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N-Complex, Graded q-Differential Algebra and N-Connection on Modules | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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N-Complex, Graded q-Differential Algebra and N-Connection on Modules

Viktor Abramov* and Olga Liivapuu

Institute of Mathematics, University of Tartu, Liivi 2, 50409 Tartu, Estonia

Corresponding Author:
Viktor Abramov
Institute of Mathematics, University of Tartu
Liivi 2, 50409 Tartu, Estonia
Tel: +3727375872
E-mail: [email protected]

Received date July 21, 2015; Accepted date August 03, 2015; Published date August 31, 2015

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

Copyright: © 2015 Abramov V, et al. This is an opeN-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

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.

Keywords

N-differential module; N-cochain complex; Cohomologies of N-cochain complex; Graded q-differential algebra; Algebra of connection form; N-connection form; Covariant N-differential; N-curvature Form; N-connection on module; Curvature of N-connection; N-connection consistent with Hermitian structure of module

Introduction

An idea to generalize the concept of a differential module and to elaborate the corresponding structures by giving the mentioned above key relation d2 = 0 a more general form dN = 0, N ≥ 2 seems to be very natural. This idea has been proposed and studied in the series of papers [1-4] 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 dN = 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 -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 [6-9]. 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 dN = 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 image and prove the power expansion formula for an n th power of the operator image. Applying this formula we show that the N th power of the covariant N-differential is the operator of multiplication by an element image, which we then define as the N-curvature form of a N-connection form A [14]. In the Section 5 we make use of a notion of a graded q-differential algebra to construct a N-connection on module introduced in [14-17], which may be viewed as a generalization of a classical connection. We define the notions such as dual N-connection, N-connection consistent with the Hermitian structure of a module. Assuming module to be a finitely generated free module we study the local structure of N-connection, define a curvature of N-connection.

N-complex

Let K be a commutative ring with a unit and E be a left K -module.

Definition 2.1: A module E endowed with an endomorphism d: E→ E is said to be a differential module with differential d if endomorphism d satisfies d2 = 0. In the case when K is a field a differential module E will be referred to as a differential vector space.

It is well known that the property d2=0 of differential image implies image, and this relation allows to measure the noN-exactness of the sequence image Econsidering the homology image of a differential module E which has the structure of a quotient module.

Let E, F be differential modules respectively with differentials

image

Definition 2.2: A homomorphism of modules φ : E→ F is said to be a homomorphism of differential modules E, F if it satisfies image

It is easy to show that if φ is a homomorphism of differential modules then image, image which means that a homomorphism of differential modules φ induces the homomorphism of homologies imageof differential modules E, F. It can be proved [4]

Proposition 2.3: For an exact sequence of differential modules

image

i.e. image, there exists a homomorphism of homologies image such that the triangle

image(1)

is exact.

Let us remind that a image-graded module E is a direct sum of sub modules image labeled by integers image, i.e. image, where an element image is said to be a homogeneous element of image-graded module of degree i.

Definition 2.4: A differential module E with differential d is said to be a cochain complex if E is a image-graded module image and its differential d has degree 1 with respect to a image-graded structure of E, i.e. d: Ei → Ei+1.

A image-graded structure of a cochain complex image induces the image -graded structure of its homology H(E), i.e. image , where image . A homology H(E) is usually referred to as a cohomology of a cochain complex E. It can be proved

Proposition 2.5: For an exact sequence of cochain complexes

image

there exists a homomorphism of homologies image such that the sequence

image

is exact.

Let us remind that a positive cochain complex is image with Ei = 0 for i < 0 or equivalently image. The well-known way of constructing this kind of complexes is based on a notion of a precosimplicial module.

Definition 2.6: A sequence of modules image together with homomorphisms imagesuch that each fi determines a sequence

image

where image is a homomorphism of modules called a coface homomorphism, is said to be a pre-cosimplicial module if

image(2)

where i, j ∈{0,1,…,n +1} and i< j.

