Lobachevskii Journal of Mathematics

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

Vol. 25, 2007, 131–160

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

H. L. Huru

QUANTIZATIONS OF BRAIDED DERIVATIONS. 2. GRADED MODULES

(submitted by V. V. Lychagin)

Abstract. For the monoidal category of graded modules 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 σ and quantizations of braidings σ_{q} of the monoidal category of graded modules. The grading is by a finite commutative monoid. We work with braidings that are symmetries, in fact, in this category are all braidings symmetries.

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

We have found explicit descriptions of all quantizations and braidings in the monoidal category of modules graded by a finite commutative monoid and they depend only on the grading, [8].

That is, first of all we have found explicit formulas for σ-symmetric graded algebras A, graded modules, graded co- and bialgebras, graded internal homomorphisms, braided derivations, braided connections and curvature. Then we have found explicit formulas for quantizations of these structures.

From [9] we have the following results. Graded internal homomorphisms has a graded braided Lie structure with respect to the braided commutator. Quantizations of the graded internal homomorphisms has the quantized braided Lie structure and can be realized within the original braided Lie structure by what we call dequantization. We shall go through this in details for graded braided derivations.

We investigate graded braided derivations in σ-symmetric graded algebras and modules. The braided 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 graded 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 graded 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, their symbols and quantizations of these. Because of the ℤ-grading of the braided symbols we can extend the notion of braiding and quantization of these to include the ℤ-grading.

This paper is the second in a trilogy.

The results here are all proved for any monoidal category, except for when grading is explicitly involved. That is, we have shown that all the results here also are true for braided derivations of algebras and modules, braided connections, braided curvature, quantizations and so on of any monoidal category. This is found in the first paper Quantizations of braided derivations. 1. Monoidal categories, [9].

In [8] 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 G. Again, we have a complete and explicit description for braided 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, [10].

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). To find this quantization we use the fact that 𝔰𝔩_{2} (ℂ) is graded by ℤ and consider the exterior algebra, hence there is a ℤ × ℤ-grading which gives nontrivial quantizers.

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

As noted, most of the proofs are found for general monoidal categories in [9], but almost all proofs will be repeated for clarity.

2. Graded modules

Let M be a finite commutative monoid. Let R be a commutative ring with unit.

Denote by mod_{R} (M) the strict monoidal category [19] of M-graded R-modules,

X = ⊕_{m∈M}X_{m}.

Denote the grading of a homogeneous element x ∈ X either by |x| ∈ M, or write x_{m}, m ∈ M. Throughout the paper is everything stated in terms of homogeneous elements.

The arrows of mod_{R} (M) are grading preserving morphisms.

The tensor product X ⊗_{R} X′ of two objects in mod_{R} (M) is defined,

(X ⊗_{R} X′)_{m} = ⊕_{i+j=m}(X_{i} ⊗_{R} X′_{j}).

Quantizations and braidings of this category is described in [12] and [8]. Recall that any quantization of the monoidal category mod_{R} (M) is realized by a 2-cocycle q ∈ Z^{2} (M, U (R)),

q (i, j) q^{−1} (i, j + k) q (i + j, k) q^{−1} (j, k) = 1,  (1)

i, j, k ∈ M. When we factor out the trivial quantizations we are left with H^{2} (M, U (R)). For homogeneous elements x ∈ X, y ∈ Y in the M-graded modules X and Y a quantization has the form

q : x ⊗ y ⟼ q (|x|, |y|) x ⊗ y.

Note that quantizations preserve associativity constraints.

Any braiding in mod_{R} (M) is realized by σ : M × M → U (R) which is a bihomomorphism,

σ (i + j, k) = σ (i, k) σ (j, k) ,
σ (i, j + k) = σ (i, j) σ (i, k) ,

and a symmetry,

σ (i, j) σ (j, i) = 1,

i, j, k ∈ M. For homogeneous elements x ∈ X, y ∈ Y in the M-graded modules X and Y a braiding has the form

σ : x ⊗ y ⟼ σ (|x|, |y|) y ⊗ x.

Any braiding σ is also a 2-cocycle gives a quantization when composed with the twist, τ ∘ σ.

A quantization by q of a braiding σ is

σ_{q} (i, j) = q^{−1} (j, i) σ (i, j) q (i, j) ,

i, j ∈ M.

2.1. Quantizations of graded algebras. An algebra A in mod_{R} (M) is called an M-graded R-algebra and is equipped with multiplication

μ : A ⊗ A → A

which maps A_{i} ⊗ A_{j} to A_{i+j}, i, j ∈ M.

Given a quantization q, a quantization of an algebra A in mod_{R} (M) is a new multiplication μ_{q} defined as follows

μ_{q} (a ⊗ b) = a ∗_{q} b = q (|a|, |b|) ab

where a, b ∈ A are homogeneous and the multiplication on the right hand side is the old multiplication.

Let σ be a braiding and let A be a σ-commutative algebra in mod_{R} (M),

ab = σ (|a|, |b|) ba

for all homogeneous a, b ∈ A

Let q be a quantization and σ_{q} be the quantized braiding. Let A be a σ-commutative algebra in mod_{R} (M). Then A_{q} is σ_{q}-commutative,

a ∗_{q} b = σ_{q} (|a|, |b|) b ∗_{q} a

for homogeneous a, b ∈ A_{q}.

2.2. Quantizations of graded modules. An A-module E in mod_{R} (M) is called a (left) M-graded A-module and is equipped with an action

ν : A ⊗ E → E

which maps A_{i} ⊗ E_{j} to E_{i+j}, i, j ∈ M and similarly for right modules.

For the category mod_{R} (M) a quantization of a left, respectively right, A-module E is

a ∗^{l} x = q (|a|, |x|) ax,

respectively

x ∗^{r} a = q (|x|, |a|) xa,

for homogeneous a ∈ A, x ∈ E.

A A-A-bimodule E is σ-symmetric if

ax = σ (|a|, |x|) xa,
xa = σ (|x|, |a|) ax,

for homogeneous a ∈ A, x ∈ E.

Note that since the braidings are symmetries will left σ-symmetric Amodule structure imply right σ-symmetric structure, hence we need only to consider left A-modules.

Let E and E′ be two σ-symmetric A-A-bimodules and E ⊗ E′ be their tensor product. Define the action of A on E ⊗ E′,

a (x ⊗ x′) = ax ⊗ x′ + σ (|a|, |x|) x ⊗ ax′,
(x ⊗ x′) a = x ⊗ x′a + σ (|x′|, |a|) xa ⊗ x′,

for homogeneous a ∈ A, x ∈ E, x′ ∈ E′, and E ⊗ E′ is σ-symmetric.

2.3. Quantizations of graded coalgebras. A coalgebra A with comultiplication Δ : A → A ⊗ A in the monoidal category mod_{R} (M) is called an M-graded R-coalgebra. The comultiplication Δ maps

A_{m} → ∑_{i+j=m} A_{i} ⊗ A_{j}.

Let σ be a braiding and let A be a σ-cocommutative coalgebra in mod_{R} (M), that is,

Δ (x) = ∑_{|x′|+|x′′|=|x|} x′ ⊗x′′ = ∑_{|x′|+|x′′|=|x|} σ^{−1} (|x′|, |x′′|) x′′ ⊗x′ = σ^{−1} ∘Δ (x)

for homogeneous x, x′, x′′ ∈ A.

A quantization of a coalgebra A in mod_{R} (M) is equipping A_{q} = A with a new comultiplication Δ_{q}, defined by

Δ_{q} (x) = q_{A,A}^{−1} ( ∑_{|x′|+|x′′|=|x|} x′ ⊗ x′′)
= ∑_{|x′|+|x′′|=|x|} q^{−1} (|x′|, |x′′|) x′ ⊗ x′′,

for homogeneous x, x′, x′′ ∈ A.

