Lobachevskii Journal of Mathematics

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

Vol. 25, 2007, 161–185

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

H. L. Huru

QUANTIZATIONS OF BRAIDED DERIVATIONS. 3. MODULES WITH ACTION BY A GROUP.

(submitted by V. V. Lychagin)

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 Ĝ-graded modules and G-modules where Ĝ is the dual of G. We have found explicit descriptions of all quantizations and braidings in the monoidal category of modules graded by Ĝ 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 Gmodules. 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.

This paper is the third in a trilogy.

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 Ĝ-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 𝔰𝔩_{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 Ĝ = Hom (G, T^{1}) be the dual of G. We shall consider the monoidal categories of Ĝ-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∈G}, and ℂ \binom{}{Ĝ} be the function algebra of Ĝ, with the basis \{θ_{χ}\}_{χ∈Ĝ},

θ_{χ} (ϕ) = \{ \begin{matrix}1, χ = ϕ \\ 0, χ ≠ ϕ \end{matrix} , χ, ϕ ∈ Ĝ.

Then the Fourier transform F is a bialgebra isomorphism

F : ℂ [G] → ℂ \binom{}{Ĝ} ,

given by

F (δ_{g}) (χ) = χ (g^{−1}) ,

and the inverse is,

F^{−1} (θ_{χ}) = \frac{1}{|G|} ∑_{g∈G} χ (g) δ_{g},

g ∈ G and χ ∈ Ĝ.

We consider the two monoidal categories of G-modules and Ĝ-graded modules.

We know that a ℂ [G]-module is also a G-module, that ℂ \binom{}{Ĝ}-module

is a Ĝ-graded module, and the other way around for both cases. Then F^{−1} constructs an isomorphism of categories of between the monoidal categories of Ĝ-graded modules and G-modules as follows.

Let X be a G-module. Then there is a Ĝ-grading on X by the projection

π_{χ} (x) = F^{−1} (χ) (x) = \frac{1}{|G|} ∑_{g∈G} χ (g) g (x) ,

x ∈ X, χ ∈ Ĝ. Let Y = ∑_{χ∈Ĝ} Y_{χ} be a Ĝ-graded ℂ-module. There is a G-module structure on Y given by

gy = ∑_{χ∈Ĝ} F (δ_{g}) (χ) y_{χ} = ∑_{χ∈Ĝ} χ (g^{−1}) y_{χ},

y = ∑_{χ∈Ĝ} y_{χ} ∈ Y, g ∈ G.

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 Ĝ-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 ∈ Z^{2} (G, U (R)),

q (g, h) = (F ⊗ F)^{−1} \binom{}{q̂} (g, h) = \frac{1}{|G|^{2}} ∑_{ϕ,χ∈Ĝ} q̂(ϕ, χ) ϕ (g) χ (h) , (1)

g, h ∈ G, where q̂ is a quantization of the monoidal category of Ĝ-graded modules. When we factor out the trivial quantizations we are left with H^{2} (M, U (R)). The quantization as an operator X ⊗ Y → X ⊗ Y of G-modules X and Y q is of the form

q = \frac{1}{|G|^{2}} ∑_{\begin{matrix}g,h∈G \\ ϕ,χ∈Ĝ \end{matrix}} q̂(ϕ, χ) ϕ (g) χ (h) δ_{g} ⊗ δ_{h}.

We use the notation

q = ∑_{g,h∈G} Q_{gh}δ_{g} ⊗ δ_{h}.

Note that δ_{g}, g ∈ G, applied to an element is the action by g.

2.2. Braidings of G-modules. Any braiding in the monoidal category of G-modules is a 2-cochain σ : G × G → U (ℂ),

σ (g, h) = (F ⊗ F)^{−1} \binom{}{σ̂} (g, h)

= τ ∘ (\frac{1}{|G|^{2}} ∑_{ϕ,χ∈Ĝ} σ̂ (ϕ, χ) ϕ (g) χ (h)) , (2)

g, h ∈ G, where σ̂ is a braiding of the monoidal category of Ĝ-graded modules. As an operator X ⊗ Y → Y ⊗ X of G-modules X and Y σ is of the form

σ = τ ∘ (\frac{1}{|G|^{2}} ∑_{\begin{matrix}g,h∈G \\ ϕ,χ∈Ĝ \end{matrix}} σ̂ (ϕ, χ) ϕ (g) χ (h) δ_{g} ⊗ δ_{h}) .

Given a quantization q then the quantization of the braiding σ is

σ_{q} = τ ∘ (\frac{1}{|G|^{2}} ∑_{\begin{matrix}g,h∈G \\ ϕ,χ∈Ĝ \end{matrix}} σ̂_{q} (ϕ, χ) ϕ (g) χ (h) δ_{g} ⊗ δ_{h}) ,

where σ̂_{q} = q^{−1}∘σ̂∘q is the quantization of the braiding σ̂ of the monoidal category of Ĝ-graded modules.

We use the notation

σ = ∑_{g,h∈G} S_{g,h}δ_{g} ⊗ δ_{h},

and given a quantization q, denote the quantization of σ by

σ_{q} = τ ∘ ∑_{g,h∈G} S_{gh}^{q}δ_{g} ⊗ δ_{h}.

Using these formulas obtained for quantizations and braidings in the monoidal category of G-modules we get an explicit description of σsymmetric 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 G-algebra A is called σ-symmetric or σ-commutative if

ab = ∑_{g,h∈G} S_{g,h}h (b) g (a) ,  (3)

a, b ∈ A.

A quantization of A is the same object A_{q} = A with the new multiplication

a ∗_{q} b = ∑_{g,h∈G} Q_{gh}g (a) h (b) ,  (4)

a, b ∈ A.

If A is σ-symmetric then A_{q} is σ^{q}-symmetric,

a ∗_{q} b = ∑_{g,h∈G} S_{g,h}^{q}h (b) ∗_{q} g (a) ,  (5)

a, b ∈ A.

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 is σ-symmetric if

ax = ∑_{g,h∈G} S_{g,h}h (x) g (a) ,  (6)

a ∈ A, x ∈ E.

A quantizations of E is the same object E = E_{q} with the new action by A_{q} = A,

a ∗_{q} x = ∑_{g,h∈G} Q_{gh}g (a) h (x) .  (7)

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

a ∗_{q} x = ∑_{g,h∈G} S_{g,h}^{q}h (x) ∗_{q} g (a) ,  (8)

a ∈ A, x ∈ E.

