N-Complex, Graded q-Differential Algebra and N-Connection on Modules

It is well known that given a differential module E with a differential d we can measure the non-exactness of this differential module by its homologies which are based on the key relation d2=0. This relation is a basis for several important structures in modern mathematics and theoretical physics to point out only two of them which are the theory of de Rham cohomologies on smooth manifolds and the apparatus of BRST-quantization in gauge field theories.


Introduction
An idea to generalize the concept of a differential module and to elaborate the corresponding structures by giving the mentioned above key relation d 2 = 0 a more general form d N = 0, N ≥ 2 seems to be very natural. This idea has been proposed and studied in the series of papers [1][2][3][4] giving rise to the structures such as differential N-complex, N-cochain complex, generalized cohomologies of N-cochain complex and graded q-differential algebra, where q is a primitive Nth root of unity. The graded q-differential algebra can be viewed as a generalization of a graded differential algebra and can be used to develop the applications of structures based on relation d N = 0 in non-commutative geometry. It is well known that the concepts of connection and its curvature are basic elements of the theory of fiber bundles and play an important role not only in a modern differential geometry but also in a modern theoretical physics namely in a gauge field theory. The development of a theory of connections has been closely related to the development of a theoretical physics. The advent of supersymmetric field theories in the 70's of the previous century gave rise to interest towards  2 -graded structures which became known in theoretical physics under the name of super structures. This direction of development has led to a concept of super connection which appeared in [5]. The emergence of non-commutative geometry in the 80's of the previous century was a powerful spur to the development of a theory of connections on modules [6][7][8][9]. A basic concept used in a theory of connections on modules is a concept of a graded differential algebra. Consequently using a concept of a graded q-differential algebra we can develop a more general theory of connections. The aim of this paper is to give a survey of several algebraic structures based on the relation d N = 0 and to show the possible applications of these structures in noncommutative geometry.
In the Section 2 we give a short overview of N-structures such as N-differential module, generalized homologies of the N-differential module, N-cochain complex. At the end of this section we give an example of a positive N-complex proposed in [3]. In the Section 3 making use of the notion of N-complex we give a generalization of a concept of a graded differential algebra, which will be referred to as a graded q-differential algebra and was introduced and studied [2,4]. As it was investigated [10,11] it is possible to construct a realization of N-differential calculus of exterior forms on a smooth finite dimensional manifold. A construction of an analog of exterior calculus with N-differential d on a non-commutative space was proposed [12]. The space we consider is a reduced quantum plane. Our approach is based on a generalized Clifford algebra with four generators equipped with the N-differential d. We study the structure of the algebra of q-differential forms on a reduce quantum plane and show that the first order calculus induced by the differential d is a first order calculus. In the Section 4 we introduce a generalization of a concept of a connection form by means of a notion of graded q-differential algebra and covariant N-differential which can be viewed as analogs of a connection form in a graded differential algebra described [13]. We begin this section with an algebra of polynomials in the variables d,a 1 ,a 2 ,... and prove the power expansion formula for an n th power of the operator 1â a = + d d . Applying this formula we show that the N th power of the covariant N-differential is the operator of multiplication by an element where i, j ∈{0,1,…,n +1} and i< j.
Given a pre-cosimplicial module ((E n ) n∈ ; f 0 ,f 1 ,…,f n+1 ) we can construct a positive cochain complex E with differential d by setting E = ⊕ n ∈  E n and 1 0 ( 1) It is easy to check that d 2 = 0 follows immediately from (2). We shall call this positive cochain complex the pre-cosimplicial complex and its differential d the simplicial differential.
It is clear that the basic relation which determines the structure of a differential module and allows to define the important characteristics of a non-exactness of a differential module such as homologies is d 2 = 0. This basic relation can be given a more general form d N =0, where N is an integer greater or equal to two. This generalization was proposed and considered in the framework of non-commutative geometry almost at the same time in the papers [1][2][3]. Now we turn to a theory which can be developed if one replaces the basic relation d 2 = 0 of a differential module by a more general one d N = 0.
Let N ≥ 2 be a positive integer and E be a left K -module.