If A is σ-cocommutative, then A_{q} is σ_{q}-cocommutative. The σ_{q}-co-commutativity is

∑_{|x′|+|x′′|=|x|} x′ ⊗_{q} x′′ = σ_{q}^{−1} ( ∑_{|x′|+|x′′|=|x|} x′′ ⊗_{q} x′)

= ∑_{|x′|+|x′′|=|x|} q (|x′′|, |x′|) σ^{−1} (|x′|, |x′′|) q^{−1} (|x′|, |x′′|) q^{−1} (|x′′|, |x′|) x′′⊗x′

= ∑_{|x′|+|x′′|=|x|} σ^{−1} (|x′|, |x′′|) q^{−1} (|x′|, |x′′|) x′′ ⊗ x′,

for homogeneous x′, x′′ ∈ A_{q}. The comultiplication in the two last lines is the comultiplication of A.

2.4. Quantizations of graded internal homomorphisms. The category mod_{R} (M) is a closed, that is internal homomorphism hom (X, Y) exists for all objects X, Y.

Remark 1. If M = G is rather a group than a monoid, then for any two objects X and Y in mod_{R} (G) there exist a grading on hom (X, Y) as f ∈ hom (X, Y), which maps X_{i} to Y_{j}, is given the grading j − i ∈ G and the internal homomorphisms can be considered as objects in mod_{R} (G).

A quantization q_{h} of all hom is a new multiplication defined by

μ_{q}^{h} = μ^{h} ∘ q_{hom(Y,Z),hom(X,Y)} : hom (Y, Z) ⊗ hom (X, Y) → hom (X, Z)

and

g ∗_{q} f = q (|g|, |f|) g ∘ f,
f ∗_{q} x = q (|f|, |x|) f ∗ x,

for homogeneous f, g and x of grading |f|, |g|, |x| ∈ G.

From [9] we have the following results. Internal homomorphisms of a σsymmetric graded module E over a σ-symmetric algebra A, hom (E, E), has a braided Lie structure with respect to the braided commutator,

[,]^{σ} = μ^{h} − μ^{h} ∘ σ.

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 shall go through this in details for braided derivations in the next section.

3. Braided derivations in graded algebras

In this section we shall discuss braided derivations in σ-commutative graded algebras and quantizations of these.

By remark 1, let from now on M = G be a finite abelian group.

Let R be a field, σ be a braiding in the monoidal category of G-graded modules and A be a G-graded σ-commutative R-algebra.

Definition 2. A σ-derivation of A of degree |∂| ∈ G is an R-linear operator ∂ : A → A such that

∂ : A_{g} → A_{g+|∂|},

g ∈ G, that satisfies the σ-Leibniz rule,

∂ (ab) = ∂ (a) b + σ (|∂|, |a|) a∂ (b) ,  (2)

where a, b ∈ A are homogeneous and a is of grading |a| ∈ G.

The set of σ-derivations of degree k is denoted by Der_{k}^{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 homogeneous a, b ∈ A, ∂ ∈ Der_{|∂|}^{σ} (A), and

a∂ ∈ Der_{|a|+|∂|}^{σ} (A) .

Definition 3. A σ-commutator (or σ-bracket) on homogeneous elements ∂_{1}, ∂_{2} ∈ Der^{σ} (A) of degree |∂_{1}| and |∂_{2}| respectively, is defined by

[∂_{1}, ∂_{2}]^{σ} = ∂_{1}∂_{2} − σ (|∂_{1}|, |∂_{2}|) ∂_{2}∂_{1}.

Proposition 4. The σ-commutator of two σ-derivations is a σ-derivation of the combined degree,

[∂_{1}, ∂_{2}]^{σ} ∈ Der_{|∂_{1}|+|∂_{2}|}^{σ} (A) ,

∂_{1}, ∂_{2} ∈ Der^{σ} (A).

Proof. The σ-commutator of two derivations satisfies the σ-Leibniz rule,

([∂_{1}, ∂_{2}]^{σ}) (ab) = [∂_{1}, ∂_{2}]^{σ} (a) b + σ (|∂_{1}| + |∂_{2}|, |a|) a [∂_{1}, ∂_{2}]^{σ} (b) ,

for homogeneous ∂_{1}, ∂_{2} ∈ Der^{σ} (A), a, b ∈ A, which is easily proved, [∂_{1}, ∂_{2}]^{σ} (ab) = ∂_{1}∂_{2} (ab) − σ (|∂_{1}|, |∂_{2}|) ∂_{2}∂_{1} (ab)

= ∂_{1} (∂_{2} (a) b + σ (|∂_{2}|, |a|) a∂_{2} (b))
− σ (|∂_{1}|, |∂_{2}|) ∂_{2} (∂_{1} (a) b + σ (|∂_{1}|, |a|) a∂_{1} (b))
= ∂_{1}∂_{2} (a) b + σ (|∂_{1}|, |∂_{2}| + |a|) ∂_{2} (a) ∂_{1} (b)
+ σ (|∂_{2}|, |a|) ∂_{1} (a) ∂_{2} (b) + σ (|∂_{1}|, |a|) σ (|∂_{2}|, |a|) a∂_{1}∂_{2} (b)

− σ (|∂_{1}|, |∂_{2}|) ∂_{2}∂_{1} (a) b − σ (|∂_{1}|, |∂_{2}|) σ (|∂_{2}|, |∂_{1}| + |a|) ∂_{1} (a) ∂_{2} (b)

− σ (|∂_{1}|, |∂_{2}|) σ (|∂_{1}|, |a|) ∂_{2} (a) ∂_{1} (b)
− σ (|∂_{1}|, |∂_{2}|) σ (|∂_{1}|, |a|) σ (|∂_{2}|, |a|) a∂_{2}∂_{1} (b)
= ∂_{1}∂_{2} (a) b + σ (|∂_{1}| + |∂_{2}|, |a|) a∂_{1}∂_{2} (b)

− σ (|∂_{1}|, |∂_{2}|) ∂_{2}∂_{1} (a) b − σ (|∂_{1}|, |∂_{2}|) σ (|∂_{1}| + |∂_{2}|, |a|) a∂_{2}∂_{1} (b)

= [∂_{1}, ∂_{2}]^{σ} (a) b + σ (|∂_{1}| + |∂_{2}|, |a|) a [∂_{1}, ∂_{2}]^{σ} (b) .

Proposition 5. The σ-bracket satisfies the conditions,

[a∂_{1}, ∂_{2}]^{σ} = a [∂_{1}, ∂_{2}]^{σ} − σ (|a| + |∂_{1}|, |∂_{2}|) ∂_{2} (a) ∂_{1},  (3)
[∂_{1}, a∂_{2}]^{σ} = σ (|∂_{1}|, |a|) a [∂_{1}, ∂_{2}]^{σ} + ∂_{1} (a) ∂_{2},  (4)

∀a ∈ A. Proof.

[a∂_{1}, ∂_{2}]^{σ} (b) = a∂_{1}∂_{2} (b) − σ (|a| + |∂_{1}|, |∂_{2}|) ∂_{2} (a∂_{1}) (b)

= a∂_{1}∂_{2} (b) − σ (|a| + |∂_{1}|, |∂_{2}|) (∂_{2} (a) ∂_{1} (b) + σ (|∂_{2}|, |a|) a∂_{2}∂_{1} (b)) = (a [∂_{1}, ∂_{2}]^{σ} − σ (|a| + |∂_{1}|, |∂_{2}|) ∂_{2} (a) ∂_{1}) (b) , [∂_{1}, a∂_{2}]^{σ} (b) = ∂_{1} (a∂_{2}) (b) − σ (|∂_{1}|, |a| + |∂_{2}|) a∂_{2}∂_{1} (b) = ∂_{1} (a) ∂_{2} (b) + σ (|∂_{1}|, |a|) a∂_{1}∂_{2} (b) − σ (|∂_{1}|, |a| + |∂_{2}|) a∂_{2}∂_{1} (b) c = (σ (|∂_{1}|, |a|) a [∂_{1}, ∂_{2}]^{σ} + ∂_{1} (a) ∂_{2}) (b) .