Given a pre-cosimplicial module image we can construct a positive cochain complex E with differential d by setting image and image. It is easy to check that d2 = 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 d2 = 0. This basic relation can be given a more general form dN=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 [1-3]. Now we turn to a theory which can be developed if one replaces the basic relation d2 = 0 of a differential module by a more general one dN = 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 dN = 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 integer 1 ≤ m ≤ N − 1 we can define the sub modules Zm(E)=Ker(dm)⊂E and Bm(E) =Im(dN-m)⊂ E. From the relation dN = 0 it follows that Bm(E) ⊂ Zm(E).

Definition 2.8: The quotient modules image are called the generalized homology of the N-differential module E.

It can be proved [4] that in the case of a N-differential module E we have the statement analogous to Proposition 2.3 which is

Proposition 2.9: If image is an exact sequence of N-differential modules then for every m ∈{1,2,…,N-1} there are homomorphisms ∂ : Hm(G)→ HN−m(E) such that for every n ∈{1,2,…,N-1} the following hexagons of homomorphisms are exact.

image

Definition 2.10: A N-differential module E with N-differential d is said to be a N-cochain complex or simply a N-complex if E is a image-graded module image and its N-differential d has degree 1, i.e. d : Ek → Ek+1.

If E is an N-complex then its cohomologies Hm(E) are image-graded modules, i.e. image , where

image

It should be noted that many notions related to N-complexes depend only on the underlying imageN -graduation, and for this purpose we define a imageN -complex to be a imageN-graded N-differential module with N-differential d of degree 1.We end this section by giving an example of a positive N-complex [3]. Let image be a Nth root of unity and image be a pre-cosimplicial module, where image are coface homomorphisms. This pre-cosimplicial module induces the positive N-complexes if we construct then positively graded or image-graded module image and equip it with the endomorphism image of degree 1 defining it by

image(3)

where

image(4)

For m = 0, 1 we get

image(5)

It can be verified that image , imagewhich means that dm is the N-differential, and we obtain the N-complexes image.

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 image be an associative unital graded image-algebra. The subspace of elements of degree zero Ω0 Ω is the subalgebra of Ω and we denote this subalgebra by image, i.e. image . Any subspace Ωk ⊂ Ω of elements of degree image is the image -bimodule.

Definition 3.1: A graded differential algebra (GDA) is an associative unital graded image-algebra equipped with a linear mapping d of degree 1 such that the sequence

image

is a cochain complex, and d is an antiderivation, i.e. it satisfies the graded Leibniz rule

image

where ω ∈ Ωk, 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.

Definition 3.2: Let A be an associative unital algebra and image be a image- bimodule. The triple image is said to be a first order differential calculus over an algebra image if image is a homomorphism of image -bimodules satisfying the Leibniz rule

image,

where image. 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 image is a first order differential calculus over the algebra image.

Definition 3.3: If Ω is a GDA with differential d and image is its subalgebra of elements of degree zero then the triple image 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 d2 = 0 of a GDA a more general form dN = 0, N ≥ 2. In the following of this section, K is the field of complex numbers image 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 (image -graded) image-algebra image q endowed with a linear mapping d of degree one such that the sequence

image

is a N-complex with N-differential d satisfying the graded q-Leibniz rule

image(6)

where image, image

Obviously the subspace of elements of degree zero image is the subalgebra of a q-GDA image. Clearly the triple image is the first order differential calculus over the algebra A. The triple (A, d, Ωq) 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 image-algebra image is the graded subspace image of A generated by the homogeneous elements image , where image, satisfying

image for any image, image. It is easy to verify that the graded q-center image is the graded subalgebra of image.

Definition 3.5: A graded q-derivation of degree image of A is a homogeneous linear mapping image satisfying the graded q-Leibniz rule

image

where

image, w ∈ A.