2.5. Operators of G-modules. Let E be a G-module and W : E → E be an operator of E. There is a induced G-action on W, defined by

g (W) = g ∘ W ∘ g^{−1},

g ∈ G.

W is called a (G, 0) = G-(intertwining) operator if

W ∘ g = g ∘ W,

and we say W ∈ Hom_{G}^{0} (E, E).

W is called a (G, α)-operator if

W ∘ g = α (g) (g ∘ W) ,

α ∈ Ĝ, that is, W is a twisted α-homomorphism, denoted by W ∈ Hom_{G}^{α} (E, E).

2.6. Quantizations of operators of G-modules. Let E be a G-module and let W : E → E be a G-operator of E.

Given a quantization q define the quantization of W, W_{q} = Q_{q} (W) by

Q_{q} (W) (x) = ∑_{g,h∈G} Q_{gh}g (W) (h (x)) ,

x ∈ E_{q}.

Q_{q} (W) is an operator of the quantized G-module E_{q}.

The quantization of all operators of E is equipped with the quantized composition of operators,

W ∗_{q} Z = ∑_{g,h∈G} S_{g,h}h (W) ∘ g (Z) .  (9)

The inverse quantization

Q_{q}^{−1} (W) = W_{c},

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-homo-geneous derivations of non-homogeneous elements of graded algebras (over a commutative ring R) is needed.

For a Ĝ-graded module E and an operator W = ∑_{χ∈Ĝ} W_{χ} of E we have,

W (x) = ∑_{ϕ,χ∈Ĝ} W_{ϕ} (x_{χ}) ,
(W (x))_{χ} = ∑_{\begin{matrix}ϕ,ψ∈Ĝ \\ ϕ+ψ=χ \end{matrix}} W_{ϕ} (x_{ψ}) ,

x = ∑_{χ∈Ĝ} x_{χ} ∈ E.

Similarly, for a σ̂-symmetric Ĝ-graded algebra A = ∑_{χ∈Ĝ} A_{χ}, and a σ̂-derivation of A,

ˆ∂ = ∑_{χ∈Ĝ} ˆ∂_{χ} : A → A

where each ˆ∂_{χ} is a derivation of degree χ ∈ Ĝ, we have

ˆ∂(a) = ∑_{ϕ,χ∈Ĝ} ˆ∂_{ϕ} (a_{χ}) ,
(ˆ∂(a))_{χ} = ∑_{\begin{matrix}ϕ,ψ∈Ĝ \\ ϕ+ψ=χ \end{matrix}} ˆ∂_{ϕ} (a_{ψ}) ,

a ∈ A, a = ∑_{χ∈Ĝ} a_{χ}.

Proposition 1. Let a, b ∈ A, a = ∑_{χ∈Ĝ} a_{χ}, b = ∑_{χ∈Ĝ} b_{χ}. Then the σ̂-derivation ˆ∂ satisfies the following σ̂-Leibniz rule

ˆ∂(ab) = ˆ∂(a) b + μ ∘ σ̂ (ˆ∂ ⊗ a) (b)
= ˆ∂(a) b + ∑_{χ,ϕ,ψ∈Ĝ} σ̂ (χ, ϕ) a_{ϕ} ˆ∂_{χ} (b_{ψ}) .

Proof.

ˆ∂(ab) = ∑_{χ,ϕ∈Ĝ} ˆ∂_{χ} ((ab)_{ϕ})

= ∑_{χ,ϕ∈Ĝ} ˆ∂_{χ} ( ∑_{\begin{matrix}ψ,κ∈Ĝ \\ ψ+κ=ϕ \end{matrix}}a_{ψ}b_{κ} )

= ∑_{χ,ψ,κ∈Ĝ} ˆ∂_{χ} (a_{ψ}b_{κ})
= ∑_{χ,ψ,κ∈Ĝ}(ˆ∂_{χ} (a_{ψ}) b_{κ} + σ̂ (χ, ψ) a_{ψ} ˆ∂_{χ} (b_{κ}))
= ˆ∂(a) b + σ̂ (ˆ∂ ⊗ a) (b) .

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 Ĝ-graded σ̂-symmetric algebra over ℂ with respect to a braiding σ̂ in the monoidal category of Ĝ-graded modules.

Let ˆ∂ be a σ̂-derivations of degree |ˆ∂| of A, ˆ∂ : A_{χ} → A_{χ+| ˆ∂|}, that is, satisfies

ˆ∂ ∘ θ_{χ} = ˆ∂ ∘ π_{χ} = π_{χ+| ˆ∂|} ∘ ˆ∂ = θ_{χ+| ˆ∂|} ∘ ˆ∂,

for a ∈ A, χ, |ˆ∂| ∈ Ĝ, θ_{χ} and θ_{χ+| ˆ∂|}, are in the basis of ℂ \binom{}{Ĝ}, and

[figure: A; ˆ∂; A; A; θ_{χ}; ˆ∂; A; θ_{χ+| ˆ∂|}]

commutes.

Proposition 2. Let ˆ∂ be a derivation of a Ĝ-graded algebra A of degree |ˆ∂| ∈ Ĝ. Then the operator ∂ = F^{−1} \binom{}{ˆ∂} of A as a G-algebra satisfies

∂ ∘ g = |ˆ∂| (g) g ∘ ∂,

that is, ∂ is an (G, |ˆ∂|)-operator.

Proof. By the commutativity of the diagram

[figure: A; ∂; A; A; F^{−1}(θ_{χ}); ∂; A; F^{−1}(θ_{χ+| ˆ∂|})],

we have that

∂ ∘ F^{−1} (θ_{χ}) = F^{−1} (θ_{χ+| ˆ∂|}) ∘ ∂,
\frac{1}{|G|} ∑_{g∈G} χ (g) ∂ ∘ ρ_{g} = \frac{1}{|G|} ∑_{g∈G} (χ + |ˆ∂|) (g) ρ_{g} ∘ ∂
= \frac{1}{|G|} ∑_{g∈G} χ (g) |ˆ∂| (g) ρ_{g} ∘ ∂,

that is, if and only if

∂ ∘ g = |ˆ∂| (g) g ∘ ∂.

3.3. Braided derivations of G-algebras. Consider a braiding σ in the monoidal category of G-modules,

σ = (F ⊗ F)^{−1} \binom{}{σ̂} ,

where σ̂ is a braiding in the monoidal category of Ĝ-graded modules. Let A be a σ-symmetric G-algebra.

By the two preceding sections, 3.1 and 3.2, we give the following definitions.