The braided derivations is a braided Lie algebra as defined in [5]. Theorem 6. Der^{σ} (A) is a G-graded σ-Lie 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_{i}^{σ} (A) , Der_{j}^{σ} (A)]^{σ} ⊆ Der_{i+j}^{σ} (A) ,  (i’)

i, j ∈ G, skew σ-symmetricity,

[∂_{1}, ∂_{2}]^{σ} = −σ (|∂_{1}|, |∂_{2}|) [∂_{2}, ∂_{1}]^{σ} ,  (ii)

the σ-Jacobi identity for derivations,

[∂_{1}, [∂_{2}, ∂_{3}]^{σ}]^{σ} = [[∂_{1}, ∂_{2}]^{σ} , ∂_{3}]^{σ} + σ (|∂_{1}|, |∂_{2}|) [∂_{2}, [∂_{1}, ∂_{3}]^{σ}]^{σ} ,  (iii)

for ∂_{1}, ∂_{2}, ∂_{3} ∈ Der^{σ} (A) of degree |∂_{1}|, |∂_{2}|, |∂_{3}| respectively.

Proof. Let’s do the proof for skew σ-symmetricity,

[∂_{1}, ∂_{2}]^{σ} = ∂_{1}∂_{2} − σ (|∂_{1}|, |∂_{2}|) ∂_{2}∂_{1}
= σ (|∂_{1}|, |∂_{2}|) ∂_{2}∂_{1} − σ (|∂_{1}|, |∂_{2}|) σ (|∂_{2}|,|∂_{1}|) ∂_{2}∂_{1}
= −σ (|∂_{1}|, |∂_{2}|) [∂_{2}, ∂_{1}]^{σ} .

3.1. Quantizations of braided derivations in graded algebras. Let A be a σ-commutative G-graded algebra. Given a quantization q and an operator ∂ : A → A of degree |∂| define it’s quantization

∂_{q} (a) = Q_{q} (∂) (a) =^{def} q (|∂|, |a|) ∂ (a) ,  (5)

for homogeneous a ∈ A_{|a|}.

Q_{q} (∂) is an operator of the quantized M-graded algebra A_{q}.

The quantization of composition is

∂_{1} ∗_{q} ∂_{2} = q (|∂_{1}|, |∂_{2}|) ∂_{1} ∘ ∂_{2}.  (6)

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

Theorem 7. Given a braiding σ, let σ_{q} be the quantization of σ. The operator

Q_{q} : (Der^{σ} (A) , [−, −]^{σ}) → (Der^{σ^{q}} (A_{q}) , [−, −]_{q}^{σ_{q}}) ,  (7)
∂ ∈ 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. The σ_{q}-Leibniz rule is satisfied

Q_{q} (∂) (a ∗_{q} b) = q (|a|, |b|) q (|∂|, |a| + |b|) ∂ (ab)
= q (|∂|, |a|) q (|∂| + |a|, |b|) (∂ (a) b + σ (|∂|, |a|) a∂ (b))
Q_{q} (∂) (a ∗_{q} b) = Q_{q} (∂) (a) ∗_{q} b + σ_{q} (|∂|, |a|) a ∗_{q} Q_{q} (∂) (b)

where Q_{q} (∂) (a) ∗_{q} b = q (|∂|, |a|) q (|∂| + |a|, |b|) ∂ (a) bσ_{q} (|∂|, |a|) a ∗_{q} Q_{q} (∂) (b) = q^{−1} (|a|, |∂|) σ (|∂|, |a|) q (|∂|, |a|) a ∗_{q} Q_{q} (∂) (b) = q^{−1} (|a|, |∂|) σ (|∂|, |a|) q (|∂|, |a|) q (|a|, |∂| + |b|) q (|∂|, |b|) a∂ (b) = q (|∂|, |a|) q (|∂| + |a|, |b|) σ (|∂|, |a|) a∂ (b) .

The σ_{q}-commutator of two derivations satisfies the σ_{q}-Leibniz rule,

[∂_{1}, ∂_{2}]_{q}^{σ_{q}} (ab) = ∂_{1} ∗_{q} ∂_{2} (ab) − σ_{q} (|∂_{1}|, |∂_{2}|) ∂_{2} ∗_{q} ∂_{1} (ab)
= (∂_{1} ∗_{q} ∂_{2}) (a) b + q (|∂_{1}|, |∂_{2}|) σ_{q} (|∂_{1}|, |∂_{2}| + |a|) ∂_{2} (a) ∂_{1} (b)
+ q (|∂_{2}|, |∂_{1}|) σ_{q} (|∂_{2}|, |a|) ∂_{1} (a) ∂_{2} (b)

+ σ_{q} (|∂_{1}| + |∂_{2}|, |a|) a∂_{1} ∗_{q} ∂_{2} (b) − σ_{q} (|∂_{1}|, |∂_{2}|) ∂_{2} ∗_{q} ∂_{1} (a) b

− q (|∂_{2}|, |∂_{1}|) σ_{q} (|∂_{1}|, |∂_{2}|) σ_{q} (|∂_{2}|, |∂_{1}| + |a|) ∂_{1} (a) ∂_{2} (b)
− q (|∂_{1}|, |∂_{2}|) σ_{q} (|∂_{1}|, |∂_{2}|) σ_{q} (|∂_{1}|, |a|) ∂_{2} (a) ∂_{1} (b)
− σ_{q} (|∂_{1}|, |∂_{2}|) σ_{q} (|∂_{1}| + |∂_{2}|, |a|) a∂_{2} ∗_{q} ∂_{1} (b)
= (∂_{1} ∗_{q} ∂_{2}) (a) b + σ_{q} (|∂_{1}| + |∂_{2}|, |a|) a (∂_{1} ∗_{q} ∂_{2}) (b)
− σ_{q} (|∂_{1}|, |∂_{2}|) (∂_{2} ∗_{q} ∂_{1}) (a) b
− σ_{q} (|∂_{1}|, |∂_{2}|) σ_{q} (|∂_{1}| + |∂_{2}|, |a|) a (∂_{2} ∗_{q} ∂_{1}) (b)
= [∂_{1}, ∂_{2}]^{σ^{q}} (a) b + σ_{q} (|∂_{1}| + |∂_{2}|, |a|) a [∂_{1}, ∂_{2}]^{σ^{q}} (b) ,

for homogeneous ∂_{1}, ∂_{2} ∈ Der^{σ^{q}} (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}} − σ_{q} (|a| + |∂_{1}|, |∂_{2}|) ∂_{2} (a) ∂_{1},  (8)
[∂_{1}, a∂_{2}]_{q}^{σ_{q}} = σ_{q} (|∂_{1}|, |a|) a [∂_{1}, ∂_{2}]_{q}^{σ_{q}} + ∂_{1} (a) ∂_{2},  (9)

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

Furthermore, Der^{σ^{q}} (A_{q}) is a σ_{q}-Lie algebra with respect to the (σ_{q}−q)- bracket, [−, −]_{q}^{σ_{q}}.

Let q_{1}, q_{2} and q be quantizations, then

Q_{q_{1}q_{2}} = Q_{q_{1}} ∘ Q_{q_{2}}

and

Q_{q^{−1}} = Q_{q}^{−1}.

The inverse of the quantization of an operator ∂ of A_{q} is denoted by

Q_{q}^{−1} (∂) = ∂_{c}.

As an object, (A_{q})_{c} = A_{q} = A.

Proposition 8. The composition satisfies

(∂_{1})_{c} ∗_{q} (∂_{2})_{c} = (∂_{1} ∘ ∂_{2})_{c} ,  (10)

for graded operators ∂_{1} and ∂_{2} on A_{q}.

Proof.

Q_{q}^{−1} (∂_{1}) ∗_{q} Q_{q}^{−1} (∂_{2}) (a)

= q (|∂_{1}|, |∂_{2}|) q^{−1} (|∂_{1}|, |∂_{2}| + |a|) q^{−1} (|∂_{2}|, |a|) ∂_{1}∂_{2} (a) = q^{−1} (|∂_{1}| + |∂_{2}|, |a|) ∂_{1}∂_{2} (a) = Q_{q}^{−1} (∂_{1} ∘ ∂_{2}) (a) ,

for homogeneous a ∈ A_{q}.

