Abstract. For the monoidal category of modules with action by a group we ﬁnd 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 or symmetries

σ

and quantizations of braidings

σ_{q}

of the monoidal category of

G

-modules where

G

is a ﬁnite abelian group.

In [8] we showed that the Fourier transform establishes an isomorphism between the categories of

\hat{G}

-graded modules and

G

-modules where

\hat{G}

is the dual of

G

. We have found explicit descriptions of all quantizations and braidings in the monoidal category of modules graded by

\hat{G}

and they depend only on the grading.

Using this we ﬁnd explicit descriptions of all quantizations and braiding also for the monoidal category of modules with action by

G

We consider

σ

-symmetric

G

-algebras

A

G

-modules and ﬁnd quantizations of these.

We investigate braided derivations of

σ

-symmetric

G

-algebras and

G

-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 quantization 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

G

-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.

We consider braided differential operators of

σ

-symmetric

G

-algebras and

G

-modules. There is a braided Lie structure on the braided differential operators. The quantizations of braided differential operators has the quantized braided Lie structure, which can be realized within the original braided Lie structure by dequantization.

Finally we increase the number of quantizations of braided symbols of differential operators by exploiting the

ℤ

-grading of the symbols.

As mentioned, we have found explicit descriptions of all quantizations and braidings in the monoidal category of modules graded by a ﬁnite commutative monoid, [8]. We have proved the same for this category, but the picture is somewhat more visible in this case. That is, we have a complete and explicit description for braided derivations of graded algebras and graded modules, braided connections, braided curvature, braided differential operators and quantizations of these structures. This is found in the second paper Quantizations of braided derivations. 2. Graded modules, [10].

All results and properties that does not concern the correspondence between the monoidal categories of

\hat{G}

-graded modules and

G

-modules are proved in general for any monoidal category in the ﬁrst paper, Quantizations of braided derivations. 1. Monoidal categories, [9]. Because of this we shall not repeat any of these proofs. We do however repeat the relevant results and show the complete and explicit description we get in the monoidal category of

G

-modules.

There are many interesting applications of these results. One of the more interesting applications is quantizations of braided Lie algebras. In the paper [11], which is to be published, we show quantizations of semisimple Lie algebras by quantizations of derivations, for example an alternative quantization of

s l_{2} (ℂ)

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

2. Modules with group action

Let

G

be a ﬁnite abelian group and let

\hat{G} = H o m (G, T^{1})

be the dual of

G

. We shall consider the monoidal categories of

\hat{G}

-graded modules and

G

-modules. The modules are over

ℂ

Let

ℂ [G]

be the group algebra of

G

, with the basis of Dirac

δ

-functions,

{\{δ_{g}\}}_{g \in G}

, and

ℂ (\hat{G})

be the function algebra of

\hat{G}

, with the basis

{\{θ_{χ}\}}_{χ \in \hat{G}}

We consider the two monoidal categories of

G

-modules and

\hat{G}

-graded modules.

We know that a

ℂ [G]

-module is also a

G

-module, that

ℂ (\hat{G})

-module is a

\hat{G}

-graded module, and the other way around for both cases. Then

F^{- 1}

constructs an isomorphism of categories of between the monoidal categories of

\hat{G}

-graded modules and

G

-modules as follows.

Let

X

be a

G

-module. Then there is a

\hat{G}

-grading on

X

by the projection

x \in X

χ \in \hat{G}

. Let

Y = \sum_{χ \in \hat{G}} Y_{χ}

be a

\hat{G}

-graded

ℂ

-module. There is a

G

-module structure on

Y

given by

We use this result to ﬁnd quantizations and braidings for

G

-modules, which can be found in [8], and in other ways compare

G

-modules with

\hat{G}

-graded modules.

2.1. Quantizations of

G

-modules. Any quantization of the monoidal category of

G

-modules is a realized by a

2

-cocycle

q \in Z^{2} (G, U (R))

g, h \in G

, where

\hat{q}

is a quantization of the monoidal category of

\hat{G}

-graded modules. When we factor out the trivial quantizations we are left with

H^{2} (M, U (R))

. The quantization as an operator

X \otimes Y \to X \otimes Y

G

-modules

X

and

Y

q

is of the form

2.2. Braidings of

G

-modules. Any braiding in the monoidal category of

G

-modules is a

2

-cochain

σ : G \times G \to U (ℂ)

g, h \in G

, where

\hat{σ}

is a braiding of the monoidal category of

\hat{G}

-graded modules. As an operator

X \otimes Y \to Y \otimes X

G

-modules

X

and

Y

σ

is of the form

where

{\hat{σ}}_{q} = q^{- 1} \circ \hat{σ} \circ q

is the quantization of the braiding

\hat{σ}

of the monoidal category of

\hat{G}

-graded modules.