It is well known that given a homogeneous element image of image we can associate to it the graded q-derivation of degree k by means of a graded q-commutator [ , ]q as follows

image(7)

where image. Clearly image, and the graded q-derivation adq(v) associated to a homogeneous element v is referred to as an inner graded q-derivation of degree k of A. Let v ∈ A1 be a homogeneous element of degree one and imagebe 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

image(8)

where w is a homogeneous element of image and

image(9)

image(10)

Making use of this Lemma we can prove [18]

Theorem 3.7: If image is an associative unital image-graded image-algebra and image is an element of degree one satisfying image , where N ≥ 2, then the inner graded q-derivation image associated to v is the N-differential of an algebra A. Hence an algebra A regarded with respect to the N-differential dv is the q-GDA.

We can apply this theorem to a generalized Clifford algebra in order to construct a q-GDA.

Definition 3.8: A generalized Clifford algebra is an algebra over the complex numbers image generated by a set of canonical generatorsimage which are subjected to the relations

image (11)

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 image . We split the set of generators of this algebra into two pairs ξ1, ξ3 and ξ2, ξ4 denoting the generators of the first pair by x, y, i.e. x = ξ1, y = ξ3, and the generators of the second pair by u, v, i.e. u = ξ2, v = ξ4. From (11) it follows

image (12)

image (13)

image (14)

image (15)

Let image be the subalgebra of the algebra image 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 image can be interpreted as the algebra of polynomial functions on a reduced quantum plane. Keeping in mind this interpretation we shall call an element image 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 image and applying the Theorem 3.7 equip it with the structure of a q-GDA in such a way that the subalgebra PN of polynomial functions will be the subalgebra of elements of degree zero of this q-GDA. We define the image-graded structure of image 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

image (16)

where image 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 image , where image is the subspace of homogeneous elements of degree i, and image .

Proposition 3.9: For any image an element image satisfies image .

Proof: For any 2 ≤ k ≤ N we have 1

image (17)

Since q is a primitive N th root of unity we have image for 1 ≤ l ≤ N −1.

Thus taking k = N in (17) we obtain

image

Now it follows from the Theorem 3.7 that the inner graded q-derivation image associated to an element image is the N –differential of the image-graded algebra image. Hence the generalized Clifford algebra with four generators image regarded with respect to the image-graded structure defined by (16) and to N-differential dω is the q-GDA, and the triple imageis the N-differential calculus over the algebra of polynomial functions image 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 dω omitting ω and writing d instead of dω:

It should be mention that the structure of the q-GDA image 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 image, image and later in this section we will show that any element of degree image of the q-GDA image can be expressed in terms of the differentials of coordinates dx, dy. Since N-differential d satisfies dN = 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 image the algebra of q-differential forms on the reduced quantum plane and its elements of degreeimage 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 d2 = 0 on the reduced quantum plane with coordinates x, y which obey

x2 = y2 =1, x y = −yx.

In analogy with exterior calculus we shall call a q-differential form θ a m-closed q-differential form, where 1≤ m ≤ N −1 , if dm θ = 0 , 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 d1 = θ. It follows from dN = 0 that each q-differential (N − m)-exact form is m-closed.

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 image , where image is the algebra of polynomial functions, d is the N-differential and image is the image -bimodule of q-differential 1-forms. Evidently image . Let us express the differentials dx, dy in terms of the generators of image . It is worth mentioning that in what follows we shall use the structure of the right image -module of image have the relations to write q-differential 1-forms in terms of differentials. We

image (18)

where image. Using these relations we obtain

image (19)

where

image (20)

It is evident that the right image -module image 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 image -module of q-differential 1-forms image is freely generated by the differentials of coordinates dx, dy.

Proof: Let image be polynomial functions on the reduced quantum plane. Using (19),(20) and the fact that {u, v} is the basis for the right image –module image we can show that the equality dx f + dy h = 0 is equivalent to the system of equations

image

image

Multiplying the second equation by q − 1 and adding it to the first equation we obtain image. As image 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 image by image . Then (dx dy) = (u v) • A, where image, and from (19) we find

image

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 xN = 1, yN = 1 they are invertible elements of the algebra image and x−1 = xN −1, y−1 = yN −1. If N ≥ 3 then the matrix A is an invertible matrix and

image

Consequently we have

image (21)

image (22)