Let γ be any braiding and p any quantization. Define the γ−p -bracket,

[−, −]_{p}^{γ} ,

on operators, for which the composition between the operators is ∗_{p},

[∂_{1}, ∂_{2}]_{p}^{γ} = ∂_{1} ∗_{p} ∂_{2} − γ (|∂_{1}|, |∂_{2}|) ∂_{2} ∗_{p} ∂_{1}.  (11)

Proposition 9. Let ∂_{1} ∈ Der_{|∂_{1}|}^{σ} (A), ∂_{2} ∈ Der_{|∂_{2}|}^{σ} (A), then

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

Proof.

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

= q (|∂_{1}|, |∂_{2}|) ∂_{1}∂_{2}

− q^{−1} (|∂_{2}|, |∂_{1}|) σ (|∂_{1}|, |∂_{2}|) q (|∂_{1}|, |∂_{2}|) q (|∂_{2}|, |∂_{1}|) ∂_{2}∂_{1}

= q (|∂_{1}|, |∂_{2}|) [∂_{1}, ∂_{2}]^{σ} .

Definition 10. Define dequantization of Der^{σ^{q}} (A_{q}) as the inverse of the set of all Q_{q}^{−1} (∂), ∂ ∈ Der^{σ^{q}} (A_{q}), equipped with the [−, −]_{q}^{σ_{q}} bracket and the A-module structure

a ∗_{q} ∂_{c} = q (|a|, |∂|) a∂_{c},  (12)

for homogeneous ∂ ∈ Der^{σ^{q}} (A_{q}) and a ∈ A_{|a|}.

The dequantization of the braided derivations operates on A in the classical manner, but satisfies somewhat different properties than the classical, as the following theorem states.

Theorem 11. The braided Lie algebra structure of Der^{σ^{q}} (A_{q}) can be realized within the classical, Der^{σ} (A), by dequantization.

For homogeneous elements ∂_{1}, ∂_{2} ∈ Der^{σ^{q}} (A_{q}), a ∈ A_{q}, the following linearity is satisfied,

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

A-module structure,

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

and for the commutator,

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

Proof. (i):

Q_{q}^{−1} (∂_{1} + ∂_{2}) = Q_{q}^{−1} (∂_{1}) + Q_{q}^{−1} (∂_{2}) ,

(ii), A-module structure: By proposition 8,

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

(iii), σ_{q}-bracket:

Q_{q}^{−1} ([∂_{1}, ∂_{2}]^{σ^{q}}) (a) = q^{−1} (|∂_{1}| + |∂_{2}|, |a|) (∂_{1}∂_{2} − σ_{q} (|∂_{1}|, |∂_{2}|) ∂_{2}∂_{1}) (a)

is equal to

[(∂_{1})_{c} , (∂_{2})_{c}]_{q}^{σ_{q}} = q^{−1} (|∂_{1}|, |∂_{2}| + |a|) q^{−1} (|∂_{2}|, |a|) q (|∂_{1}|, |∂_{2}|) ∂_{1}∂_{2} (a)

− q^{−1} (|∂_{2}|, |∂_{1}| + |a|) q^{−1} (|∂_{1}|, |a|) q (|∂_{2}|, |∂_{1}|) σ_{q} (|∂_{1}|, |∂_{2}|) ∂_{2}∂_{1} (a) = q^{−1} (|∂_{1}| + |∂_{2}|, |a|) q^{−1} (|∂_{1}|, |∂_{2}|) q (|∂_{1}|, |∂_{2}|) ∂_{1}∂_{2} (a)

− q^{−1} (|∂_{1}| + |∂_{2}|, |a|) q^{−1} (|∂_{2}|, |∂_{1}|) q (|∂_{2}|, |∂_{1}|) σ_{q} (|∂_{1}|, |∂_{2}|) ∂_{2}∂_{1} (a) = q^{−1} (|∂_{1}| + |∂_{2}|, |a|) (∂_{1}∂_{2} − σ_{q} (|∂_{1}|, |∂_{2}|) ∂_{2}∂_{1}) (a) ,

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

3.2. 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 12. Let A be σ-commutative algebra and ∂ ∈ Der_{|∂|}^{σ} (A), a ∈ A_{|a|}. Then the evaluation of ∂_{c} on homogeneous a ∈ A is

∂ (a) = [∂, a]^{σ} .

Let

∂_{q} ∈ Der_{|∂|}^{σ_{q}} (A_{q}) .

Then the evaluation of the derivation on some homogeneous a ∈ (A_{q})_{|a|} is equal to taking the σ_{q} − q-bracket of ∂_{q} and a,

∂_{q} (a) = [∂_{q}, a]_{q}^{σ_{q}} .

Proof. Let ∂ ∈ Der_{|∂|}^{σ} (A) and a ∈ A_{|a|}, b ∈ A_{|b|}. By the σ-Leibniz rule

∂ (ab) = ∂ (a) b + σ (|∂|, |a|) a∂ (b) ,

and clearly, by rearranging,

∂ (a) = [∂, a]^{σ} = ∂a − σ (|∂|, |a|) a∂.

For proof of the second half of the proposition let ∂_{q} ∈ Der_{|∂|}^{σ_{q}} (A_{q}) and a ∈ A_{|a|}, b ∈ A_{|b|}. By the σ-Leibniz rule

∂_{q} (ab) = ∂_{q} (a) ∗_{q} b + σ_{q} (|∂|, |a|) a ∗_{q} ∂_{q} (b) ,

and by rearranging,

∂_{q} (a) = [∂_{q}, a]_{q}^{σ_{q}} = ∂_{q} ∗_{q} a − σ_{q} (|∂|, |a|) a ∗_{q} ∂_{q}.

4. Braided derivations in graded modules

Let R be a field, σ be a braiding in the monoidal category of graded modules, A be a G-graded σ-commutative R-algebra and E a G-graded σ-symmetric A-module. Let

∂_{A} : A → A

be a G-graded σ-derivation of A.

Definition 13. An operator of E, ∂ : E → E is said to be a graded σ-derivation over ∂_{A} of degree |∂| ∈ G if ∂ is R-linear,

∂ : E_{g} → E_{g+|∂|},

g ∈ G

|∂| = |∂_{A}|,

and satisfy the σ-Leibniz rule with respect to ∂_{A},

∂ (ax) = ∂_{A} (a) x + σ (|∂|, |a|) a∂ (x) ,

for homogeneous x ∈ E and a ∈ A.

The 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.

The set of all σ-derivations of E over A of degree g ∈ G is denoted by Der_{g}^{(σ,A)} (E) and the set of all σ-derivations of E over A (equipped with the quantization of the composition) is denoted by Der^{(σ,A)} (E) .

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

(a∂) (b) = a (∂ (x)) ,  (13)

and

a∂ ∈ Der_{|a|+|∂|}^{(σ,A)} (E) ,  (14)

for homogeneous a ∈ A, x ∈ E ∂ ∈ Der_{|∂|}^{(σ,A)} (E).

The σ-commutator is defined as for σ-derivations of A.

Proposition 14. The σ-commutator of two σ-derivations is a σ-derivation of the combined degree,

[∂_{1}, ∂_{2}]^{σ} ∈ Der_{|∂_{1}|+|∂_{2}|}^{(σ,A)} (E) ,

for homogeneous ∂_{1}, ∂_{2} ∈ Der^{(σ,A)} (E).