Using these formulas obtained for quantizations and braidings in the monoidal category of

G

-modules we get an explicit description of

σ

-symmetric

G

-algebras,

G

-modules,

G

-co- and bimodules and internal homomorphisms. We shall not describe quantizations of

σ

-symmetric

G

-co- and bimodules, these are described for any monoidal category in [9].

2.3. Quantizations of braided

G

-algebras. First the description of quantization of

σ

-symmetric

G

-algebras.

A quantization of

A

is the same object

A_{q} = A

with the new multiplication

2.4. Quantizations of braided

G

-modules. We also get an explicit description of

σ

-symmetric

G

-modules over

σ

-symmetric

G

-algebras.

Let

E

be a

σ

-symmetric

G

-module over a

σ

-symmetric

G

-algebra

A

E

σ

-symmetric if

A quantizations of

E

is the same object

E = E_{q}

with the new action by

A_{q} = A

2.5. Operators of

G

-modules. Let

E

be a

G

-module and

W : E \to E

be an operator of

E

. There is a induced

G

-action on

W

, deﬁned by

α \in \hat{G}

, that is,

W

is a twisted

α

-homomorphism, denoted by

W \in H o m_{G}^{α} (E, E)

2.6. Quantizations of operators of

G

-modules. Let

E

be a

G

-module and let

W : E \to E

be a

G

-operator of

E

Given a quantization

q

deﬁne the quantization of

W

W_{q} = Q_{q} (W)

The quantization of all operators of

E

is equipped with the quantized composition of operators,

is called the dequantization of

W

. The dequantization of

W

is an operator of the original or classical module

E

3. Braided derivations of

G

-algebras

We shall describe braided derivations of

G

-algebras, but ﬁrst we will show how braided derivations of non-homogeneous elements of graded algebras look like and what the notion of degree is for braided derivations of

G

-algebras.

3.1. Braided derivations of non-homogeneous elements of graded algebras. To ﬁnd derivations of

G

-algebras a description of non-homogeneous derivations of non-homogeneous elements of graded algebras (over a commutative ring

R

) is needed.

For a

\hat{G}

-graded module

E

and an operator

W = \sum_{χ \in \hat{G}} W_{χ}

E

we have,

Similarly, for a

\hat{σ}

-symmetric

\hat{G}

-graded algebra

A = \sum_{χ \in \hat{G}} A_{χ}

, and a

\hat{σ}

-derivation of

A

where each

{\hat{\partial}}_{χ}

is a derivation of degree

χ \in \hat{G}

, we have

3.2. Degrees of braided derivations. Also, it is nice to know the equivalent to degrees for

σ

-derivations of

G

-algebras.

Let

A

be a

\hat{G}

-graded

\hat{σ}

-symmetric algebra over

ℂ

with respect to a braiding

\hat{σ}

in the monoidal category of

\hat{G}

-graded modules.

Let

\hat{\partial}

be a

\hat{σ}

-derivations of degree

∣ \hat{\partial} ∣

A

\hat{\partial} : A_{χ} \to A_{χ + ∣ \hat{\partial} ∣}

, that is, satisﬁes

for

a \in A

χ, ∣ \hat{\partial} ∣ \in \hat{G}

θ_{χ}

and

θ_{χ + ∣ \hat{\partial} ∣}

, are in the basis of

ℂ (\hat{G})

, and

3.3. Braided derivations of

G

-algebras. Consider a braiding

σ

in the monoidal category of

G

-modules,

where

\hat{σ}

is a braiding in the monoidal category of

\hat{G}

-graded modules. Let

A

be a

σ

-symmetric

G

-algebra.