Definition 3. A σ-derivation ∂ of A is an operator A → A that satisfies the σ-Leibniz rule

∂ (ab) = ∂ (a) b + ∑_{g,h∈G} S_{g,h}h (a) (g ∘ ∂) (b)

a, b ∈ A.

Definition 4. A σ-derivation ∂ of A is said to be of degree |∂| ∈ Ĝ if it is an (G, |∂|)-operator A → A, that is

∂ ∘ g = |∂| (g) g ∘ ∂.  (10)

We call ∂ homogeneous if it has a degree.

A σ-derivation ∂_{χ} of degree χ_{∂} ∈ Ĝ satisfies the simplified σ-Leibniz rule

∂ (ab) = ∂ (a) b + \frac{1}{|G|^{2}} ∑_{\begin{matrix}g,h∈G \\ ϕ∈Ĝ \end{matrix}} σ̂ (χ_{∂}, ϕ) ϕ (h) h (a) (∂ ∘ g) (b) ,

a, b ∈ A.

The set of σ-derivations of degree χ is denoted by Der_{χ}^{σ} (A) and the set of all σ-derivations by Der^{σ} (A).

A left A-module structure on Der^{σ} (A) is defined by

(a∂) (b) = a (∂ (b)) ,

for a, b ∈ A, ∂ ∈ Der^{σ} (A).

A σ-commutator (or σ-bracket) of elements ∂_{1}, ∂_{2} ∈ Der^{σ} (A) is defined by

[∂_{1}, ∂_{2}]^{σ} = ∂_{1}∂_{2} − ∑_{g,h∈G} S_{g,h}h (∂_{2}) g (∂_{1}) .

The σ-bracket satisfies the A-module conditions,

[a∂_{1}, ∂_{2}]^{σ} = a [∂_{1}, ∂_{2}]^{σ} − ∑_{g,h∈G} S_{g,h} (h∂_{2} (ga)) g∂_{1},  (11)
[∂_{1}, a∂_{2}]^{σ} = ∑_{g,h∈G} S_{g,h}h (a) [g∂_{1}, ∂_{2}]^{σ} + ∂_{1} (a) ∂_{2},  (12)

∀a ∈ A, ∂_{1}, ∂_{2} ∈ Der^{σ} (A).

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

[Der^{σ} (A) , Der^{σ} (A)]^{σ} ⊆ Der^{σ} (A) ,  (i)
[Der_{ϕ}^{σ} (A) , Der_{χ}^{σ} (A)]^{σ} ⊆ Der_{ϕ+χ}^{σ}(A) ,  (i’)

ϕ, χ ∈ Ĝ, skew σ-symmetricity,

[∂_{1}, ∂_{2}]^{σ} = − ∑_{g,h∈G} S_{gh} [h∂_{2}, g∂_{1}]^{σ} ,  (ii)

and the σ-Jacobi identity for σ-derivations,

[∂_{1}, [∂_{2}, ∂_{3}]^{σ}]^{σ} = [[∂_{1}, ∂_{2}]^{σ} , ∂_{3}]^{σ} + ∑_{g,h∈G} S_{gh} [h∂_{2}, [g∂_{1}, ∂_{3}]^{σ}]^{σ} ,  (iii)

for braided derivations ∂_{1}, ∂_{2} and ∂_{3}.

Proof. Except from (i′) all those above are proved for any monoidal category in [9]. Now

[Der_{ϕ}^{σ} (A) , Der_{χ}^{σ} (A)]^{σ} ⊆ Der_{ϕ+χ}^{σ}(A)

if and only if

[∂_{1}, ∂_{2}]^{σ} ∘ g = (ϕ + χ) (g) g ∘ [∂_{1}, ∂_{2}]^{σ} ,

which is is easy to prove,

[∂_{1}, ∂_{2}]^{σ} ∘ g = ∂_{1} ∘ ∂_{2} ∘ g − ∑_{h,h′∈G} S_{h,h′}h′ (∂_{2}) ∘ h (∂_{1}) ∘ g

= χ (g) ϕ (g) g ∘ ∂_{1} ∘ ∂_{2} − ϕ (g) χ (g) g ∘ ∑_{h,h′∈G} S_{h,h′}h′ (∂_{2}) ∘ h (∂_{1})

= (ϕ + χ) (g) g ∘ [∂_{1}, ∂_{2}]^{σ} , ∂_{1} ∈ Der_{ϕ}^{σ} (A) , ∂_{2} ∈ Der_{χ}^{σ} (A) .

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, ∂, by

Q_{q} (∂) (a) = ∂_{q} (a) =^{def} ∑_{g,h∈G} Q_{gh}g (∂) (h (a)) ,

a ∈ A.

Q_{q} (∂) is an operator of the quantized G-algebra A_{q}. Let us denote by Der^{σ^{q}} (A_{q}) the quantization of all σ-derivations of A equipped with the quantization of composition,

∂_{1} ∗_{q} ∂_{2} = ∑_{g,h∈G} Q_{gh}g (∂_{1}) ∘ h (∂_{2}) ,

∂_{1}, ∂_{2} ∈ Der^{σ^{q}} (A_{q}). We define the σ_{q} − q-bracket by

[∂_{1}, ∂_{2}]_{q}^{σ_{q}} = ∂_{1} ∗_{q} ∂_{2} − ∑_{g,h∈G} S_{g,h}^{q}h (∂_{2}) ∗_{q} g (∂_{1}) .

The operator Q_{q}

Q_{q} : (Der^{σ} (A) , [,]^{σ}) → (Der^{σ^{q}} (A_{q}) , [,]_{q}^{σ_{q}}) ,  (13)
∂ ∈ Der^{σ} (A) ⟼ Q_{q} (∂) ∈ Der^{σ^{q}} (A_{q}) ,

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

Der^{σ^{q}} (A_{q}) is a σ_{q}-symmetric A_{q}-module,

[a∂_{1}, ∂_{2}]_{q}^{σ_{q}} = a [∂_{1}, ∂_{2}]_{q}^{σ_{q}} − ∑_{g,h∈G} S_{g,h}^{q} (h∂_{2} (ga)) g∂_{1}  (14)
[∂_{1}, a∂_{2}]_{q}^{σ_{q}} = ∑_{g,h∈G} S_{g,h}^{q}h (a) [g∂_{1}, ∂_{2}]_{q}^{σ_{q}} + ∂_{1} (a) ∂_{2},  (15)

∀a ∈ A, ∂_{1}, ∂_{2} ∈ Der^{σ^{q}} (A_{q}).