Proof. The σ-commutator of two derivations satisfies the σ-Leibniz rule over A,

([∂_{1}, ∂_{2}]^{σ}) (ax) = [(∂_{1})_{A} , (∂_{2})_{A}]^{σ} (a) x + σ (|∂_{1}| + |∂_{2}|, |a|) a [∂_{1}, ∂_{2}]^{σ} (x) ,

for homogeneous a ∈ A, which is proved as follows,

[∂_{1}, ∂_{2}]^{σ} (ax) = ∂_{1}∂_{2} (ax) − σ (|∂_{1}|, |∂_{2}|) ∂_{2}∂_{1} (ax) = (∂_{1})_{A} (∂_{2})_{A} (a) x + σ (|∂_{1}|, |∂_{2}| + |a|) (∂_{2})_{A} (a) ∂_{1} (x)

+ σ (|∂_{2}|, |a|) (∂_{1})_{A} (a) ∂_{2} (x) + σ (|∂_{1}|, |a|) σ (|∂_{2}|, |a|) a∂_{1}∂_{2} (x)
− σ (|∂_{1}|, |∂_{2}|) (∂_{2})_{A} (∂_{1})_{A} (a) x
− σ (|∂_{1}|, |∂_{2}|) σ (|∂_{2}|, |∂_{1}| + |a|) (∂_{1})_{A} (a) ∂_{2} (x)
− σ (|∂_{1}|, |∂_{2}|) σ (|∂_{1}|, |a|) (∂_{2})_{A} (a) ∂_{1} (x)
− σ (|∂_{1}|, |∂_{2}|) σ (|∂_{1}|, |a|) σ (|∂_{2}|, |a|) a∂_{2}∂_{1} (x)

= (∂_{1})_{A} (∂_{2})_{A} (a) x + σ (|∂_{1}| + |∂_{2}|, |a|) a∂_{1}∂_{2} (x) − σ (|∂_{1}|, |∂_{2}|) (∂_{2})_{A} (∂_{1})_{A} (a) x − σ (|∂_{1}|, |∂_{2}|) σ (|∂_{1}| + |∂_{2}|, |a|) a∂_{2}∂_{1} (x) = [(∂_{1})_{A} , (∂_{2})_{A}]^{σ} (a) x + σ (|∂_{1}| + |∂_{2}|, |a|) a [∂_{1}, ∂_{2}]^{σ} (x) .

Proposition 15. The σ-bracket satisfies

[a∂_{1}, ∂_{2}]^{σ} = a [∂_{1}, ∂_{2}]^{σ} − σ (|a| + |∂_{1}|, |∂_{2}|) ∂_{2} (a) ∂_{1},  (15)
[∂_{1}, a∂_{2}]^{σ} = σ (|∂_{1}|, |a|) a [∂_{1}, ∂_{2}]^{σ} + ∂_{1} (a) ∂_{2},  (16)

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

Theorem 16. Der^{(σ,A)} (E) is a G-graded σ-Lie algebra with respect to the σ-bracket. That is, the following properties are satisfied,

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

and [∂_{1}, ∂_{2}]^{σ} is a σ-derivation over [(∂_{1})_{A} , (∂_{2})_{A}]^{σ}, the σ-bracket is skew σ-symmetric,

[∂_{1}, ∂_{2}]^{σ} = −σ (|∂_{1}|, |∂_{2}|) [∂_{2}, ∂_{1}]^{σ} ,  (ii)

and the σ-Jacobi identity for derivations is satisfied,

[∂_{1}, [∂_{2}, ∂_{3}]^{σ}]^{σ} = [[∂_{1}, ∂_{2}]^{σ} , ∂_{3}]^{σ} + σ (|∂_{1}|, |∂_{2}|) [∂_{2}, [∂_{1}, ∂_{3}]^{σ}]^{σ} ,  (iii)

for homogeneous ∂_{1}, ∂_{2}, ∂_{3} ∈ Der^{(σ,A)} (E).

We get the exact sequence of graded A-modules and G-graded Lie algebras

0 → End_{A}^{σ} (E) → Der^{(σ,A)} (E) →^{π} Der^{σ} (A)  (17)

where End_{A}^{σ} (E) is the σ-symmetric (graded) endomorphisms of E over A.

4.1. Quantizations of braided derivations in graded modules. Let A be a σ-commutative G-graded algebra and E a σ-commutative G-graded A-module

Given a quantization q and an operator

∂ : E → E

of degree |∂|, define the quantization of ∂,

∂_{q} (x) = Q_{q} (∂) (x) = q (|∂|, |x|) ∂ (x) ,  (18)

x ∈ E.

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

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

Theorem 17. The operator Q_{q} is an A-module isomorphism between the σ-derivations of E over A and the σ_{q}-derivations of E_{q} over A_{q},

Q_{q} : (Der^{(σ,A)} (E) , [−, −]^{σ}) → (Der^{(σ_{q},A_{q})} (E_{q}) , [−, −]_{q}^{σ_{q}}) , (19)
∂ ∈ Der_{|∂|}^{(σ,A)} (E) ⟼ Q_{q} (∂) ∈ Der_{|∂|}^{(σ_{q},A_{q})} (E_{q}) .

That is, Q_{q} (∂) satisfies the σ_{q}-Leibniz rule with respect to Q_{q} (∂_{A})

Q_{q} (∂) (ax) = Q_{q} (∂_{A}) (a) x + σ_{q} (|∂|, |a|) aQ_{q} (∂) (x) ,  (20)

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

Der^{(σ_{q},A_{q})} (E_{q}) is a σ_{q}-symmetric module, satisfying (8) and (9), and is a σ_{q}-Lie algebra with respect to the σ_{q} − q-bracket, [−, −]_{q}^{σ_{q}}.

Theorem 18. The Lie algebra structure of Der^{(σ_{q},A)} (E_{q}) can be realized within the classical, Der^{(σ,A)} (E), by dequantization. That is the following is satisfied. The linearity,

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

for homogeneous ∂_{1}, ∂_{2} ∈ Der^{(σ_{q},A)} (E_{q}), A-module structure,

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

and the commutator,

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

for homogeneous ∂_{1}, ∂_{2} ∈ Der^{(σ_{q},A)} (E_{q}), a ∈ A_{q}.

We get the following commutative diagram exact sequences of graded A-modules and G-graded Lie algebras

(21)

where

π_{q} =^{def} Q_{q} ∘ π ∘ Q_{q}^{−1}.  (22)

5. Braided connections and curvature in graded modules

Let ∂_{1}, ∂_{2} ∈ Der^{σ} (A) be homogeneous.

Definition 19. A σ-connection in a σ-symmetric graded module E is a σ-symmetric graded module homomorphism ∇ of degree 0

∇ : Der_{|∂_{1}|}^{σ} (A) → Der_{|∂_{1}|}^{(σ,A)} (E)

such that

π ∘ ∇ = Id.

Definition 20. A σ-connection ∇ is flat if it is a σ-Lie algebra homomorphism, that is,

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

for all ∂_{1}, ∂_{2} ∈ Der^{σ} (A). Definition 21. In general, define the σ-curvature of ∇ to be

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

Theorem 22. Given homogeneous ∂_{1} and ∂_{2}, the σ-curvature

K_{∇} : Der^{σ} (A) ⊗ Der^{σ} (A) → End_{A} (E) ,

applied to ∂_{1} and ∂_{2} is a σ-symmetric endomorphism of E, that is,

K_{∇} (∂_{1}, ∂_{2}) (a∂_{1}) = σ (|∂_{1}| + |∂_{2}|, a) aK_{∇} (∂_{1}, ∂_{2}) (∂_{1}) ,  (i)

and K_{∇} is skew σ-symmetric,

K_{∇} (∂_{1}, ∂_{2}) = −σ (|∂_{1}|, |∂_{2}|) K_{∇} (∂_{2}, ∂_{1}) .  (ii)

Furthermore K_{∇} satisfies the σ-symmetric A-module homomorphisms

K_{∇} (a∂_{1}, ∂_{2}) = aK_{∇} (∂_{1}, ∂_{2}) ,  (iii)
K_{∇} (∂_{1}, a∂_{2}) = σ (|∂_{1}|, |a|) aK_{∇} (∂_{1}, ∂_{2}) ,  (iv)

a ∈ A. Proof. (i):

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

= [(∇∂_{1})_{A} , (∇∂_{2})_{A}]^{σ} (a) x + σ (|∂_{1}| + |∂_{2}|, a) a [∇∂_{1}, ∇∂_{2}]^{σ} (x)

− (∇ ([∂_{1}, ∂_{2}]^{σ}))_{A} (a) x − σ (|∂_{1}| + |∂_{2}|, |a|) a (∇ ([∂_{1}, ∂_{2}]^{σ})) (x) = [∂_{1}, ∂_{2}]^{σ} (a) x + σ (|∂_{1}| + |∂_{2}|, a) a [∇∂_{1}, ∇∂_{2}]^{σ} (x)

− [∂_{1}, ∂_{2}]^{σ} (a) x − σ (|∂_{1}| + |∂_{2}|, |a|) a (∇ ([∂_{1}, ∂_{2}]^{σ})) (x) = σ (|∂_{1}| + |∂_{2}|, a) (a [∇∂_{1}, ∇∂_{2}]^{σ} (x) − a (∇ ([∂_{1}, ∂_{2}]^{σ})) (x))

= σ (|∂_{1}| + |∂_{2}|, a) aK_{∇} (∂_{1}, ∂_{2}) (x) .