σ

-derivation

\partial_{χ}

of degree

χ_{\partial} \in \hat{G}

satisﬁes the simpliﬁed

σ

-Leibniz rule

The set of

σ

-derivations of degree

χ

is denoted by

D e r_{χ}^{σ} (A)

and the set of all

σ

-derivations by

D e r^{σ} (A)

σ

-commutator (or

σ

-bracket) of elements

\partial_{1}, \partial_{2} \in D e r^{σ} (A)

is deﬁned by

Theorem 5. $D e r^{σ} (A)$ is a $σ$ -Lie $G$ -algebra with respect to the $σ$ -bracket, that is, the following properties are satisﬁed;

{[D e r^{σ} (A), D e r_{}^{σ} (A)]}^{σ} \subseteq D e r_{}^{σ} (A),

(i)

{[D e r_{φ}^{σ} (A), D e r_{χ}^{σ} (A)]}^{σ} \subseteq D e r_{φ + χ}^{σ} (A),

(i’)

$φ, χ \in \hat{G}$ , skew $σ$ -symmetricity,

{[\partial_{1}, \partial_{2}]}^{σ} = - \sum_{g, h \in G} S_{g h} {[h \partial_{2}, g \partial_{1}]}^{σ},

(ii)

and the $σ$ -Jacobi identity for $σ$ -derivations,

{[\partial_{1}, {[\partial_{2}, \partial_{3}]}^{σ}]}^{σ} = {[{[\partial_{1}, \partial_{2}]}^{σ}, \partial_{3}]}^{σ} + \sum_{g, h \in G} S_{g h} {[h \partial_{2}, {[g \partial_{1}, \partial_{3}]}^{σ}]}^{σ},

(iii)

for braided derivations $\partial_{1}$ , $\partial_{2}$ and $\partial_{3}$ .

Proof. Except from $(i^{'})$ all those above are proved for any monoidal category in [9]. Now

{[D e r_{φ}^{σ} (A), D e r_{χ}^{σ} (A)]}^{σ} \subseteq D e r_{φ + χ}^{σ} (A)

if and only if

{[\partial_{1}, \partial_{2}]}^{σ} \circ g = (φ + χ) (g) g \circ {[\partial_{1}, \partial_{2}]}^{σ},

which is is easy to prove,

\begin{array}{l} {[\partial_{1}, \partial_{2}]}^{σ} \circ g = \partial_{1} \circ \partial_{2} \circ g - \sum_{h, h^{'} \in G} S_{h, h^{'}} h^{'} (\partial_{2}) \circ h (\partial_{1}) \circ g \\ = χ (g) φ (g) g \circ \partial_{1} \circ \partial_{2} - φ (g) χ (g) g \circ \sum_{h, h^{'} \in G} S_{h, h^{'}} h^{'} (\partial_{2}) \circ h (\partial_{1}) \\ = (φ + χ) (g) g \circ {[\partial_{1}, \partial_{2}]}^{σ}, \partial_{1} \in D e r_{φ}^{σ} (A), \partial_{2} \in D e r_{χ}^{σ} (A) . \end{array}

3.4. Quantizations of

σ

-derivations of

G

-algebras. Consider derivations of a

G

-algebra

A

Given a quantization

q

deﬁne the quantization of a

σ

-derivation of

A

\partial

, by

Q_{q} (\partial)

is an operator of the quantized

G

-algebra

A_{q}

. Let us denote by

D e r^{σ_{q}} (A_{q})

the quantization of all

σ

-derivations of

A

equipped with the quantization of composition,

\partial_{1}, \partial_{2} \in D e r^{σ_{q}} (A_{q})

. We deﬁne the

σ_{q} - q

-bracket by

is an isomorphism of vector spaces between the

σ

-derivations of

A

and the

σ_{q}

-derivations of

A_{q}

Theorem 6. $D e r^{σ_{q}} (A_{q})$ is a $σ_{q}$ -Lie $G$ -algebra with respect to the $σ_{q} - q$ -bracket and the quantized combination, that is the following properties are satisﬁed,

{[D e r^{σ_{q}} (A_{q}), D e r_{}^{σ_{q}} (A_{q})]}_{q}^{σ_{q}} \subseteq D e r_{}^{σ_{q}} (A_{q}),

(i)

{[D e r_{φ}^{σ_{q}} (A_{q}), D e r_{χ}^{σ_{q}} (A_{q})]}_{q}^{σ_{q}} \subseteq D e r_{φ + χ}^{σ_{q}} (A_{q}),