Definition 2.7:
A left K -module E is said to be a N-differential module with N-differential d if d: E→ E is an endomorphism of E satisfying d N = 0. In the case when K is a field a N-differential module E will be referred to as a N-differential vector space.
Obviously N-differential module can be viewed as a generalization of a concept of differential module to any integer N ≥ 2. Now for each

Definition 2.8: The quotient modules
For m = 0, 1 we get It can be verified that d m : which means that d m is the N-differential, and we obtain the N-complexes (E,d 0 ), (E,d 1 ),…, (E,d n+1 ).

Graded q-differential algebra
It is well known that if we endow a cochain complex with an additional structure which is an associative unital law of multiplication in such a way that a differential of a cochain complex satisfies the graded Leibniz rule with respect to this multiplication then we obtain a concept of a graded differential algebra. Analogously if we equip a N-complex with an associative unital law of multiplication in such a way that a N-differential of a N-complex satisfies the q-graded Leibniz rule, where q is a primitive Nth root of unity, then we obtain a generalization of a graded differential algebra which is called graded q-differential algebra. The aims of this section are to remind a concept of a graded differential algebra, to give a definition of a graded q-differential algebra, to show in what a way an associative graded algebra can be endowed with a structure of a graded q-differential algebra and finally to show a relation of a graded q-differential algebra with noncommutative geometry by giving a construction based on a generalized Clifford algebra which leads to a realization of a graded q-differential algebra by analogs of differential forms on a reduced quantum plane.
Let Ω = ⊕ n∈ Ω n be an associative unital graded -algebra. The subspace of elements of degree zero Ω 0 Ω is the subalgebra of Ω and we denote this subalgebra by A, i.e. A = Ω 0 . Any subspace Ω k ⊂ Ω of elements of degree k ∈  is the (A, A) -bimodule. Definition 3.1: A graded differential algebra (GDA) is an associative unital graded -algebra equipped with a linear mapping d of degree 1 such that the sequence is a cochain complex, and d is an antiderivation, i.e. it satisfies the graded Leibniz rule where ω ∈ Ω k , θ ∈ Ω.
Let us mention that if Ω is a GDA then Ker d is the graded unital subalgebra of Ω whereas Im d is the graded two-sided ideal of Ker d. Hence the cohomology H(Ω) is the unital associative graded algebra. A M is said to be a first order differential calculus over an algebra A if : → d A M is a homomorphism of (A, A) -bimodules satisfying the Leibniz rule where a, b ∈ A. A homomorphism d is referred to as a differential of a first order differential calculus.
If Ω is a GDA with differential d then clearly the triple A, d, Ω q is a first order differential calculus over the algebra A.

Definition 3.3:
If Ω is a GDA with differential d and A is its subalgebra of elements of degree zero then the triple (A, d, Ω q ) will be referred to as the differential calculus over the algebra A. Now we are going to describe a generalization of a GDA introduced and studied [2,4] by giving the basic condition d 2 = 0 of a GDA a more general form d N = 0, N ≥ 2. In the following of this section, K is the field of complex numbers  and q is a primitive N th root of unity, where N ≥ 2.

Definition 3.4:
A graded q-differential algebra (q-GDA) is an associative unital -graded ( N -graded) -algebra Ω q = ⊕ k∈ Ω k q endowed with a linear mapping d of degree one such that the sequence is a N-complex with N-differential d satisfying the graded q-Leibniz rule where ω ∈ Ω k q , θ ∈ Ω q .
Obviously the subspace of elements of degree zero is the subalgebra of a q-GDA Ω q . Clearly the triple is the first order differential calculus over the algebra A. The triple (A, d, Ω q ) will be referred to as a N-differential calculus over the algebra A.
Let us remind that a graded q-center of an associative unital graded -algebra  = ⊕ k∈  k is the graded subspace Z() = ⊕ k∈ Z k ()of A generated by the homogeneous elements v∈ k , where k ∈ , satisfying vw = q km wv for any w∈ m , m∈ . It is easy to verify that the graded q-center Z() is the graded subalgebra of .

