}^{l} ∘ (ev_{A,A} ⊗ 1)
= ν_{q}^{l} ∘ (1 ⊗ c_{σ_{q}}^{q}) ∘ (σ_{q} ⊗ 1) ∘ (1 ⊗ ∇_{q} ⊗ ∇_{q}) − ∇_{q} ∘ ν_{q}^{l} ∘ (1 ⊗ c_{σ_{q}}^{q}) ∘ (σ_{q} ⊗ 1)
= ν_{q}^{l} ∘ K_{∇_{q}}^{q} ∘ (σ_{q} ⊗ 1) .

We have the following condition for the braided curvature of dequantizations of braided derivations.

Theorem 35 The σ_{q}-curvature of the connection ∇_{q} defined by

K_{∇_{q}} = c_{σ_{q}} ∘ (∇_{q} ⊗ ∇_{q}) − ∇_{q} ∘ c_{σ_{q}},  (28)

and the σ_{q} − q-curvature K_{∇}^{q} of the σ-connection ∇ of E defined by

K_{∇}^{q} = c_{σ_{q}}^{q} ∘ (∇ ⊗ ∇) − ∇ ∘ c_{σ_{q}}^{q},  (29)

are related as follows,

Q_{q}^{−1} ∘ K_{∇_{q}} = K_{∇}^{q} ∘ (Q_{q}^{−1} ⊗ Q_{q}^{−1}) .  (30)

Proof. Proof of (30):

K_{∇_{q}} = c_{σ_{q}} ∘ (∇_{q} ⊗ ∇_{q}) − ∇_{q} ∘ c_{σ_{q}}
= Q_{q} ∘ c_{σ_{q}}^{q} ∘ (Q_{q}^{−1} ⊗ Q_{q}^{−1}) ∘ ((Q_{q} ∘ ∇ ∘ Q_{q}^{−1}) ⊗ (Q_{q} ∘ ∇ ∘ Q_{q}^{−1}))
−Q_{q} ∘ ∇ ∘ Q_{q}^{−1} ∘ Q_{q} ∘ c_{σ_{q}}^{q} ∘ (Q_{q}^{−1} ⊗ Q_{q}^{−1})
= Q_{q} ∘ c_{σ_{q}}^{q} ∘ (∇ ⊗ ∇) ∘ (Q_{q}^{−1} ⊗ Q_{q}^{−1}) − Q_{q} ∘ ∇ ∘ c_{σ_{q}}^{q} ∘ (Q_{q}^{−1} ⊗ Q_{q}^{−1})
= Q_{q} ∘ K_{∇}^{q} ∘ (Q_{q}^{−1} ⊗ Q_{q}^{−1}) .

6 Braided differential operators

We shall see how the picture is for braided differential operators.

6.1 Braided differential operators in algebras

Let σ be a braiding and A be a σ-symmetric algebra.

Recall that the module

(hom (A, A))_{σ}^{(k)} = Diff_{k}^{σ} (A, A)

is called the braided or σ-differential operators in A order at most k.

An equivalent and more familiar way to define a σ-differential operator of order at most k is the linear map

f : A → A,

such that

c_{σ} ∘ (1 ⊗ c_{σ}) ∘ ⋯ ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ c_{σ}) (a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{k} ⊗ f) = 0  (31)

∀a_{0}, …, a_{k} ∈ A. Denote by Diff_{k}^{σ} (A, A) the set of σ-differential operators of order at most k. Note,

f ∈ Diff_{k}^{σ} (A, A) ⇔ c_{σ} (f ⊗ a) ∈ Diff_{k−1}^{σ} (A, A) , ∀a ∈ A.

Let Diff^{σ} (A, A) = ∪Diff_{k}^{σ} (A, A).

From [15] we have the following two results.

MONOIDAL CATEGORIES 37

Proposition 36 The σ-commutator of two σ-differential operators:

f ∈ Diff_{i}^{σ} (A, A) and g ∈ Diff_{j}^{σ} (A, A)

is a σ-differential operator of order at most i + j − 1,

c_{σ} (f ⊗ g) ∈ Diff_{i+j−1}^{σ} (A, A) .

The next result also follows from theorem 9. Corollary 37 If σ is a symmetry and an algebra A is σ-symmetric then Diff^{σ} (A, A) is a σ-Lie algebra. Proposition 38 There is an A − A-module structure on Diff^{σ} (A, A) defined by

ν_{a}^{l} (f) (b) = ν^{l} (a ⊗ f) (b) = af (b) ,  (32)
ν_{a}^{r} (f) (b) = ν^{r} (f ⊗ a) (b) = f (ab) ,  (33)

a, b ∈ A, f ∈ Diff^{σ} (A, A), and ν^{l} (a ⊗ f), ν^{r} (f ⊗ a) ∈ Diff_{k}^{σ} (A, A), for f ∈ Diff_{k}^{σ} (A, A). Proof. Let