(i’)

$φ, χ \in \hat{G}$ , skew $σ_{q}$ -symmetricity,

{[\partial_{1}, \partial_{2}]}_{q}^{σ_{q}} = - \sum_{g, h \in G} S_{g h}^{q} {[h \partial_{2}, g \partial_{1}]}_{q}^{σ_{q}},

(ii)

and the $σ_{q}$ -Jacobi identity for $σ_{q}$ -derivations,

{[\partial_{1}, {[\partial_{2}, \partial_{3}]}_{q}^{σ_{q}}]}_{q}^{σ_{q}} = {[{[\partial_{1}, \partial_{2}]}_{q}^{σ_{q}}, \partial_{3}]}_{q}^{σ_{q}} + \sum_{g, h \in G} S_{g h}^{q} {[h \partial_{2}, {[g \partial_{1}, \partial_{3}]}_{q}^{σ_{q}}]}_{q}^{σ_{q}},

(iii)

for $\partial_{1}, \partial_{2}, \partial_{3} \in D e r^{σ_{q}} (A_{q})$ .

The

σ_{q}

-Lie algebra structure of

D e r^{σ_{q}} (A_{q})

can be realized within the classical one by dequantization,

\partial_{1}, \partial_{2} \in D e r_{∣ W ∣}^{σ_{q}} (A_{q})

A_{q}

-module structure,

for

\partial_{1}, \partial_{2} \in D e r^{σ_{q}} (A_{q})

a \in A_{q}

, and the

σ_{q}

-Jacobi identity is satisﬁed and can be found using (18).

4. Braided derivations of

G

-modules

Let

A

be a

σ

-symmetric

G

-algebra and

E

be a

σ

-symmetric

G

-module over

A

Such a pair

(\partial, \partial_{A})

is called a

σ

-derivation of

E

over

A

The morphism

π : (\partial, \partial_{A}) \to \partial_{A}

we call the projection from the

σ

-derivations of

E

over

A

to the

σ

-derivations of

A

σ

-derivation

\partial

E

over

\partial_{A}

is said to be of degree

∣ \partial ∣ \in \hat{G}

if it is an

(G, ∣ \partial ∣)

-operator.

The set of

σ

-derivations of degree

χ

is denoted by

D e r_{χ}^{(σ, A)} (E)

and the set of all

σ

-derivations by

D e r^{(σ, A)} (E)

The

σ

-bracket satisﬁes the

A

-module conditions (11) and (12) for

σ

-derivations of

E

over

A

D e r^{(σ, A)} (E)

4.1. Quantization of braided derivations of

G

-modules. Consider derivations of a

σ

-symmetric

G

-algebra

A

and a

σ

-symmetric

A

-module

E

Let

(\partial, \partial_{A}) \in D e r^{(σ, A)} (E)

. Given a quantizer

q

deﬁne the quantization of

\partial

x \in E

. If

x \in A

, this is the quantization of the derivation of

A

\partial_{A}

Denote by

D e r^{(σ_{q}, A_{q})} (E_{q})

the quantization of all

σ

-derivations of

E

over

A

equipped with the quantization of composition.

is an isomorphism of modules between the

σ

-derivations of

E

over

A

and the

σ_{q}

-derivations of

E_{q}

over

A_{q}

D e r^{(σ_{q}, A_{q})} (E_{q})

is a

σ_{q}

-Lie

G

-algebra with respect to the

σ_{q} - q

-bracket, that is the following properties are satisﬁed,

φ, χ \in \hat{G}

, skew

σ_{q}

-symmetricity,

{[,]}_{q}^{σ_{q}} = - {[,]}_{q}^{σ_{q}} \circ σ_{q}

, and the

σ_{q}

-Jacobi identity, see (ii) and (iii) in theorem 6.

The

σ_{q}

-Lie algebra of the structure of

(D e r^{(σ_{q}, A_{q})} (E_{q}), {[,]}_{q}^{σ_{q}})

can be realized within the classical one by dequantization,

\partial \in D e r^{(σ_{q}, A_{q})} (E_{q})

, where we have the linearity (16), the

A_{q}

-module structure (17), the commutator satisﬁes (18) and the braided Jacobi identity is satisﬁed and can be written in terms of the

σ_{q} - q

-bracket using (18).