Definition 3.5:
A graded q-derivation of degree k ∈  of A is a homogeneous linear mapping D : → m+k satisfying the graded q-Leibniz rule It is well known that given a homogeneous element v∈ k of  we can associate to it the graded q-derivation of degree k by means of a graded q-commutator [ , ] q as follows where w∈ m . Clearly ad q (v) : → m+k , and the graded q-derivation ad q (v)associated to a homogeneous element v is referred to as an inner graded q-derivation of degree k of A. Let v ∈ A 1 be a homogeneous element of degree one and d v = ad q (v) : →  m+k be the inner graded q-derivation associated to v. The kth power of this inner graded q-derivation can be expanded as follows:

Lemma 3.6: For any integer k ≥ 2 it holds
where w is a homogeneous element of  and Making use of this Lemma we can prove [18] Theorem 3.7: If  is an associative unital -graded -algebra and where N ≥ 2, then the inner graded q-derivation d v = ad q (v) :  k → k+1 associated to v is the N-differential of an algebra A. Hence an algebra A regarded with respect to the N-differential d v is the q-GDA.
We can apply this theorem to a generalized Clifford algebra in order to construct a q-GDA.
where sg(k) is the usual sign function, and 1 is the identity element of an algebra.
Since our aim in this section is to construct the analogs of differential forms on a reduced quantum plane we shall consider the generalized Clifford algebra with four generators, i.e. p = 4. Let us denote the generalized Clifford algebra with four generators by N C . We split the set of generators of this algebra into two pairs ξ 1 , ξ 3 and ξ 2 , ξ 4 denoting the generators of the first pair by x, y, i.e. x = ξ 1 , y = ξ 3 , and the generators of the second pair by u, v, i.e. u = ξ 2 , v = ξ 4 . From (11) it follows , Let N P be the subalgebra of the algebra N C generated by x, y. The relations (12) show that the generators x, y can be interpreted as coordinates of a reduced quantum plane [19], and correspondingly the subalgebra N P can be interpreted as the algebra of polynomial functions on a reduced quantum plane. Keeping in mind this interpretation we shall call an element ∈ N f P a polynomial function on a reduced quantum plane. Our next step is to construct a N-differential calculus on a reduced quantum plane. For this purpose we take the generalized Clifford algebra N C and applying the Theorem 3.7 equip it with the structure of a q-GDA in such a way that the subalgebra N P of polynomial functions will be the subalgebra of elements of degree zero of this q-GDA. We define the  N-graded structure of N C as follows: we assign the degree zero to the generators x, y and the degree one to the generators u, v. Hence denoting the degree of an element w by |w| we can write where 0, 1 are the residue classes of 0, 1 modulo N. As usual the degree of any monomial composed of generators x, y, u, v equals to the sum of degrees of its components. Obviously Since q is a primitive N th root of unity we have Now it follows from the Theorem 3.7 that the inner graded q-derivation d ω = ad q (ω) associated to an element Hence the generalized Clifford algebra with four generators N C regarded with respect to the  N-graded structure defined by (16) and to N-differential d ω is the q-GDA, and the triple ( ) C is the N-differential calculus over the algebra of polynomial functions N P of the reduced quantum plane. In what follows of this section we consider ω as a fixed element, and keeping this in mind we simplify the symbol for N-differential d ω omitting ω and writing d instead of d ω : It should be mention that the structure of the q-GDA N C depends on a choice of element ω, and consequently the numbers λ, µ can be considered as the parameters of this structure.
The N-differential d induces the differentials of coordinates dy and later in this section we will show that any element of degree 0 > k of the q-GDA N C can be expressed in terms of the differentials of coordinates dx, dy. Since N-differential d satisfies d N = 0 it can be viewed as an analog of exterior differentiation of higher order on the reduced quantum plane. Proceeding with this analogy we will call the q-GDA N C the algebra of q-differential forms on the reduced quantum plane and its elements of degree 0 > k expressed in terms of the differentials dx,dy the q-differential k-forms. Let us mention that in the particular case of N = 2 the above construction yields an analog of exterior calculus with exterior differential d satisfying d 2 = 0 on the reduced