σ^{i,i+1} = (\underbrace{1 ⊗ ⋯ ⊗ 1}_{i−1} ⊗ σ ⊗ \underbrace{1 ⊗ ⋯ ⊗ 1}_{(k+3)−(i+1)}) ,

than the left action on a braided differential operator again is a braided differential operator,

c_{σ} ∘ (1 ⊗ c_{σ}) ∘ ⋯ ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ c_{σ}) ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k+1} ⊗ ν^{l})

= c_{σ} ∘ ⋯ ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k−1} ⊗ c_{σ}) ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ ((ν^{l} + ev_{A,A} ∘ σ) ∘ (1 ⊗ ν^{l})))
= c_{σ} ∘ ⋯ ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ ((ν^{l} ∘ (μ ⊗ 1) + μ ∘ (1 ⊗ ev_{A,A}) ∘ (1 ⊗ σ)) ∘ (σ ⊗ 1)))
= c_{σ} ∘ ⋯ ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ ν^{l}) ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k+1} ⊗ c_{σ}) ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ σ ⊗ 1)

⋮

= ν^{l} ∘ (1 ⊗ c_{σ}) ∘ ⋯ ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k+1} ⊗ c_{σ}) ∘ σ^{1,2} ∘ ⋯ ∘ σ^{k+1,k+2}
= 0,

when applied to a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{k} ⊗ b ⊗ f, and also the right action on a braided differential operator again is a braided differential operator,

c_{σ} ∘ (1 ⊗ c_{σ}) ∘ ⋯ ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ c_{σ}) ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k+1} ⊗ ν^{r})

= c_{σ} ∘ ⋯ ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k−1} ⊗ c_{σ}) ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ ((ν^{l} + ev_{A,A} ∘ σ) ∘ (1 ⊗ ν^{r})))

= c_{σ} ∘ ⋯ ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ (ν^{r} ∘ (ν^{l} ⊗ 1) + ev_{A,A} ∘ (1 ⊗ μ) ∘ (1 ⊗ σ) ∘ (σ ⊗ 1)))

= c_{σ} ∘ ⋯ ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ ν^{r}) ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ c_{σ} ⊗ 1)

⋮

= ν^{r} ∘ (c_{σ} ⊗ 1) ∘ ⋯ ∘ (\underbrace{1 ⊗ ⋯ ⊗ 1}_{k} ⊗ c_{σ} ⊗ 1)
= 0,

when applied to a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{k} ⊗ f ⊗ b, a_{0}, …, a_{k}, b ∈ A, f ∈ Diff_{k}^{σ} (A, A).

Consider the symbol of the differential operators which is the leading part with respect to derivatives,

Smbl_{k}^{σ} (A, A) = Diff_{k}^{σ} (A, A) /Diff_{k−1}^{σ} (A, A) ,

then we have the ℤ-graded object

Smbl^{σ} (A, A) = ∑ Smbl_{k}^{σ} (A, A) .

The class of [f, g]^{σ} ∈ Diff_{i+j−1}^{σ} (A, A),

\overline{[f, g]^{σ}} ∈ Smbl_{i+j−1}^{σ} (A, A) ,

depends on the class of the two σ-differential operators f ∈ Diff_{i}^{σ} (A, A) and g ∈ Diff_{j}^{σ} (A, A), hence there is a σ-Poisson structure on the braided symbol algebra, [15].

6.2 Braided differential operators in modules

Let σ be a braiding, A be a σ-symmetric algebra and let E be a σ-symmetric Amodule.

Definition 39 The module

(hom (E, E))_{σ}^{(k)} = Diff_{k}^{σ} (E, E)

is called the braided or σ-differential operators in order at most k of E.

Denote the σ-differential operators of order at most k in E by Diff_{k}^{(σ,A)} (E, E) and we consider Diff^{(σ,A)} (E, E) = ∪Diff_{k}^{(σ,A)} (E, E).

From [15] we have the following two results.

Proposition 40 The σ-commutator of two σ-differential operators

f ∈ Diff_{i}^{(σ,A)} (E, E) and g ∈ Diff_{j}^{(σ,A)} (E, E) ,

is given by

[f, g]^{σ} ∈ Diff_{i+j}^{(σ,A)} (E, E) ,

and so has order at most i + j.

Corollary 41 If σ is a symmetry and an algebra A and a left A-module E are σsymmetric then Diff^{σ} (E, E) is a σ-Lie algebra.

We consider the A − A-module structure on Diff^{σ} (E, E).

Proposition 42 There is an A − A-module structure on Diff^{σ} (E, E) defined by

ν_{a}^{l} (f) (x) = ν^{l} (a ⊗ f) (x) = af (x) ,
ν_{a}^{r} (f) (x) = ν^{r} (f ⊗ a) (x) = f (ax) ,

a ∈ A, x ∈ E, f ∈ Diff^{σ} (E, E).

Proof. The proof is the same as for proposition 38.

Consider the symbol of the differential operators which is the leading part with respect to derivatives,

Smbl_{k}^{σ} (E, E) = Diff_{k}^{σ} (E, E) /Diff_{k−1}^{σ} (E, E) ,

then we have the ℤ-graded object

Smbl^{σ} (E, E) = ∑ Smbl_{k}^{σ} (E, E) .