(G, σ)

-connections

The properties of

K_{\nabla}

are the following.

K_{\nabla}

satisﬁes linearity,

for

a \in A

x \in E

and braided derivations

\partial_{1}, \partial_{2} \in D e r^{σ} (A)

5.1. Quantizations of

(G, σ)

-connections and curvatures. Let

\nabla

be a

G

-invariant

(G, σ)

-connection.

Given a quantization

q

, deﬁne the quantization of the

(G, σ)

-connection

\nabla

The connection

\nabla_{q}

is a

σ_{q}

-connection in

E_{q}

which is easy to see as

π_{q} = Q_{q} \circ π \circ Q_{q}^{- 1}

. Furthermore is the

σ_{q}

-connection

\nabla_{q}

(G, σ_{q})

-connection, that is,

The quantization of the

σ

-curvature, the

σ_{q} - q

-curvature

K_{\nabla_{q}}^{q}

, is deﬁned

and satisﬁes the

A_{q}

-module homomorphism conditions with respect to

σ_{q}

for

a \in A_{q}

x \in E_{q}

and braided derivations

\partial_{1}, \partial_{2} \in D e r^{σ_{q}} (A_{q})

Theorem 10. The $σ_{q} - q$ -curvature $K_{\nabla}^{q}$ of the $σ$ -connection $\nabla$ of $E$ deﬁned by

K_{\nabla}^{q} ({(\partial_{1})}_{c}, {(\partial_{2})}_{c}) = {[\nabla {(\partial_{1})}_{c}, \nabla {(\partial_{2})}_{c}]}_{q}^{σ_{q}} - \nabla ({[{(\partial_{1})}_{c}, {(\partial_{2})}_{c}]}_{q}^{σ_{q}}),

(33)

and the $σ_{q}$ -curvature of the $σ_{q}$ -connection $\nabla_{q}$ deﬁned by,

K_{\nabla_{q}} (\partial_{1}, \partial_{2}) = {[\nabla_{q} \partial_{1}, \nabla_{q} \partial_{2}]}^{σ_{q}} - \nabla_{q} ({[\partial_{1}, \partial_{2}]}^{σ_{q}}),

(34)

are related as follows

{(K_{\nabla_{q}} (\partial_{1}, \partial_{2}))}_{c} = K_{\nabla}^{q} ({(\partial_{1})}_{c}, {(\partial_{2})}_{c}),

(35)

$\partial_{1}, \partial_{2} \in D e r^{σ_{q}} (A_{q})$ .

6. Braided differential operators

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

G

be a ﬁnite abelian group.

6.1. Braided differential operators in

G

-algebras. Let

A

be a

σ

-symmetric

G

-algebra.

Deﬁne a

σ

-differential operator

f

of order at most

k

as the linear map

σ

-differential operator

f

of order at most

k

is of degree

∣ f ∣ \in \hat{G}

if it is an

(G, ∣ f ∣)

-operator

A \to A

, that is

Denote by

D i f f_{k, χ}^{σ} (A, A)

the

σ

-differential operators of order at most

k

and degree

χ \in \hat{G}

and by

D i f f_{k}^{σ} (A, A)

the set of all of order at most

k .

From [16] we have the two following results. The

σ

-commutator of two

σ

-differential operators

f_{1} \in D i f f_{i}^{σ} (A, A)

and

f_{2} \in D i f f_{j}^{σ} (A, A)

is a

σ

-differential operator of order at most

i + j - 1

for homogeneous

f_{1} \in D i f f_{i, ∣ f_{1} ∣}^{σ} (A, A)

and

f_{2} \in D i f f_{j, ∣ f_{2} ∣}^{σ} (A, A)

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

depends on the class of the two

σ

-differential operators

f_{1} \in D i f f_{i}^{σ} (A, A)

and

f_{2} \in D i f f_{j}^{σ} (A, A)

, and there is a

σ

-Poisson structure on the braided symbol algebra, see [16].

6.2. Braided differential operators in

G

-modules. Let

A

be a

σ

-symmetric

G

-algebra and let

E

be a

σ

-symmetric

A

-module.