(ii):

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

148 H. L. HURU (iii): K_{∇} (a∂_{1}, ∂_{2}) = [∇ (a∂_{1}) , ∇∂_{2}]^{σ} − ∇ ([a∂_{1}, ∂_{2}]^{σ})

= a ( \begin{matrix}[(∇∂_{1}) , ∇∂_{2}]^{σ} − σ (|a| + |∂_{1}|, |∂_{2}|) ∇ (∂_{2}) (a) ∇ (∂_{1}) \\ −a∇ [∂_{1}, ∂_{2}]^{σ} + σ (|a| + |∂_{1}|, |∂_{2}|) ∇ (∂_{2} (a) ∂_{1}) \end{matrix} )

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

(iv):

K_{∇} (∂_{1}, a∂_{2}) = [∇∂_{1}, ∇ (a∂_{2})]^{σ} − ∇ ([∂_{1}, a∂_{2}]^{σ})
= [∇∂_{1}, a (∇∂_{2})]^{σ} − σ (|∂_{1}|, |a|) ∇ (a [∂_{1}, ∂_{2}]^{σ})
= σ (|∂_{1}|, |a|) a [∇∂_{1}, (∇∂_{2})]^{σ} + ∇ (∂_{1}) (a) ∇ (∂_{2})
−σ (|∂_{1}|, |a|) a∇ ([∂_{1}, ∂_{2}]^{σ}) + ∇ (∂_{1} (a) ∂_{2})
= σ (|∂_{1}|, |a|) a ([∇∂_{1}, ∇∂_{2}]^{σ} − ∇ [∂_{1}, ∂_{2}]^{σ}) .

5.1. Quantization of braided connections and curvature. Definition 23. Let ∇ be a σ-connection in E. The quantization of ∇,

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

is defined by

∇_{q} =^{def} Q_{q} ∘ ∇ ∘ Q_{q}^{−1},  (23)

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}].

Hence, ∇_{q} is a splitting of the lower sequence in (21). Proposition 24. The quantization of a connection ∇, ∇_{q}, is a σ_{q}- connection in E_{q}.

Let ∇_{q} be a σ_{q}-connection in E_{q}. Then the σ_{q} − q-curvature of ∇_{q}

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

is defined by

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

Theorem 25. The σ_{q} − q-curvature satisfies

K_{∇_{q}}^{q} (∂_{1}, ∂_{2}) (ax) = σ_{q} (|∂_{1}| + |∂_{2}|, a) aK_{∇_{q}}^{q} (∂_{1}, ∂_{2}) (x) ,  (i)

and is skew σ_{q}-symmetric,

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

Furthermore, the σ_{q} − q-curvature satisfies the σ_{q}-symmetric A_{q}-module homomorphisms

K_{∇_{q}}^{q} (a∂_{1}, ∂_{2}) =aK_{∇_{q}}^{q} (∂_{1}, ∂_{2}) ,  (ii)
K_{∇_{q}}^{q} (∂_{1}, a∂_{2}) =σ_{q} (|∂_{1}|, |a|) aK_{∇_{q}}^{q} (∂_{1}, ∂_{2}) ,  (iii)

a ∈ A_{q}.

We get the following picture for dequantizations of braided derivations.

Theorem 26. 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}}) , (25)

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}})  (26)

are related as follows,

(K_{∇_{q}} (∂_{1}, ∂_{2}))_{c} = K_{∇}^{q} ((∂_{1})_{c} , (∂_{2})_{c}) ,  (27)

∂_{1}, ∂_{2} ∈ Der^{σ^{q}} (A_{q}). If ∇ is flat σ-connection in E with respect to the σ-curvature K_{∇}, then is ∇ is flat in E with respect to the σ_{q}−q-curvature K_{∇}^{q} and ∇_{q} is a flat σ_{q}-connection in E_{q} with respect to the σ_{q}-curvature K_{∇_{q}}.

Proof. Proof of (27):

K_{∇_{q}} (∂_{1}, ∂_{2})

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

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

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

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

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

If K_{∇} (∂_{1}, ∂_{2}) = 0, then

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

= q (|∂_{1}|, |∂_{2}|) ([∇ (∂_{1})_{c} , ∇ (∂_{2})_{c}]^{σ} − ∇ ([(∂_{1})_{c} , (∂_{2})_{c}]^{σ})) = 0.

The formula (27) means that

K_{∇_{q}} (∂_{1}, ∂_{2}) (x) = Q_{q} (K_{∇}^{q} ((∂_{1})_{c} , (∂_{2})_{c})) (x)
= q (|∂_{1}|, |∂_{2}|) Q_{q} (K_{∇} ((∂_{1})_{c} , (∂_{2})_{c})) (x)
= q (|∂_{1}| + |∂_{2}|, |x|) q (|∂_{1}|, |∂_{2}|) K_{∇} ((∂_{1})_{c} , (∂_{2})_{c}) (x) ,

for x ∈ E.

6. Application to γ-densities and γ-forms on ℝ.

Consider the real line, ℝ, and γ-densities on ℝ,

θ = f (x) (|dx|)^{γ} ,

and γ-forms on ℝ

θ′ = f (x) (dx)^{γ} ,

γ ∈ ℝ and f is a function on ℝ. Denote by Ω_{γ} the set of all γ-densities on ℝ.

If γ = 1, then we have differential 1-forms on ℝ, if γ = −1, then we have vector fields on ℝ and if γ = 0, then we have functions on ℝ.

For a function on y, g (y), a γ-change of variable is

g (y) dy ⟼ g (F (x)) |F′|^{γ} (dx)^{γ} ,

if y = F (x).

Ω = ⊕_{γ∈ℝ}Ω_{γ} is an algebra with the multiplication

Ω_{β} ⊗ Ω_{γ} → Ω_{β+γ},
f (x) (|dx|)^{α} ⊗ g (x) (|dx|)^{β} ↦ fg (x) (|dx|)^{α+β} .

Definition 27. Define a bracket on Ω by

[f_{β}|dx|^{β}, g_{γ}|dx|^{γ}] = [f_{β}, g_{γ}] |dx|^{β+γ−1},

where

[g_{β}, f_{γ}] = γg′_{β}f_{γ} − βg_{β}f′_{γ},

g_{β} ∈ Ω_{β} and f_{γ} ∈ Ω_{γ}.

Clearly,

[Ω_{β}, Ω_{γ}] ⊆ Ω_{β+γ−1},

since f′_{γ} ∈ Ω_{γ−1} when f_{γ} ∈ Ω_{γ}.

Proposition 28. Ω is a ℝ-graded Lie algebra. That is,

[g_{β}, f_{γ}] = − [f_{β}, g_{γ}] ,

and the Jacobi identity is satisfied,

[h_{α}, [g_{β}, f_{γ}]] = [[h_{α}, g_{β}] , f_{γ}] + [g_{β}, [h_{α}, f_{γ}]] ,

h_{α} ∈ Ω_{α}, g_{β} ∈ Ω_{β}, f_{γ} ∈ Ω_{γ}.

Now consider a collection of n-tuples

Γ = {γ = (γ_{1}, …, γ_{n})} ⊂ ℝ^{n},

such that for γ, γ′ ∈ Γ the sum γ + γ′ ∈ Γ and γ + γ′ − 1_{i} ∈ Γ for all i = 1, …, n, where 1_{i} = (0, …, 1, …, 0), 1 in the ith place.