The class of [f, g]^{σ} ∈ Diff_{i+j}^{σ} (E, E),

\overline{[f, g]^{σ}} ∈ Smbl_{i+j}^{σ} (E, E) ,

depends on the class of the two σ-differential operators f ∈ Diff_{i}^{σ} (E, E) and g ∈ Diff_{j}^{σ} (E, E), hence there is a σ-Poisson structure on the braided symbol algebra.

6.3 Quantizations of braided differential operators in algebras

We can define quantization of σ-differential operators in algebras. Let A be a σcommutative algebra.

Definition 43 Given a quantization q and f ∈ Diff^{σ} (A, A) define the quantization of f by

Q_{q} (f) (a) = f_{q} (a) =^{def} ev_{A,A} ∘ q (f ⊗ a) ,

a ∈ A.

Q_{q} (f) is an operator of the quantized algebra A_{q}. From [14] we have the following theorem.

Theorem 44 Given a braiding σ, let σ_{q} be the quantization of σ. The operator

Q_{q} : (Diff^{σ} (A, A) , c_{σ}) → (Diff^{σ^{q}} (A_{q}, A_{q}) , c_{σ_{q}}^{q}) ,  (34)
f ∈ Diff^{σ} (A, A) ⟼ Q_{q} (f) ∈ Diff^{σ^{q}} (A_{q}, A_{q}) ,

is an isomorphism of modules.

The symbol of Q_{q} is an isomorphism of modules

Smbl (Q_{q}) : (Smbl^{σ} (A, A) , c_{σ}) → (Smbl^{σ^{q}} (A_{q}, A_{q}) , c_{σ_{q}}^{q}) ,  (35)
f ∈ Smbl^{σ} (A, A) ⟼ Smbl (Q_{q}) (f) ∈ Smbl^{σ^{q}} (A_{q}, A_{q}) .

By proposition 38 is Diff^{σ^{q}} (A_{q}, A_{q}) a σ_{q}-symmetric module. By corollary 37, if σ is a symmetry then Diff^{σ^{q}} (A_{q}, A_{q}) is a σ_{q}-Lie algebra with respect to the σ_{q} − qbracket and the quantized composition. Furthermore there is a σ_{q}-Poisson structure on the quantized braided symbol algebra Smbl^{σ^{q}} (A_{q}, A_{q}).

By theorem 13 the σ_{q}-Lie algebra structure of (Diff^{σ^{q}} (A_{q}, A_{q}) , c_{σ_{q}}^{q}) can be realized within the classical one by dequantization.

6.4 Quantizations of braided differential operators in modules

Let A be a σ-symmetric algebra and let E be a σ-symmetric A-module.

Definition 45 Given a quantization q and f ∈ Diff^{(σ,A)} (E, E) define the quantization of f by

Q_{q} (f) (x) = f_{q} (x) =^{def} ev_{E,E} ∘ q (f ⊗ x) ,

x ∈ E.

Q_{q} (f) is an operator of the quantized module E_{q}.

Theorem 46 Given a braiding σ, let σ_{q} be the quantization of σ. The operator

Q_{q} : (Diff^{(σ,A)} (E, E) , c_{σ}) → (Diff^{(σ_{q},A_{q})} (E_{q}, E_{q}) , c_{σ_{q}}^{q}) ,  (36)
f ∈ Diff^{(σ,A)} (E, E) ⟼ Q_{q} (f) ∈ Diff^{(σ_{q},A_{q})} (E_{q}, E_{q}) ,

is an isomorphism of modules.

The symbol of Q_{q} is an isomorphism of modules

Smbl (Q_{q}) : (Smbl^{(σ,A)} (E, E) , c_{σ}) → (Smbl^{(σ_{q},A_{q})} (E_{q}, E_{q}) , c_{σ_{q}}^{q}) ,  (37)
f ∈ Smbl^{(σ,A)} (E, E) ⟼ Smbl (Q_{q}) (f) ∈ Smbl^{(σ_{q},A_{q})} (E_{q}, E_{q}) .

Proof. The isomorphism as σ-differential operators and σ-symbols is shown in [14].

By proposition 42 is Diff^{(σ_{q},A_{q})} (E_{q}, E_{q}) a σ_{q}-symmetric module. By corollary 41, if σ is a symmetry then Diff^{(σ_{q},A_{q})} (E_{q}, E_{q}) is a σ_{q}-Lie algebra with respect to the σ_{q} − q-bracket and the quantized composition. Furthermore there is a σ_{q}-Poisson structure on the quantized braided symbol algebra, Smbl^{(σ_{q},A_{q})} (E_{q}, E_{q}).

By theorem 13 the σ_{q}-Lie algebra structure of (Diff^{(σ_{q},A_{q})} (E_{q}, E_{q}) , c_{σ_{q}}^{q}) can be realized within the classical one by dequantization.

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Department of Mathematics, The University of Tromsoe, N-9037 Tromsoe, Norway E-mail address: Hilja.Huru@matnat.uit.no

Received November 7, 2006