Deﬁne a

σ

-differential operator

f

E

of order at most

k

as the linear map

σ

-differential operator

f

E

of order at most

k

is of degree

∣ f ∣ \in G

if it is an

(G, ∣ f ∣)

-operator

A \to A

, that is

Denote by

D i f f_{k, χ}^{σ} (E, E)

the

σ

-differential operators of order at most

k

and degree

χ \in \hat{G}

, the

σ

-differential operators in order at most

k

E

D i f f_{k}^{(σ, A)} (E, E)

and we consider

D i f f^{(σ, A)} (E, E) = \cup D i f f_{k}^{(σ, A)} (E, E)

From [16] we have the two following results. The

σ

-commutator of two

σ

-differential operators

f_{1} \in D i f f_{i}^{σ} (E, E)

and

f_{2} \in D i f f_{j}^{σ} (E, E)

is a

σ

-differential operator of order at most

i + j

for homogeneous

f_{2} \in D i f f_{i, ∣ f_{1} ∣}^{σ} (E, E)

and

f_{2} \in D i f f_{j, ∣ f_{2} ∣}^{σ} (E, E)

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

depends on the class of the two

σ

-differential operators

f_{1} \in D i f f_{i}^{σ} (E, E)

and

f_{2} \in D i f f_{j}^{σ} (E, E)

, and there is a

σ

-Poisson structure on the braided symbol algebra, see [16].

6.3. Quantizations of braided differential operators in

G

-algebras. We can deﬁne quantization of

σ

-differential operators in

G

-algebras. Let

A

be a

σ

-commutative

G

-algebra.

The operator

Q_{q} (f)

is an operator of the quantized

G

-algebra

A_{q}

. Let us denote by

D i f f^{σ_{q}} (A_{q}, A_{q})

the quantization of all

σ

-differential operators of

A

equipped with the quantization of composition.

From [15] we have the following result. Given a braiding

σ

, let

σ_{q}

be the quantization of

σ

. The operator

is an isomorphism of modules. The symbol of

Q_{q}

is an isomorphism of modules

By proposition 11 is

D i f f^{σ_{q}} (A_{q}, A_{q})

σ_{q}

-symmetric module and a

σ_{q}

-Lie

G

-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.

The

σ_{q}

-Lie algebra structure of

D i f f^{σ_{q}} (A_{q}, A_{q})

can be realized within the classical one by dequantization, where we have the conditions for linearity (16), the

A_{q}

-module structure (17) and the commutator (18), with

D e r

replaced by

D i f f

6.4. Quantizations of braided differential operators in

G

-modules. Let

A

be a

σ

-symmetric

G

-algebra and let

E

be a

σ

-symmetric

A

-module.

The operator

Q_{q} (f)

is an operator of the quantized

G

-module

E_{q}

. Denote by

D i f f^{(σ_{q}, A_{q})} (E_{q}, E_{q})

the quantization of all

σ

-differential operators of

E

equipped with the quantization of composition.

is an isomorphism of modules. The symbol of

Q_{q}

is an isomorphism of modules

D i f f^{(σ_{q}, A_{q})} (E_{q}, E_{q})

is a

σ_{q}

-symmetric module and a

σ_{q}

-Lie

G

-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,

S m b l^{(σ_{q}, A_{q})} (E_{q}, E_{q})

The

σ_{q}

-Lie algebra structure of

D i f f^{(σ_{q}, A_{q})} (E_{q}, E_{q})

can be realized within the classical one by dequantization, where we have the conditions for linearity (16), the

A_{q}

-module structure (17) and the commutator (18), with

D e r

replaced by

D i f f

6.5. Braided symbol and

G \oplus S^{1}

-modules. Any

\hat{G}

-graded differential operator has a ﬁbration by

ℤ

. However, the symbol of the braided differential operators of

\hat{G}

-graded algebras

A

and modules

E

S m b l^{\hat{σ}} (A, A)

and

S m b l^{(\hat{σ}, A)} (E, E)

, is a graded by

\hat{G} \oplus ℤ

. Hence there is an action of the dual of

\hat{G} \oplus ℤ

G \oplus S^{1}

on the symbol of the braided differential operators of

G

-algebras

A

and

G

-modules

E

S m b l^{σ} (A, A)

and

S m b l^{(σ, A)} (E, E)

In [11] and [10] we consider quantizations and braidings with respect to

\hat{G} \oplus ℤ

-grading. Here, instead of considering quantizations and braidings with respect to the

G

-action, we consider such with respect to

G \oplus S^{1}

-action.