quantum plane with coordinates x, y which obey In analogy with exterior calculus we shall call a q-differential form θ a m-closed q-differential form, where , and we shall call a q-differential n-form θ a q-differential l-exact form if there exists a q-differential (n − l)-form ρ such that d 1 Our next aim is to describe the structure of the algebra of q-differential forms in terms of the differentials of coordinate's dx, dy. We begin with the first order differential calculus which is the triple , ∈ N dx dy C . Let us express the differentials dx, dy in terms of the generators of N C . It is worth mentioning that in what follows we shall use the structure of the right N P -module of 1 N C have the relations to write q-differential 1-forms in terms of differentials. We where 1 1 , Using these relations we obtain It is evident that the right N P -module 1 N C is a free right module and {u, v} is the basis for this right module. Proposition 3.10: For any integer N ≥ 3 the right N P -module of q-differential 1-forms 1 N C is freely generated by the differentials of coordinates dx, dy.
Proof: Let , ∈ N f h P be polynomial functions on the reduced quantum plane. Using (19), (20) and the fact that {u, v} is the basis for the right N P -module 1 N C we can show that the equality dx f + dy h = 0 is equivalent to the system of equations Multiplying the second equation by q − 1 and adding it to the first equation we obtain (q-q -1 )h = 0. As q-q -1 ≠ 0 for N ≥ 3 we conclude that h = 0. In the same way we show that f = 0, and this proves that the differentials dx, dy are linearly independent q-differential 1-forms.
In order to prove that any q-differential 1-form is a linear combination of differentials we find the transition matrix from the basis {u, v} to the basis {dx, dy}. Let us denote the algebra of square matrices of order 2 whose entries are the elements of the algebra N P by It should be noted that the transition matrix depends on the coordinates of a point of a reduced quantum plane. As the coordinates of a reduced quantum plane obey the relations x N = 1, y N = 1 they are invertible elements of the algebra N P and x −1 = x N −1 , y −1 = y N −1 . If N ≥ 3 then the matrix A is an invertible matrix and Consequently we have Now any q-differential 1-form can be expressed in terms of the differentials, and this ends the proof.
From the Proposition 3.10 it follows that the first order differential C is the coordinate calculus with coordinate differential d [20]. If we have a coordinate calculus then a coordinate differential of this calculus induces the partial derivatives which satisfy the twisted Leibniz rule. The second term in the right hand side of the twisted Leibniz rule for a partial derivative depends on the homomorphism from the algebra N P to the algebra of (2 × 2)-matrices with entries from N P , which we denote by for ∈ N f P we have 11 12 12 22 Since N P is the algebra of polynomial functions in two variables x, y which are subjected to the commutation relation xy = q yx and the relations x N = 1, y n = 1 it is sufficient to find the explicit formula for the homomorphism R applied to coordinates x, y. Taking firstly f = x and then f = y in (24), (25) we find Putting the entries of these matrices into the relations (24), (25) we get 1 , The partial derivatives induced by the N-differential d are defined by It can be shown [20] that the partial derivatives satisfy the twisted Leibniz rule 11 12 ( ) ( ) ( ) Using the twisted Leibniz rule and (26) we calculate ( 1) Using these formulae we can calculate the partial derivatives of any polynomial function ∈ N f P which can be written as We remind that the set of generators {x,u,y,v} of the generalized Clifford algebra N C has been split into two parts, where the first part {x, y} generates the algebra of polynomial functions N P , and the second part {u, v} generates the N-differential calculus ( ) , , C . We already have proved that any 1-form θ = u f + v h can be uniquely expressed in terms of the differentials dx, dy. Now if we take x, y, dx, dy as the generators for the algebra N C then we may divide the algebraic relations between the new generators into three parts. The first part contains the relations The third part will contain the relations between the differentials dx, dxy. It is evident that the commutation relation uv = qvu will give us a quadratic relation for the differentials dx, dy, and the relations u N = v N = 1will give rise to two relations of degree N with respect to differentials.