Theorem 6. Der^{σ^{q}} (A_{q}) is a σ_{q}-Lie G-algebra with respect to the σ_{q} −qbracket and the quantized combination, that is the following properties are satisfied,

[Der^{σ^{q}} (A_{q}) , Der^{σ_{q}} (A_{q})]_{q}^{σ_{q}} ⊆ Der^{σ_{q}} (A_{q}) ,  (i)
[Der_{ϕ}^{σ_{q}} (A_{q}) , Der_{χ}^{σ_{q}} (A_{q})]_{q}^{σ_{q}} ⊆ Der_{ϕ+χ}^{σ_{q}}(A_{q}) ,  (i’)

ϕ, χ ∈ Ĝ, skew σ_{q}-symmetricity,

[∂_{1}, ∂_{2}]_{q}^{σ_{q}} = − ∑_{g,h∈G} S_{gh}^{q} [h∂_{2}, g∂_{1}]_{q}^{σ_{q}} ,  (ii)

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

[∂_{1}, [∂_{2}, ∂_{3}]_{q}^{σ_{q}}]_{q}^{σ_{q}} = [[∂_{1}, ∂_{2}]_{q}^{σ_{q}} , ∂_{3}]_{q}^{σ_{q}} + ∑_{g,h∈G} S_{gh}^{q} [h∂_{2}, [g∂_{1}, ∂_{3}]_{q}^{σ_{q}}]_{q}^{σ_{q}} , (iii)

for ∂_{1}, ∂_{2}, ∂_{3} ∈ Der^{σ^{q}} (A_{q}).

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

Q_{q}^{−1} (∂) = ∂_{c},

∂ ∈ Der^{σ^{q}} (A_{q}), where we have the following linearity,

(∂_{1} + ∂_{2})_{c} = (∂_{1})_{c} + (∂_{2})_{c} ,  (16)

∂_{1}, ∂_{2} ∈ Der_{|W |}^{σ_{q}} (A_{q}), A_{q}-module structure,

(a ∘ ∂_{1})_{c} = a ∗_{q} (∂_{1})_{c} ,  (17)

the commutator satisfies,

([∂_{1}, ∂_{2}]^{σ^{q}})_{c} = [(∂_{1})_{c} , (∂_{2})_{c}]_{q}^{σ_{q}} ,  (18)

for ∂_{1}, ∂_{2} ∈ Der^{σ^{q}} (A_{q}), a ∈ 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.

Definition 7. Let ∂_{A} ∈ Der^{σ} (A). A σ-derivation ∂ of E over ∂_{A} is an operator E → E that satisfies the σ-Leibniz-rule over A,

∂ (ax) = ∂_{A} (a) x + ∑_{g,h∈G} S_{g,h}h (a) (g ∘ ∂) (x)

a ∈ A, x ∈ E.

Such a pair (∂, ∂_{A}) is called a σ-derivation of E over A.

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

A σ-derivation ∂ of E over ∂_{A} is said to be of degree |∂| ∈ Ĝ if it is an (G, |∂|)-operator.

The set of σ-derivations of degree χ is denoted by Der_{χ}^{(σ,A)} (E) and the set of all σ-derivations by Der^{(σ,A)} (E).

An A-module structure on Der^{(σ,A)} (E) is defined by

(a∂) (x) = a (∂ (x)) ,

for a ∈ A, x ∈ E, ∂ ∈ Der^{(σ,A)} (E).

The σ-bracket satisfies the A-module conditions (11) and (12) for σderivations of E over A, Der^{(σ,A)} (E).

Theorem 8. Der^{(σ,A)} (E) is a σ-Lie G-algebra with respect to the σbracket, that is the following properties are satisfied for σ-derivations of E over A;

[Der^{(σ,A)} (E) , Der^{(σ,A)} (E)]^{σ} ⊆ Der^{(σ,A)} (E) ,  (i)
[Der_{ϕ}^{(σ,A)} (E) , Der_{χ}^{(σ,A)} (E)]^{σ} ⊆ Der_{ϕ+χ}^{(σ,A)} (E) ,  (i’)

ϕ, χ ∈ Ĝ, skew σ-symmetricity, [,]^{σ} = − [,]^{σ} ∘ σ, and the σ-Jacobi identity, see (ii) and (iii) in theorem 5.

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 (∂, ∂_{A}) ∈ Der^{(σ,A)} (E). Given a quantizer q define the quantization of ∂ by

∂_{q} (x) = Q_{q} (∂) (x) =^{def} ∑_{g,h∈G} Q_{gh}g (∂) (h (x)) ,  (19)

x ∈ E. If x ∈ A, this is the quantization of the derivation of A, ∂_{A}.

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

Denote by Der^{(σ_{q},A_{q})} (E_{q}) the quantization of all σ-derivations of E over A equipped with the quantization of composition.

The operator Q_{q},

Q_{q} : (Der^{(σ,A)} (E) , [,]^{σ}) → (Der^{(σ_{q},A_{q})} (E_{q}) , [,]_{q}^{σ_{q}}) , (20)
∂ ∈ 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}.

Der^{(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,

[Der^{σ^{q}} (A_{q}) , Der^{σ_{q}} (A_{q})]_{q}^{σ_{q}} ⊆ Der^{σ_{q}} (A_{q}) ,  (i)
[Der_{ϕ}^{σ_{q}} (A_{q}) , Der_{χ}^{σ_{q}} (A_{q})]_{q}^{σ_{q}} ⊆ Der_{ϕ+χ}^{σ_{q}}(A_{q}) ,  (i’)

ϕ, χ ∈ Ĝ, skew σ_{q}-symmetricity, [,]_{q}^{σ_{q}} = − [,]_{q}^{σ_{q}} ∘ σ_{q}, and the σ_{q}-Jacobi identity, see (ii) and (iii) in theorem 6.

The σ_{q}-Lie algebra of the structure of (Der^{(σ_{q},A_{q})} (E_{q}) , [,]_{q}^{σ_{q}}) can be realized within the classical one by dequantization,

Q_{q}^{−1} (∂) = ∂_{c},

∂ ∈ Der^{(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).

5. (G, σ)-connections

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

Definition 9. The map

∇ : Der^{σ} (A) → Der^{(σ,A)} (E) ,

is a σ-connection in E if

π ∘ ∇ = Id.