Let

\bar{G} = \hat{G} \oplus ℤ

, denote its elements by

\bar{χ} = (χ, χ_{ℤ})

and

\tilde{G} = G \oplus S^{1}

and denote elements by

\tilde{g} = (g, g_{S})

Any symmetry in the monoidal category of

\tilde{G} = G \oplus S^{1}

-modules is of the form

is a symmetry of the monoidal category of

\bar{G} = \hat{G} \oplus ℤ

-graded modules deﬁned by

Any quantization in the monoidal category of

\tilde{G} = G \oplus S^{1}

-modules is of the form

where

\bar{q}

is quantization in the monoidal category of

\bar{G} = \hat{G} \oplus ℤ

-graded modules,

Considering

S m b l^{σ} (A, A)

and

S m b l^{(σ, A)} (E, E)

\tilde{G}

-modules they are equipped with a symmetry

\tilde{σ}

\tilde{G}

given by

\bar{σ}

where

\bar{σ} ∣_{(\{0\} \oplus ℤ) \times (\{0\} \oplus ℤ)} = τ

and

\bar{σ} ∣_{(\hat{G} \oplus \{0\}) \times (\{0\} \oplus ℤ)} = γ

trivial, that is

\bar{σ} = \hat{σ}

Remark 15. If we quantize $S m b l^{σ} (A, A)$ and $S m b l^{(σ, A)} (E, E)$ by the quantizer $\tilde{q}$ given by $γ = \bar{σ} ∣_{(\hat{G} \oplus \{0\}) \times (\{0\} \oplus ℤ)}$ then the resulting algebra is $S m b l^{\tilde{σ}} (A, A)$ and $S m b l^{(\tilde{σ}, A)} (E, E)$ , which are $\tilde{σ}$ -Poisson algebras with respect to the braiding $\tilde{σ}$ given by the braiding

\bar{σ} (\bar{χ}, {\bar{χ}}^{'}) = σ (χ, χ^{'}) γ (χ_{,} χ_{ℤ}^{'}) γ^{- 1} (χ^{'}, χ_{ℤ})

in the category of $\bar{G}$ -graded modules.

We show the quantized braided Poisson structure for the quantization of $S m b l^{\tilde{σ}} (A, A)$ and $S m b l^{(\tilde{σ}, A)} (E, E)$ for a general braiding $\tilde{σ}$ in theorems 16 and 17.

Note that for any braiding $\tilde{σ}$ , which is given by some

\bar{σ}, \bar{σ} ∣_{(\{0\} \oplus ℤ) \times (\{0\} \oplus ℤ)} = τ

always will be trivial since the structure arises from (quantizations of ) $S m b l^{σ} (A, A)$ and $S m b l^{(σ, A)} (E, E)$ .

Assume we have a braided symbols

S m b l^{\tilde{σ}} (A, A)

and

S m b l^{(\tilde{σ}, A)} (E, E)

with respect to a symmetry

\tilde{σ}

, given by a symmetry

\bar{σ}

in the category of

\bar{G}

-graded modules,

\bar{σ} ∣_{(\{0\} \oplus ℤ) \times (\{0\} \oplus ℤ)} = τ = 1

A quantization of

f \in S m b l_{k}^{\tilde{σ}} (A, A)

f \in S m b l_{k}^{(\tilde{σ}, A)} (E, E)

We obtain the following properties for the quantization of

S m b l^{\tilde{σ}} (A, A)

Theorem 16. $S m b l^{{\tilde{σ}}_{\tilde{q}}} (A_{\tilde{q}}, A_{\tilde{q}})$ is a ${\tilde{σ}}_{\tilde{q}}$ -Poisson algebra with respect to the ${\tilde{σ}}_{\tilde{q}} - \tilde{q}$ -bracket, that is, the following properties are satisﬁed

{[S m b l^{{\tilde{σ}}_{\tilde{q}}} (A_{\tilde{q}}, A_{\tilde{q}}), S m b l^{{\tilde{σ}}_{\tilde{q}}} (A_{\tilde{q}}, A_{\tilde{q}})]}_{\tilde{q}}^{{\tilde{σ}}_{q}} \subseteq S m b l^{{\tilde{σ}}_{\tilde{q}}} (A_{\tilde{q}}, A_{\tilde{q}}),