Proposition 3.11:
The commutation relation uv = qvu for the generators u, v written in terms of the differentials dx, dy takes on the form where 4 3 This proposition can be proved by straightforward computation with the help of the formulae (21) ( ) , ,( ) dx dydx dy . Hence any q-differential k-form θ on the reduced quantum plane can be uniquely expressed as follows: The peculiar property of the q-analog of exterior calculus on a reduced quantum plane is an appearance of the higher order differentials of coordinates, and this gives us a possibility to construct one more basis for the module of 2-forms. Indeed as dk ≠ 0for k running integers from 2 to N − 1 we have the set of higher order differentials of coordinates d 2 x, d 2 y,….,d N-1 x, d N-1 y, and we can use these higher order differentials to construct a basis for k N C . In this paper we shall describe a case of the module of q-differential 2-forms. The elements ω, ω 1 , ω − ω 1 can be written as q-differential 1-forms as follows: Differentiating ω we obtain where dω is the q-differential 2-form. Now we can write the second order differential d 2 x as a q-differential 2-form as follows: Expressing the second order differential d 2 y in terms of (dx) 2 , dy dx, (dy) 2 we prove the following proposition: Proposition 3.12: The second order differential d 2 y can be written as follows: Proof: As We point out that from the Proposition 3.12 it follows that the relation (29) can be written by means of the second order differential d 2 y in a more symmetric form 3 2 2 We end this section by considering the structure of algebra of q-differential forms on a reduced quantum plane at cubic root of unity, i.e in the case of N = 3. In this case we have the algebra of q-differentials forms on a reduced quantum plane at cubic root of unity with differential d satisfying d 3 = 0. It can be verified that now the right hand sides of the formulae (33), (34) are the 1-closed q-differential 2-forms, i.e. The last term in the relation (29) vanishes because q is a primitive cube root of unity and satisfies q 3 − 1 = 0. Making use of the relation 1 + q + q 2 = 0 we can write the coefficient γ 1 as follows: Hence the relation for the differentials (29) with respect to the basis  2 takes on the form The above relation is similar to the anticommutativity of differentials in a classical algebra of differential forms on a plane and meaning this analogy we can say that the algebra of q-differential forms on a reduced quantum plane at cubic root of unity with differential satisfying d 3 = 0 may be viewed as a deformation of the classical algebra of differential forms with parameter

N-connection form and its curvature
In this section we propose a generalization of a concept of connection form by means of a notion of q-GDA [15][16][17]. We begin this section with an algebra of polynomials in two variables which we will use later in this section to prove propositions describing the structure of the curvature of a connection form.
Let d, a 1 , a 2 ,..., a n ,... be the set of variables.   variables d, a 1 , a 2 , . . . , a n    Proof: It is suffice to show that in the case of the algebra of N-connection form we have a n = 0 for any integer n ≥ N. Indeed making use of the commutation relations (37) we can express any variable a n , n ≥ 2 in the terms of d, a 1 as polynomial ( 1) 2 1 ( 1) Now assuming n = N we see that the first and last terms in (39) vanish because of d N = 0, other terms are zero because of vanishing of q-binomial coefficients provided q is a primitive Nth root of unity. Hence for any integer n ≥ N we have a n = 0. In what follows we will denote by (1) 0, ( ) , (ˆˆ) Our aim now is to find a power expansion for polynomials p (n) with respect to variables d, a 1 ,a 2 ,…,a n . It is obvious that making use of the commutation relations (37) we can rearrange the factors in each summand of this expansion by removing all d's to the right.   Let Ω q be a  N -graded q-differential algebra with N-differential d, where q is a Nth primitive root of unity, and 0 = Ω q A be the subalgebra of elements of degree zero.  = Ω q A a N-connection form in a q-GDA Ω q . The linear operator of degree one 1 : will be referred to as a covariant N-differential induced by a N-connection form A.
Since d is a N-differential which means that d n ≠ 0 for 1 ≤ n ≤ N − 1 if we successively apply it to a N-connection form A we get the sequence of elements A, dA, d 2 A, . . . , d N −1 A, where. Let us denote by Ω q [A] the graded subalgebra of Ω q generated by these elements A, dA, d 2 A, . . . , d N −1 A. For any integer n = 1, 2, . . . , N