Furthermore, ∇ is called a (G, σ)-connection in Der^{(σ,A)} (E) if the following diagram commutes

[figure: Der^{σ,A}(E); g; Der^{σ,A}(E); Der^{σ}(A); ∇; g; Der^{σ}(A); ∇].  (21)

Define G-action on ∇ by

g (∇) = g ∘ ∇ ∘ g^{−1}.  (22)

Then a (G, σ)-connection in E is G-invariant,

g (∇) = ∇,

and ∇ preserves the degree on the σ-derivations,

∇ : Der_{χ}^{σ} (A) → Der_{χ}^{σ} (E) .

Assume that ∇ is a (G, σ)-connection. We say that the σ-connection ∇ is flat if

[∇∂_{1}, ∇∂_{2}]^{σ} = ∇ ([∂_{1}, ∂_{2}]^{σ}) ,  (23)

for all ∂_{1}, ∂_{2} ∈ Der^{σ} (A).

Define the σ-curvature of ∇,

K_{∇} (∂_{1}, ∂_{2}) = [∇∂_{1}, ∇∂_{2}]^{σ} − ∇ ([∂_{1}, ∂_{2}]^{σ}) ,  (24)

written in full,

K_{∇} (∂_{1}, ∂_{2}) = ∇ (∂_{1}) ∇ (∂_{2}) − ∇ (∂_{1}∂_{2})

+ ∑_{g,h∈G} S_{g,h} (∇ (h (∂_{2}) g (∂_{1})) − h (∇ (∂_{2})) g (∇ (∂_{1}))) .

The properties of K_{∇} are the following. K_{∇} satisfies linearity,

K_{∇} (∂_{1}, ∂_{2}) (ax) = ∑_{g,h∈G} S_{gh}h (a) (gK_{∇} (∂_{1}, ∂_{2})) (x) ,  (25)

is skew σ-symmetric,

K_{∇} (∂_{1}, ∂_{2}) = − ∑_{g,h∈G} S_{gh}K_{∇} (h (∂_{2}) , g (∂_{1})) ,

and satisfies the A-module homomorphism conditions with respect to σ,

K_{∇} (a∂_{1}, ∂_{2}) = aK_{∇} (∂_{1}, ∂_{2}) ,  (26)
K_{∇} (∂_{1}, a∂_{2}) = ∑_{g,h∈G} S_{gh}h (a) K_{∇} (g∂_{1}, ∂_{2}) ,  (27)

for a ∈ A, x ∈ E and braided derivations ∂_{1}, ∂_{2} ∈ Der^{σ} (A).

5.1. Quantizations of (G, σ)-connections and curvatures. Let ∇ be a G-invariant (G, σ)-connection.

Given a quantization q, define the quantization of the (G, σ)-connection ∇,

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

by

∇_{q} = Q_{q} ∘ ∇ ∘ Q_{q}^{−1}.

The connection ∇_{q} is a σ_{q}-connection in E_{q} which is easy to see as π_{q} = Q_{q} ∘ π ∘ Q_{q}^{−1}. Furthermore is the σ_{q}-connection ∇_{q} a (G, σ_{q})-connection, that is,

[figure: Der^{σ_{q},A_{q}}(E_{q}); g; Der^{σ_{q},A_{q}}(E_{q}); Der^{σ}(A_{q}); ∇_{q}; g; Der_{q}^{σ}(A_{q}); ∇_{q}]  (28)

commutes, and is G-invariant.

The quantization of the σ-curvature, the σ_{q} − q-curvature K_{∇_{q}}^{q}, is defined

K_{∇_{q}}^{q} (∂_{1}, ∂_{2}) = [∇_{q}∂_{1}, ∇_{q}∂_{2}]_{q}^{σ_{q}} − ∇_{q} ([∂_{1}, ∂_{2}]_{q}^{σ_{q}}) ,

∂_{1}, ∂_{2} ∈ Der^{σ^{q}} (A_{q}).

The σ_{q} − q-curvature satisfies linearity,

K_{∇_{q}}^{q} (∂_{1}, ∂_{2}) (ax) = ∑_{g,h∈G} S_{gh}^{q}h (a) (gK_{∇_{q}}^{q} (∂_{1}, ∂_{2})) (x) ,  (29)

is skew σ_{q}-symmetric,

K_{∇_{q}}^{q} (∂_{1}, ∂_{2}) = − ∑_{g,h∈G} S_{gh}^{q}K_{∇_{q}}^{q} (h (∂_{2}) , g (∂_{1})) ,  (30)

and satisfies the A_{q}-module homomorphism conditions with respect to σ_{q},

K_{∇_{q}}^{q} (a∂_{1}, ∂_{2}) = aK_{∇_{q}}^{q} (∂_{1}, ∂_{2}) ,  (31)
K_{∇_{q}}^{q} (∂_{1}, a∂_{2}) = ∑_{g,h∈G} S_{gh}^{q}h (a) K_{∇_{q}}^{q} (g∂_{1}, ∂_{2}) ,  (32)

for a ∈ A_{q}, x ∈ E_{q} and braided derivations ∂_{1}, ∂_{2} ∈ Der^{σ^{q}} (A_{q}).

We have the following picture for dequantizations of braided derivations.

Theorem 10. The σ_{q} − q-curvature K_{∇}^{q} of the σ-connection ∇ of E defined by

K_{∇}^{q} ((∂_{1})_{c} , (∂_{2})_{c}) = [∇ (∂_{1})_{c} , ∇ (∂_{2})_{c}]_{q}^{σ_{q}} − ∇ ([(∂_{1})_{c} , (∂_{2})_{c}]_{q}^{σ_{q}}) , (33)

and the σ_{q}-curvature of the σ_{q}-connection ∇_{q} defined by,

K_{∇_{q}} (∂_{1}, ∂_{2}) = [∇_{q}∂_{1}, ∇_{q}∂_{2}]^{σ^{q}} − ∇_{q} ([∂_{1}, ∂_{2}]^{σ^{q}}) ,  (34)

are related as follows

(K_{∇_{q}} (∂_{1}, ∂_{2}))_{c} = K_{∇}^{q} ((∂_{1})_{c} , (∂_{2})_{c}) ,  (35)

∂_{1}, ∂_{2} ∈ Der^{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 σsymmetric G-algebra.

Define a σ-differential operator f of order at most k as the linear map

f : A → A,

such that

[a_{0}, [a_{1}, … [a_{k}, f]^{σ} ⋯]^{σ}]^{σ} = 0,  (36)

∀a_{0}, …, a_{k} ∈ A.

A σ-differential operator f of order at most k is of degree |f| ∈ Ĝ if it is an (G, |f|)-operator A → A, that is

f ∘ g = |f| (g) g ∘ f,  (37)

g ∈ G.