(i)

{[S m b l_{i}^{{\tilde{σ}}_{\tilde{q}}} (A_{\tilde{q}}, A_{\tilde{q}}), S m b l_{j}^{{\tilde{σ}}_{\tilde{q}}} (A_{\tilde{q}}, A_{\tilde{q}})]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}} \subseteq S m b l_{i + j - 1}^{{\tilde{σ}}_{\tilde{q}}} (A_{\tilde{q}}, A_{\tilde{q}}),

(i’)

skew ${\tilde{σ}}_{\tilde{q}}$ -symmetricity,

{[f_{1}, f_{2}]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}} = - \sum_{\tilde{g}, \tilde{h} \in \tilde{G}} {\bar{S}}_{\tilde{g} \tilde{h}} {[\tilde{h} f_{2}, \tilde{g} f_{1}]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}},

(ii)

the ${\tilde{σ}}_{\tilde{q}}$ -Jacobi identity,

{[f_{1}, {[f_{2}, f_{3}]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}}]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}} = {[{[f_{1}, f_{2}]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}}, f_{3}]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}} + \sum_{\tilde{g}, \tilde{h} \in \tilde{G}} {\bar{S}}_{\tilde{g} \tilde{h}} {[\tilde{h} f_{2}, {[\tilde{g} f_{1}, f_{3}]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}}]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}},

(iii)

and

{[f_{1}, f_{2} f_{3}]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}} = {[f_{1}, f_{2}]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}} h + \sum_{\tilde{g}, \tilde{h} \in \tilde{G}} {\bar{S}}_{\tilde{g} \tilde{h}} \tilde{h} f_{2} {[\tilde{g} f_{1}, f_{3}]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}},

(iv)

for $f_{1} \in S m b l_{i}^{{\tilde{σ}}_{\tilde{q}}} (A_{\tilde{q}}, A_{\tilde{q}})$ , $f_{2} \in S m b l_{j}^{{\tilde{σ}}_{\tilde{q}}} (A_{\tilde{q}}, A_{\tilde{q}})$ , $f_{3} \in S m b l_{k}^{{\tilde{σ}}_{\tilde{q}}} (A_{\tilde{q}}, A_{\tilde{q}})$ .

Except for (i’) we obtain the same properties for the quantization of

S m b l^{(\tilde{σ}, A)} (E, E)

Theorem 17. $S m b l^{({\tilde{σ}}_{\tilde{q}}, A_{\tilde{q}})} (E_{\tilde{q}}, E_{\tilde{q}})$ is a $\bar{G} = \hat{G} \oplus ℤ$ -graded ${\tilde{σ}}_{\tilde{q}}$ -Poisson algebra with respect to the ${\tilde{σ}}_{\tilde{q}} - \tilde{q}$ -bracket, that is, the properties (i), (ii), (iii) and (iv) of theorem 16 are satisﬁed when replacing $A_{\tilde{q}}$ by $E_{\tilde{q}}$ and $S m b l^{{\tilde{σ}}_{\tilde{q}}}$ by $S m b l^{({\tilde{σ}}_{\tilde{q}}, A_{\tilde{q}})}$ , and

\begin{matrix} {[S m b l_{i}^{({\tilde{σ}}_{\tilde{q}}, A_{\tilde{q}})} (E_{\tilde{q}}, E_{\tilde{q}}), S m b l_{j}^{({\tilde{σ}}_{\tilde{q}}, A_{\tilde{q}})} (E_{\tilde{q}}, E_{\tilde{q}})]}_{\tilde{q}}^{{\tilde{σ}}_{\tilde{q}}} \\ \subseteq S m b l_{i + j}^{({\tilde{σ}}_{\tilde{q}}, A_{\tilde{q}})} (E_{\tilde{q}}, E_{\tilde{q}}) . & (48) \end{matrix}

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1. Introduction

2. Modules with group action

3. Braided derivations of G-algebras

4. Braided derivations of G-modules

5. G,σ-connections

6. Braided differential operators

References

3. Braided derivations of $G$ -algebras

4. Braided derivations of $G$ -modules

5. $(G, σ)$ -connections