For each γ = (γ_{1}, …, γ_{n}) ∈ Γ consider Ω_{γ} = ⊕_{γ_{i}}Ω_{γ_{i}} and the Γ-graded module Ω_{Γ} = ⊕_{γ}Ω_{γ}. Clearly is Ω_{Γ} a Γ-graded Lie algebra.

Any quantization of Ω_{Γ} is given by

q (γ, γ′) = exp (i ⟨P γ, γ′⟩) ,

where P is a skew symmetric n × n-matrix and the quantization of the standard twist, τ_{q} = q^{−1} ∘ τ ∘ q, is realized by

q^{−1} (γ′, γ) q (γ, γ′) = exp (2i ⟨P γ, γ′⟩) .

Then the quantization of Ω_{Γ} by q is a Γ-graded τ_{q}-Lie algebra, that is,

[g_{γ}, f_{γ}′] = − exp (2i ⟨P γ, γ′⟩) [f_{γ}′, g_{γ}] ,

and the Jacobi identity is satisfied,

[h_{γ}, [g_{γ}′, f_{γ}′′]] = [[h_{γ}, g_{γ}′] , f_{γ}′′] + exp (2i ⟨P γ, γ′⟩) [g_{γ}′, [h_{γ}, f_{γ}′′]] ,

h_{γ} ∈ Ω_{γ}, g_{γ}′ ∈ Ω_{γ}′, f_{γ}′′ ∈ Ω_{γ}′′.

Note that even if Γ is infinite there is no problem to extend the theory in this paper to this case as long as there is some minimal grading ς such that there is no f ∈ Ω_{Γ} with grading by γ < ς.

7. Braided differential operators

We shall see how the picture is for braided differential operators. Let G be a finite abelian group.

7.1. Braided differential operators in graded algebras. Let A be a σ-symmetric G-graded algebra.

Define a graded σ-differential operator f of degree |f| ∈ G and order at most k as the linear map f : A → A, such that

f : A_{g} → A_{g+|f|},

g ∈ G, and

[a_{0}, [a_{1}, … [a_{k}, f]^{σ} ⋯]^{σ}]^{σ} = 0,  (28)

∀a_{0}, …, a_{k} ∈ A.

Denote by Diff_{k,i}^{σ} (A, A) the σ-differential operators of order at most k and degree i 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 algebra. Furthermore, 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).

Proposition 29. 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,|f|+|a|}^{σ} (A, A), for homogeneous f ∈ Diff_{k,|f|}^{σ} (A, A).

Proof.

[a_{0}, [a_{1}, … [a_{k}, bf]^{σ} ⋯]^{σ}]^{σ}

= [a_{0}, [a_{1}, … [a_{k−1}, (a_{k} (bf) − σ (|a_{k}|, |b| + |f|) (bf) a_{k})]^{σ} ⋯]^{σ}]^{σ} = [a_{0}, [a_{1}, … [a_{k−1}, (σ (|a_{k}|, |b|) (ba_{k}) f − σ (|a_{k}|, |b| + |f|) bf (a_{k}))]^{σ} ⋯]^{σ}]^{σ}

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

⋮ = σ (|a_{k}| + ⋯ + |a_{0}|, |b|) b [a_{0}, [a_{1}, … [a_{k}, f]^{σ} ⋯]^{σ}]^{σ} = 0 Also the right action on a braided differential operator again is a braided differential operator,

[a_{0}, [a_{1}, … [a_{k}, fb]^{σ} ⋯]^{σ}]^{σ}
= [a_{0}, [a_{1}, … [a_{k−1}, (a_{k} (fb) − σ (|a_{k}|, |f| + |b|) (fb) a_{k})]^{σ} ⋯]^{σ}]^{σ}
= [a_{0}, [a_{1}, … [a_{k−1}, ((a_{k}f) b − σ (|a_{k}|, |f|) f (a_{k}b))]^{σ} ⋯]^{σ}]^{σ}
= [a_{0}, [a_{1}, … [a_{k−1}, [a_{k}, f]^{σ} b]^{σ} ⋯]^{σ}]^{σ}
⋮
= [a_{0}, [a_{1}, … [a_{k}, f]^{σ} ⋯]^{σ}]^{σ} b = 0,

for homogeneous 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) = ∑_{k∈ℤ}Smbl_{k}^{σ} (A, A) .

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

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

depends on the class of the two homogeneous σ-differential operators f_{1} ∈ Diff_{i,|f_{1}|}^{σ} (A, A) and f_{2} ∈ Diff_{j,|f_{2}|}^{σ} (A, A), and there is a graded σ-Poisson structure on the braided symbol algebra.

7.2. Braided differential operators in graded modules. Let A be a σ-symmetric algebra and let E be a σ-symmetric A-module.

Define a graded σ-differential operator f of E of degree |f| ∈ G and order at most k as the linear map f : E → E, such that

f : E_{g} → E_{g+|f|},

g ∈ G, and

[x, [a_{0}, … [a_{k−1}, f]^{σ} ⋯]^{σ}]^{σ} = 0,  (29)

∀a_{0}, …, a_{k−1} ∈ A, x ∈ E.

Denote by Diff_{k,i}^{σ} (E, E) the σ-differential operators of order at most k and degree i, 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 algebra. Furthermore,

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

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

Proposition 30. 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,|f|+|a|}^{σ} (E, E), for homogeneous a ∈ A and f ∈ Diff_{k,|f|}^{σ} (E, E).

Proof. The proof is the same as for proposition 29.

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,|f_{1}|+|f_{2}|}^{σ} (E, E),

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

depends on the class of the two homogeneous σ-differential operators f_{1} ∈ Diff_{i,|f_{1}|}^{σ} (E, E) and f_{2} ∈ Diff_{j,|f_{2}|}^{σ} (E, E), and there is a graded σ-Poisson structure on the braided symbol algebra.

7.3. Quantizations of braided differential operators in algebras. We define quantization of σ-differential operators in algebras.

Definition 31. Given a quantization q and f ∈ Diff_{|f|}^{σ} (A, A) define the quantization of f by

Q_{q} (f) (a) = q (|f|, |a|) f (a) ,

for homogeneous a ∈ A. Sometimes we use the notation f_{q} = Q_{q} (f).

Q_{q} (f) is an operator of the quantized graded algebra A_{q}. 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 A be a σ-commutative algebra. 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}}) , (30)
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}}) , (31)
f ∈ Smbl^{σ} (A, A) ⟼ Smbl (Q_{q}) (f) ∈ Smbl^{σ^{q}} (A_{q}, A_{q}) .

By proposition 29 is Diff^{σ^{q}} (A_{q}, A_{q}) a σ_{q}-symmetric module and 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.

The braided differential operators of A satisfy theorem 11 with Der replaced by Diff and the σ_{q}-Lie algebra structure of Diff^{σ^{q}} (A_{q}, A_{q}) can be realized within the classical one by dequantization.

7.4. Quantizations of braided differential operators in modules. Let A be a σ-symmetric algebra and let E be a σ-symmetric A-module.

Definition 32. Let q be a quantization and f ∈ Diff_{|f|}^{(σ,A)} (E, E) be homogeneous. Then the quantization of f is defined by

Q_{q} (f) (x) = q (|f|, |x|) f (x) ,

where x ∈ E is homogeneous.

Sometimes we use the notation f_{q} = Q_{q} (f).

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

Given a braiding σ, let E be a σ-commutative A-module. 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}}) , (32)
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}}) ,(33)
f ∈ Smbl^{(σ,A)} (E, E) ⟼ Smbl (Q_{q}) (f) ∈ Smbl^{(σ_{q},A_{q})} (E_{q}, E_{q}) .

This is shown in [15].

Diff^{(σ_{q},A_{q})} (E_{q}, E_{q}) a σ_{q}-symmetric module and 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}).

The braided differential operators of E satisfy theorem 18 with Der replaced by Diff so 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.