Particularly if n = N then the Nth power of the covariant N-differential d A is the operator of multiplication by the element ( )
The Theorem 4.7 allows us to define the curvature of a N-connection form A as follows It is easy to see that in the particular case of a graded differential algebra (N = 2, q = −1) with differential d satisfying d 2 = 0 the above definition yields a connection form A and its curvature (2) 2 = + A F dA A as elements of degree respectively one and two of a graded differential algebra [13].

Proposition 4.9: For any N-connection form A in a q-GDA Ω q the N-curvature form
Proof: Indeed we have [ ] From (41)-(44) we obtain the expressions for N-curvature form (3) 2 3 = + + +

N-connection on Modules
This section is devoted to the definition and the study of a connection in non-commutative geometry. Our main purpose is to describe a concept of N-connection on a module which may be considered as a generalization of a classical connection [15][16][17]. In our approach we generalize a concept of Ω-connection on modules proposed in reference [9]. In order to have an algebraic model of differential forms with values in vector bundle we introduce a left module over the subalgebra of elements of degree zero of a q-GDA. We study the structure of a N-connection, define its curvature and prove the Bianchi identity. We show that every projective module admits a N-connection. In a last part of this section we study the local structure of a N-connection. Assuming that left module is a finitely generated free module we introduce a matrix of N-connection and the curvature matrix. Finally we find the expressions for components of the curvature in terms of the components of a N-connection.
Let A be an unital associative -algebra and Ω q be an N-differential calculus over A, i.e. Ω q is a q-GDA with N-differential d and . Let ε be a left A -module. Considering algebra Ω q as the (A, A)-bimodule we take the tensor product of modules which clearly has the structure of left A -module. Taking into account that an algebra Ω q can be viewed as the direct sum of (A, A)-bimodules Ω i q we can split the left A -module F into the direct sum of the left A -modules [21], i.e. = ⊕ i i F F , which means that F inherits the graded structure of algebra Ω q , and F is the graded left A -module. It is worth noting that the left A submodule 0 ε = ⊗ A F A of elements of degree zero is isomorphic to a left A -module ε, where isomorphism 0 : ϕ ε → F can be defined by ( ) ϕ ξ ξ = ⊗ e A , where e is the identity element of algebra A. Since an algebra Ω q can be considered as the

( )
, Ω q A -bimodule the left A -module F can be also viewed as the left Ω q-module [21] and we will use this structure to describe a concept of N-connection.
The tensor product F is also the vector space over  where this vector space is the tensor product of the vector spaces Ω q and ε. It is evident that F is a graded vector space, i.e.
. Let us denote the F by ( ) L F . The structure of the graded vector space of F induces the structure of a graded vector space on ( ) L F , and we shall denote the subspace of homogeneous linear operators of degree i by ( ) i L F .

Definition 5.1: A N-connection on the left Ω q-module F is a linear operator
: where ω ∈ Ω k q , ξ ε ∈ , and |ω| is the degree of the homogeneous element of algebra Ω q .
Similarly one can define a N-connection on right modules. If  Making use of the previously introduced notations we can write 1 ( ) ∇ ∈ q L F . Let us note that if N = 2 then q = −1, and in this particular case the Definition 5.1 gives us the notion of a classical connection. Hence a concept of a N-connection can be viewed as a generalization of a classical connection. Let ε be a left A -module. The set of all homomorphisms of ε into A has the structure of the dual module of the left A -module ε and is denoted by ε * . It is evident that ε * is a right A -module.