Denote by Diff_{k,χ}^{σ} (A, A) the σ-differential operators of order at most k and degree χ ∈ Ĝ and by Diff_{k}^{σ} (A, A) the set of all of order at most k.

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

From [16] we have the two following results. The σ-commutator of two σ-differential operators f_{1} ∈ Diff_{i}^{σ} (A, A) and f_{2} ∈ Diff_{j}^{σ} (A, A) is a σ-differential operator of order at most i + j − 1,

[f_{1}, f_{2}]^{σ} ∈ Diff_{i+j−1}^{σ} (A, A) .

and Diff^{σ} (A, A) is a σ-Lie G-algebra. Clearly,

[f_{1}, f_{2}]^{σ} ∈ Diff_{i+j−1,|f_{1}|+|f_{2}|}^{σ} (A, A) ,

for homogeneous f_{1} ∈ Diff_{i,|f_{1}|}^{σ} (A, A) and f_{2} ∈ Diff_{j,|f_{2}|}^{σ} (A, A).

We consider A − A-module structure on Diff^{σ} (A, A), [9].

Proposition 11. There is an A − A-module structure on Diff^{σ} (A, A) defined by

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

a, b ∈ A, f ∈ Diff^{σ} (A, A) and

af, fa ∈ Diff_{k}^{σ} (A, A) ,

for 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) = ∑_{k∈ℤ}Smbl_{k}^{σ} (A, A) .

The class of [f_{1}, f_{2}]^{σ} ∈ Diff_{i+j−1}^{σ} (A, A),

\overline{[f_{1}, f_{2}]^{σ}} ∈ Smbl_{i+j−1}^{σ} (A, A) ,

depends on the class of the two σ-differential operators f_{1} ∈ Diff_{i}^{σ} (A, A) and f_{2} ∈ Diff_{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 σsymmetric G-algebra and let E be a σ-symmetric A-module.

Define a σ-differential operator f of E of order at most k as the linear map

f : E → E,

such that

[x, [a_{0}, … [a_{k−1}, f]^{σ} ⋯]^{σ}]^{σ} = 0,  (38)

∀a_{0}, …, a_{k−1} ∈ A, x ∈ E.

A σ-differential operator f of E of order at most k is of degree |f| ∈ G if it is an (G, |f|)-operator A → A, that is

f ∘ g = |f| (g) g ∘ f,  (39)

g ∈ G.

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

From [16] we have the two following results. The σ-commutator of two σ-differential operators f_{1} ∈ Diff_{i}^{σ} (E, E) and f_{2} ∈ Diff_{j}^{σ} (E, E) is a σ-differential operator of order at most i + j,

[f_{1}, f_{2}]^{σ} ∈ Diff_{i+j}^{σ} (E, E) .

and Diff^{σ} (E, E) is a σ-Lie G-algebra. Furthermore,

[f_{1}, f_{2}]^{σ} ∈ Diff_{i+j,|f_{1}|+|f_{2}|}^{σ} (E, E) ,

for homogeneous f_{2} ∈ Diff_{i,|f_{1}|}^{σ} (E, E) and f_{2} ∈ Diff_{j,|f_{2}|}^{σ} (E, E).

Proposition 12. There is an A − A-module structure on Diff^{σ} (E, E) defined by

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

and

af, fa ∈ Diff_{k}^{σ} (E, E) ,

for a ∈ A and f ∈ Diff_{k}^{σ} (E, E).

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) = ∑_{k∈ℤ}Smbl_{k}^{σ} (E, E) .

The class of [f_{1}, f_{2}]^{σ} ∈ Diff_{i+j}^{σ} (E, E),

\overline{[f_{1}, f_{2}]^{σ}} ∈ Smbl_{i+j}^{σ} (E, E) ,

depends on the class of the two σ-differential operators f_{1} ∈ Diff_{i}^{σ} (E, E) and f_{2} ∈ Diff_{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.

Definition 13. Given a quantization q and f ∈ Diff^{σ} (A, A) define the quantization of f by

Q_{q} (f) (a) = f_{q} (a) =^{def} ∑_{g,h∈G} Q_{gh}g (f) (h (a)) ,

for a ∈ A.

The operator Q_{q} (f) is an operator of the quantized G-algebra A_{q}. Let us denote by Diff^{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

Q_{q} : (Diff^{σ} (A, A) , [,]^{σ}) → (Diff^{σ^{q}} (A_{q}, A_{q}) , [,]_{q}^{σ_{q}}) , (40)
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) , [,]^{σ}) → (Smbl^{σ^{q}} (A_{q}, A_{q}) , [,]_{q}^{σ_{q}}) , (41)
f ∈ Smbl^{σ} (A, A) ⟼ Smbl (Q_{q}) (f) ∈ Smbl^{σ^{q}} (A_{q}, A_{q}) .

By proposition 11 is Diff^{σ^{q}} (A_{q}, A_{q}) 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.

The σ_{q}-Lie algebra structure of Diff^{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 Der replaced by Diff.

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.

Definition 14. 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} ∑_{g,h∈G} Q_{gh}g (f) (h (x)) ,

where x ∈ E.

The operator Q_{q} (f) is an operator of the quantized G-module E_{q}. Denote by Diff^{(σ_{q},A_{q})} (E_{q}, E_{q}) the quantization of all σ-differential operators of E equipped with the quantization of composition.

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

Q_{q} : (Diff^{(σ,A)} (E, E) , [,]^{σ}) → (Diff^{(σ_{q},A_{q})} (E_{q}, E_{q}) , [,]_{q}^{σ_{q}}) , (42)

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) , [,]^{σ}) → (Smbl^{(σ_{q},A_{q})} (E_{q}, E_{q}) , [,]_{q}^{σ_{q}}) , (43)

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

Diff^{(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, Smbl^{(σ_{q},A_{q})} (E_{q}, E_{q}).

The σ_{q}-Lie algebra structure of Diff^{(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 Der replaced by Diff.

6.5. Braided symbol and G⊕S^{1}-modules. Any Ĝ-graded differential operator has a fibration by ℤ. However, the symbol of the braided differential operators of Ĝ-graded algebras A and modules E, Smblσ̂ (A, A) and Smbl^{(σ̂,A)} (E, E), is a graded by Ĝ⊕ℤ. Hence there is an action of the dual of Ĝ ⊕ ℤ, G ⊕ S^{1} on the symbol of the braided differential operators of G-algebras A and G-modules E, Smbl^{σ} (A, A) and Smbl^{(σ,A)} (E, E).