7.5. Braided symbol and G ⊕ ℤ-grading. Any G-graded differential operator has a fibration by ℤ. However, the symbol of the braided differential operators of G-graded algebras A and modules E, Smbl^{σ} (A, A) and Smbl^{(σ,A)} (E, E), is ℤ-graded and so there is a grading by G ⊕ ℤ.

Instead of only considering quantizations and braidings with respect to the G-grading, we consider such with respect to the grading G ⊕ ℤ. In [11] we consider such quantizations and braidings in connection with exterior and symmetric algebras. We recall the following description of symmetries and quantizations for this case.

Let Ḡ = G ⊕ ℤ and denote its elements by ḡ = (g, g_{ℤ}).

Any symmetry

σ̄ : (G ⊕ ℤ) × (G ⊕ ℤ) → U (ℂ)

of the monoidal category of Ḡ = G ⊕ ℤ-graded modules is defined by

σ̄ (ḡ, ḡ′) = σ (g, g′) τ (g_{ℤ}, g′_{ℤ}) γ (g_{,}g′_{ℤ}) γ^{−1} (g′, g_{ℤ}) ,  (34)

where we have a symmetry of Ḡ-graded modules,

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

a symmetry of ℤ-graded modules,

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

and a bihomomorphism,

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

A quantization q̄ of Ḡ = G × ℤ-graded modules is of the form

q̄(ḡ, ḡ′) = q (g, g′) κ (g, g′_{ℤ}) κ^{−1} (g′, g_{ℤ}) p (g_{ℤ}, g′_{ℤ})
= q (g, g′) κ (g, g′_{ℤ}) κ^{−1} (g′, g_{ℤ}) ,  (35)

where ḡ, ḡ′ ∈ Ḡ,

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

is a bihomomorphism,

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

is quantization of G-graded modules and p, which is a representative of the second cohomology of ℤ, is trivial.

Considering Smbl^{σ} (A, A) and Smbl^{(σ,A)} (E, E) as Ḡ-graded, they are equipped with a symmetry σ̄ of Ḡ, where σ̄|_{({0}⊕ℤ)×({0}⊕ℤ)} = τ and σ̄|_{(G⊕{0})×({0}⊕ℤ)} = γ trivial, that is σ̄ = σ.

Remark 33. If we quantize Smbl^{σ} (A, A) and Smbl^{(σ,A)} (E, E) by the quantizer γ = σ̄|_{(G⊕{0})×({0}⊕ℤ)} then the resulting algebra is Smblσ̄ (A, A) and Smbl^{(σ̄,A)} (E, E) that are σ̄-Poisson algebras with respect to the braiding

σ̄ (ḡ, ḡ′) = σ (g, g′) γ (g_{,}g′_{ℤ}) γ^{−1} (g′, g_{ℤ}) .

We show the quantized braided Poisson structure for the quantization of Smblσ̄ (A, A) and Smbl^{(σ̄,A)} (E, E) for a general braiding σ̄ in theorems 34 and 35.

Note that σ̄|_{({0}⊕ℤ)×({0}⊕ℤ)} = τ always will be trivial since the structure arises from Smbl^{σ} (A, A) and Smbl^{(σ,A)} (E, E).

Assume we have a braided symbols Smblσ̄ (A, A) and Smbl^{(σ̄,A)} (E, E) with respect to a symmetry σ̄, σ̄|_{({0}⊕ℤ)×({0}⊕ℤ)} = τ = 1.

We use quantizations of the form (35). A quantization of a symbol

f ∈ Smblσ̄ ; k,|f| (A, A) or f ∈ Smbl_{k,|f|}^{(σ̄,A)} (E, E) is

fq̄ (x) = q̄((|f|, k) , (|x|, 1)) f (x) ,

where the homogeneous x (in either E or A) is given the grading (|x|, 1), |x| ∈ G.

The Qq̄ is an isomorphism of modules

Smbl (Qq̄) : (Smblσ̄ (A, A) , [,]σ̄) → (Smbl^{σ̄q̄} (Aq̄, Aq̄) , [,]σ̄q̄ ; q̄) , (36)
f ∈ Smblσ̄ (A, A) ⟼ Smbl (Qq̄) (f) ∈ Smbl^{σ̄q̄} (Aq̄, Aq̄) ,

and

Smbl (Qq̄) : (Smbl^{(σ̄,A)} (E, E) , [,]σ̄) → (Smbl^{(σ̄q̄,Aq̄)} (Eq̄, Eq̄) , [,]σ̄q̄ ; q̄) (37)

f ∈ Smbl^{(σ̄,A)} (E, E) ⟼ Smbl (Qq̄) (f) ∈ Smbl^{(σ̄q̄,Aq̄)} (Eq̄, Eq̄) .

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

Theorem 34. Smbl^{σ̄q̄} (Aq̄, Aq̄) is a Ḡ = G⊕ℤ-graded σ̄q̄-Poisson algebra with respect to the σ̄q̄ − q̄-bracket, that is the following properties are satisfied:

[Smbl^{σ̄q̄} (Aq̄, Aq̄) , Smbl^{σ̄q̄} (Aq̄, Aq̄)]σ̄q̄ ; q̄ ⊆ Smbl^{σ̄q̄} (Aq̄, Aq̄) ,  (i)
[Smbl_{i,|f|}^{σ̄q̄} (Aq̄, Aq̄) , Smbl_{j,|g|}^{σ̄q̄} (Aq̄, Aq̄)]σ̄_{q} ; q̄
⊆ Smbl_{i+j−1,|f|+|g|}^{σ̄q̄} (Aq̄, Aq̄) ,  (i’)

skew σ̄q̄-symmetricity,

[f_{1}, f_{2}]σ̄q̄ ; q̄ = −σ̄q̄ ((|f_{1}|, i) , (|f_{2}|, j)) [f_{2}, f_{1}]σ̄q̄ ; q̄ ,  (ii)

the σ̄q̄-Jacobi identity,

[f_{1}, [f_{2}, f_{3}]σ̄q̄ ; q̄]σ̄q̄ ; q̄

= [[f_{1}, f_{2}]σ̄q̄ ; q̄ , f_{3}]σ̄q̄ ; q̄ + σ̄q̄ ((|f_{1}|, i) , (|f_{2}|, j)) [f_{2}, [f_{1}, f_{3}]σ̄q̄ ; q̄]σ̄q̄ ; q̄ , (iii)

and

[f_{1}, f_{2}f_{3}]σ̄q̄ ; q̄ = [f_{1}, f_{2}]σ̄q̄ ; q̄ f_{3} + σ̄q̄ ((|f_{1}|, i) , (|f_{2}|, j)) f_{2} [f_{1}, f_{3}]σ̄q̄ ; q̄ ,  (iv)

for f_{1} ∈ Smbl_{i,|f_{1}|}^{σ̄q̄} (Aq̄, Aq̄), f_{2} ∈ Smbl_{j,|f_{2}|}^{σ̄q̄} (Aq̄, Aq̄), f_{3} ∈ Smbl_{k,|f_{3}|}^{σ̄q̄} (Aq̄, Aq̄).

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

Theorem 35. Smbl^{(σ̄q̄,Aq̄)} (Eq̄, Eq̄) is a Ḡ = G ⊕ ℤ-graded σ̄q̄-Poisson algebra with respect to the σ̄q̄ − q̄-bracket, that is the properties (i), (ii), (iii) and (iv) of theorem 34 are satisfied when replacing Aq̄ by Eq̄ and Smbl^{σ̄q̄} by Smbl^{(σ̄q̄,Aq̄)}, and

[Smbl_{i,g}^{(σ̄q̄,Aq̄)} (Eq̄, Eq̄) , Smbl_{j,h}^{(σ̄q̄,Aq̄)} (Eq̄, Eq̄)]σ̄q̄ ; q̄
⊆ Smbl_{i+j,g+h}^{(σ̄q̄,Aq̄)} (Eq̄, Eq̄) ,  (i’)

g, h ∈ G.

QUANTIZATIONS OF BRAIDED DERIVATIONS 159

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160 H. L. HURU

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

Received November 7, 2006