Definition 5.2: A linear map
where ξ ε ∈ , η ε * ∈ and ∇ q is a N-connection on ε, is said to be the dual connection of ∇ q .
It is easy to verify that ∇ * q has a structure of N-connection on the right module ε * . Indeed, for any , , In order to define a Hermitian structure on a right A -module ε we assume A to be a graded q-differential algebra with involution* such that the largest linear subset contained in the convex cone ∈ C A generated by a * a is equal to zero, i.e.
( ) 0 ∩ − = C C c. The right A -module ε is called a Hermitian module if ε is endowed with a sesquilinear map , and ( , ) 0 0 ξ ξ ξ ε ξ ξ ξ We have used the convention for sesquilinear map to take the second argument to be linear, therefore we define a Hermitian structure on right modules. In a similar manner one can define a Hermitian structure on left modules.

Definition 5.3: A N-connection ∇ q on a Hermitian right A -module ε is said to be consistent with a Hermitian structure of ε if it satisfies
where , Our next aim to define a notion of a curvature of N-connection. We start with the following Proposition 5.4: The N-th power of any N-connection ∇ q is the endomorphism of degree N of the left Ω q-module F .
Proof. It suffices to verify that for any homogeneous element ω ∈ Ω q an endomorphism Let us expand the k-th power of ∇ q as follows and this proves that ∇ N q is the endomorphism of the left Ω q -module F .
This proposition allows us to define the curvature of N-connection as follows of degree N of the left Ω q-module F is said to be the curvature of a N-connection ∇ q .
The graded vector space L(F) can be endowed with the structure of a graded algebra if one takes the product A ○ B of two linear operators A, B of the vector space F as an algebra multiplication. We can extend a N-connection ∇ q to the linear operator on the vector space L(F) by means of the graded q-commutator as follows where A is a homogeneous linear operator. Obviously ∇ q is the linear operator of degree one on the vector space L(F) , i.e.
The following theorem shows that not every left A-module admits a N-connection [14]. In analogy with the theory of Ω-connection [9] we can prove that there is an N-connection on every projective module, and for this we need the following proposition. Proposition 5.7 : If ε = A⊗V is a free A-module, where V is a -vector space, then ∇ q =d ⊗ I v is N-connection on ε and this connection is flat, i.e. its curvature vanishes.
Where f,g ∈ A, v ∈V. Because of d N = 0 and q is the primitive N th root of unity, we get i. e. the curvature of such a N-connection vanishes.

Theorem 5.8 : Every projective module admits a N-connection.
Proof : Let  be a projective module. From the theory of modules it is known that a module  is projective if and only if there exists a module  such that ε =  ⊗  is a free module. It is well known that free Citation: Abramov V, Liivapuu O (2015) N-Complex, Graded q-Differential Algebra and N-Connection on Modules. J Generalized Lie Theory Appl S1: 006. doi:10.4172/1736-4337.S1-006 where f ∈ A, m ∈ It is well known that a connection on the vector bundle of finite rank over a finite dimensional smooth manifold can be studied locally by choosing a local trivialization of the vector bundle and this leads to the basis for the module of sections of this vector bundle. Let us concentrate now on an algebraic analog of the local structure of a N-connection ∇ q . For this purpose we assume ε to be a finitely generated free left A-module. Let In order to simplify the notations and assuming that N is fixed we shall denote f , and it clearly demonstrates that ∇ N q is the endomorphism of degree N of the left Ω q -module F . Let us consider the form of the curvature matrix of a N-connection in three special cases when N = 2, N = 3 and N = 4. If N = 2 then q = −1, and Ω q is a GDA with differential d satisfying d 2 = 0. This is a