In [11] and [10] we consider quantizations and braidings with respect to Ĝ ⊕ ℤ-grading. Here, instead of considering quantizations and braidings with respect to the G-action, we consider such with respect to G ⊕ S^{1}- action.

Let Ḡ = Ĝ ⊕ ℤ, denote its elements by χ̄ = (χ, χ_{ℤ}) and G̃ = G ⊕ S^{1} and denote elements by g̃ = (g, g_{S}).

Any symmetry in the monoidal category of G̃ = G ⊕ S^{1}-modules is of the form

σ̃ = τ̃ ∘ (\frac{1}{|G|^{2}} ∑_{\begin{matrix}g̃,h̃∈G̃ \\ χ̄,ϕ̄∈Ḡ \end{matrix}} σ̄ (ϕ̄, χ̄) ϕ̄ \binom{}{g} χ̄ \binom{}{h} δg̃ ⊗ δh̃) ,  (44)

where

σ̄ : (Ĝ ⊕ ℤ) × (Ĝ ⊕ ℤ) → U (ℂ)

is a symmetry of the monoidal category of Ḡ = Ĝ ⊕ ℤ-graded modules defined by

σ̄ (χ̄, χ̄′) = σ (χ, χ′) τ (χ_{ℤ}, χ′_{ℤ}) γ (χ_{,}χ′_{ℤ}) γ^{−1} (χ′, χ_{ℤ}) ,  (45)

χ̄, χ̄′ ∈ Ḡ, for which

σ̄|_{(Ĝ⊕{0})×(Ĝ⊕{0})} = σ : Ĝ × Ĝ → U (R) ,

is a symmetry of Ḡ-graded modules,

σ̄|_{({0}⊕ℤ)×({0}⊕ℤ)} = τ : ℤ × ℤ → U (R) ,

is a symmetry of ℤ-graded modules and

σ̄|_{(Ĝ⊕{0})×({0}⊕ℤ)} = γ : Ĝ × ℤ → U (R) ,

is a bihomomorphism.

Denote by

σ̃ = ∑_{g̃,h̃∈G̃} S̄_{g̃h̃}δg̃ ⊗ δh̃.

Any quantization in the monoidal category of G̃ = G ⊕ S^{1}-modules is of the form

q̃ = \frac{1}{|G|^{2}} ∑_{\begin{matrix}g̃,h̃∈G̃ \\ χ̄,ϕ̄∈Ḡ \end{matrix}} q̄(ϕ̄, χ̄) ϕ̄ \binom{}{g} χ̄ \binom{}{h} δg̃ ⊗ δh̃,  (46)

where q̄ is quantization in the monoidal category of Ḡ = Ĝ ⊕ ℤ-graded modules,

q̄(χ̄, χ̄′) = q (χ, χ′) κ (χ, χ′_{ℤ}) κ^{−1} (χ′, χ_{ℤ}) ,  (47)

where χ̄, χ̄′ ∈ Ḡ,

q̄|_{(Ĝ⊕{0})×({0}⊕ℤ)} = κ : Ĝ × ℤ → U (R)

is a bihomomorphism and

q̄|_{(Ĝ⊕{0})×(Ĝ⊕{0})} = q : Ĝ × Ĝ → U (R) ,

is quantization of Ĝ-graded modules.

Denote by

q̃ = ∑_{g̃,h̃∈G̃} Q̄_{g̃h̃}δg̃ ⊗ δh̃.

Considering Smbl^{σ} (A, A) and Smbl^{(σ,A)} (E, E) as G̃-modules they are equipped with a symmetry σ̃ of G̃ given by σ̄ where σ̄|_{({0}⊕ℤ)×({0}⊕ℤ)} = τ and σ̄|_{(Ĝ⊕{0})×({0}⊕ℤ)} = γ trivial, that is σ̄ = σ̂.

Remark 15. If we quantize Smbl^{σ} (A, A) and Smbl^{(σ,A)} (E, E) by the quantizer q̃ given by γ = σ̄|_{(Ĝ⊕{0})×({0}⊕ℤ)} then the resulting algebra is Smblσ̃ (A, A) and Smbl^{(σ̃,A)} (E, E), which are σ̃-Poisson algebras with respect to the braiding σ̃ given by the braiding

σ̄ (χ̄, χ̄′) = σ (χ, χ′) γ (χ_{,}χ′_{ℤ}) γ^{−1} (χ′, χ_{ℤ})

in the category of Ḡ-graded modules.

We show the quantized braided Poisson structure for the quantization of Smblσ̃ (A, A) and Smbl^{(σ̃,A)} (E, E) for a general braiding σ̃ in theorems 16 and 17.

Note that for any braiding σ̃, which is given by some

σ̄, σ̄|_{({0}⊕ℤ)×({0}⊕ℤ)} = τ

always will be trivial since the structure arises from (quantizations of ) Smbl^{σ} (A, A) and Smbl^{(σ,A)} (E, E).

Assume we have a braided symbols Smblσ̃ (A, A) and Smbl^{(σ̃,A)} (E, E) with respect to a symmetry σ̃, given by a symmetry σ̄ in the category of Ḡ-graded modules, σ̄|_{({0}⊕ℤ)×({0}⊕ℤ)} = τ = 1.

A quantization of f ∈ Smblσ̃ ; k (A, A) or f ∈ Smbl_{k}^{(σ̃,A)} (E, E) is

Qq̃(f) (x) = fq̃(x) =^{def} ∑_{g̃,h̃∈G̃} Q̄_{g̃h̃}g̃f (h̃x) .

where x is in either Eq̃ or Aq̃.

The operator Qq̃ is an isomorphism of modules

Smbl (Qq̃) : (Smblσ̃ (A, A) , [,]σ̃) → (Smbl^{σ̃q̃} (Aq̃, Aq̃) , [,]σ̃q̃ ; q̃) ,
f ∈ Smblσ̃ (A, A) ⟼ Smbl (Qq̃) (f) ∈ Smbl^{σ̃q̃} (Aq̃, Aq̃) .

and

Smbl (Qq̃) : (Smbl^{(σ̃,A)} (E, E) , [,]σ̃) → (Smbl^{(σ̃q̃,Aq̃)} (Eq̃, Eq̃) , [,]σ̃q̃ ; q̃) ,

f ∈ Smbl^{(σ̃,A)} (E, E) ⟼ Smbl (Qq̃) (f) ∈ Smbl^{(σ̃q̃,Aq̃)} (Eq̃, Eq̃) .

We obtain the following properties for the quantization of Smblσ̃ (A, A).

Theorem 16. Smbl^{σ̃q̃} (Aq̃, Aq̃) is a σ̃q̃-Poisson algebra with respect to the σ̃q̃ − q̃-bracket, that is, the following properties are satisfied

[Smbl^{σ̃q̃} (Aq̃, Aq̃) , Smbl^{σ̃q̃} (Aq̃, Aq̃)]σ̃_{q} ; q̃ ⊆ Smbl^{σ̃q̃} (Aq̃, Aq̃) ,  (i)
[Smbl_{i}^{σ̃q̃} (Aq̃, Aq̃) , Smbl_{j}^{σ̃q̃} (Aq̃, Aq̃)]σ̃q̃ ; q̃ ⊆ Smbl_{i+j−1}^{σ̃q̃} (Aq̃, Aq̃) , (i’)

184 H. L. HURU

skew σ̃q̃-symmetricity,

[f_{1}, f_{2}]σ̃q̃ ; q̃ = − ∑_{g̃,h̃∈G̃} S̄_{g̃h̃} [h̃f_{2}, g̃f_{1}]σ̃q̃ ; q̃ ,  (ii)

the σ̃q̃-Jacobi identity,

[f_{1}, [f_{2}, f_{3}]σ̃q̃ ; q̃]σ̃q̃ ; q̃ = [[f_{1}, f_{2}]σ̃q̃ ; q̃ , f_{3}]σ̃q̃ ; q̃ + ∑_{g̃,h̃∈G̃} S̄_{g̃h̃} [h̃f_{2}, [g̃f_{1}, f_{3}]σ̃q̃ ; q̃]σ̃q̃ ; q̃ , (iii)

and

[f_{1}, f_{2}f_{3}]σ̃q̃ ; q̃ = [f_{1}, f_{2}]σ̃q̃ ; q̃ h + ∑_{g̃,h̃∈G̃} S̄_{g̃h̃}h̃f_{2} [g̃f_{1}, f_{3}]σ̃q̃ ; q̃ ,  (iv)

for f_{1} ∈ Smbl_{i}^{σ̃q̃} (Aq̃, Aq̃), f_{2} ∈ Smbl_{j}^{σ̃q̃} (Aq̃, Aq̃), f_{3} ∈ Smbl_{k}^{σ̃q̃} (Aq̃, Aq̃).

Except for (i’) we obtain the same properties for the quantization of Smbl^{(σ̃,A)} (E, E).

Theorem 17. Smbl^{(σ̃q̃,Aq̃)} (Eq̃, Eq̃) is a Ḡ = Ĝ ⊕ ℤ-graded σ̃q̃-Poisson algebra with respect to the σ̃q̃ − q̃-bracket, that is, the properties (i), (ii), (iii) and (iv) of theorem 16 are satisfied when replacing Aq̃ by Eq̃ and Smbl^{σ̃q̃} by Smbl^{(σ̃q̃,Aq̃)}, and

[Smbl_{i}^{(σ̃q̃,Aq̃)} (Eq̃, Eq̃) , Smbl_{j}^{(σ̃q̃,Aq̃)} (Eq̃, Eq̃)]σ̃q̃ ; q̃

⊆ Smbl_{i+j}^{(σ̃q̃,Aq̃)} (Eq̃, Eq̃) . (i’)

References

[1] Henri Cartan, Samuel Eilenberg. Homological algebra, Princeton University Press, 1956.

[2] V. Chari, A. Pressley. A Guide to Quantum Groups, Cambridge University Press, 1994.

[3] S. Eilenberg, S. Mac Lane. Cohomology Theory in Abstract Groups 1, Vol. 48, No.1 of Annals of Mathematics, 1947.

[4] C. Faith. Algebra: Rings, Modules and Categories I, Band 190 of Die Grundlehren der mathemathischen Wissenschaften in Einzeldarstellungen. Springer-Verlag, 1973.

[5] D. Gurevich. The Yang Baxter equation and generalizations of formal Lie theory, Soviet Math. Dokl. 33, 758-762, 1986.

[6] D. Gurevich. Algebraic aspects of quantum Yang Baxter equation, Algebra and Analysis, 2, 4, 1990.

[7] D. Gurevich, A. Radul, V. Rubtsov. Non-commutative differential geometry and Yang-Baxter equation, Intitute des Hautes Etudies Scientifiques, 88, 1991.

QUANTIZATIONS OF BRAIDED DERIVATIONS 185

[8] H. L. Huru. Associativity constraints, braidings and quantizations of modules with grading and action. Vol. 23, Lobachevskii Journal of Mathematics, http://ljm.ksu.ru/vol23/110.html, 2006.

[9] H. L. Huru. Quantization of braided algebras. 1. Monoidal categories. Lobachevskii Journal of Mathematics, Vol. 24 (2006), 13–24.

[10] H. L. Huru. Quantization of braided algebras. 2. Graded Modules. Lobachevskii Journal of Mathematics, Vol 25, 2007.

[11] H. L. Huru. Braided symmetric and exterior algebras and quantizations of braided Lie algebras.

[12] H. L. Huru, V. V. Lychagin. Quantization and classical non-commutative and non-associative algebras, preprint, Institut Mittag-Leffler, Stockholm, 2005.

[13] P. K. Jakobsen, V. Lychagin. The Categorical Theory of Relations and Quantizations, 2001.

[14] Cathrine V. Jensen. Linear ordinary differential equations and D-modules, solving and reduction methods, Dr.Scient. thesis, The University of Tromsø, Nov. 2004.

[15] V. V. Lychagin. Quantizations of Braided Differential Operators, Erwin Schrödinger International Institute of Mathematical Physics, Wien, and Sophus Lie Center, Moscow, 1991.

[16] V. V. Lychagin. Differential operators and quantizations, Preprint series in Pure Mathematics, Matematisk institutt, Universitetet i Oslo, No. 44, 1993.

[17] V. V. Lychagin. Calculus and Quantizations Over Hopf Algebras, Acta Applicandae Mathematicae, 1-50, 1998.

[18] V. V. Lychagin. Quantizations of Differential Equations, Pergamon Nonlinear Analysis 47, 2621-2632, 2001.

[19] Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics. Springer, 1998.

[20] M. A. Naimark, A. I. Stern. Theory of Group Representations, Grundlehren der mathemathischen Wissenschaften; 246. Springer-Verlag, 1975/1982.

[21] J.-P. Serre. Linear Representations of Finite Groups, volume 42 of Graduate Texts in Mathematics. Springer-Verlag, 1977.

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

Received November 7, 2006