Now any q-differential 1-form image , where image , 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 calculus image 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 image to the algebra of (2 × 2)-matrices with entries from image , which we denote by image , and this homomorphism is determined by the relation between the right and left module structures of the bimodule of q-differential 1-forms image . Let us denote this homomorphism by image, i.e. for image we have

image

and

image (24)

image (25)

Since image is the algebra of polynomial functions in two variables x, y which are subjected to the commutation relation xy = q yx and the relations xN = 1, yn = 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

image(26)

Putting the entries of these matrices into the relations (24), (25) we get

image(27)

image(28)

The partial derivatives induced by the N-differential d are defined by

image

It can be shown [20] that the partial derivatives satisfy the twisted Leibniz rule

image

image

Using the twisted Leibniz rule and (26) we calculate

image

image

Using these formulae we can calculate the partial derivatives of any polynomial function image which can be written as image . For instant the derivative with respect to x of f is

image

We remind that the set of generators {x,u,y,v} of the generalized Clifford algebra image has been split into two parts, where the first part {x, y} generates the algebra of polynomial functions image , and the second part {u, v} generates the N-differential calculus image . 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 image then we may divide the algebraic relations between the new generators into three parts. The first part contains the relations

image

which determine the structure of the algebra image of polynomial functions on the reduced quantum plane. The second part consists of the relations (27),(28) between coordinates x, y and their differentials

image

image

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 uN = vN = 1 will 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

image (29)

where

image (30)

This proposition can be proved by straightforward computation with the help of the formulae (21), (22) and the relations (27), (28).

The relation (29) allows us to choose the ordered set of monomials B,

where

image

as the basis for the right image -module of q-differential k-forms image . For example the right image -module of q-differential 2-forms image is spanned by the monomials (dx)2,dydx,(dy)2 . Hence any q-differential k-form θ on the reduced quantum plane can be uniquely expressed as follows:

image

where imageare polynomial functions. 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 ≠ 0 for k running integers from 2 to N − 1 we have the set of higher order differentials of coordinates image, and we can use these higher order differentials to construct a basis for image . In this paper we shall describe a case of the module of q-differential 2-forms. The elements ω, ω1, ω − ω1 can be written as q-differential 1-forms as follows:

image (31)

Differentiating ω we obtain

image (32)

where dω is the q-differential 2-form. Now we can write the second order differential d2x as a q-differential 2-form as follows:

image(33)

Expressing the second order differential d2y in terms of (dx)2, dy dx, (dy)2 we prove the following proposition:

Proposition 3.12: The second order differential d2y can be written as follows:

image(34)

Proof: As image we have

image

Now applying the formula (31) and the multiplication rules (27), (28) we get the expression (34).

The Propositions 3.11, 3.12 show that we can change the basis image by the basis ' image in the right image -module of q-differential 2-forms image , where

image, image

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 d2y in a more symmetric form

, image(35)

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 d3 = 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.

image

The last term in the relation (29) vanishes because q is a primitive cube root of unity and satisfies q3 − 1 = 0. Making use of the relation 1 + q + q2= 0 we can write the coefficient γ1 as follows:

image

Hence the relation for the differentials (29) with respect to the basis image takes on the form

image(36)

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 d3 = 0 may be viewed as a deformation of the classical algebra of differential forms with parameter image and the additional term proportional to the form image.

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 [15-17]. 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 image be the set of variables.

Definition 4.1: The algebra of polynomials image with coefficients in image generated by the variables imageis said to be the algebra of connection form if the variables are subjected to the relations

image(37)

where q ≠ 0 is a complex number.

Let image be the subalgebra of image generated by the variable image and image be the subalgebra of image freely generated by the variables a1, a2, . . . , an, . . .. We define the image-graduation of the algebra of connection form image by assigning degree one to the generator d and degree i to a generator ai. It should be mentioned that this image-graded structure of image induces the image-graded structures of the subalgebras image .

Definition 4.2: The algebra of connection form image is said to be the algebra of N-connection form if q is a primitive N th root of unity, and the variable d obeys the additional relation image. The algebra of N-connection form will be denoted by image .

Proposition 4.3: The algebra of N-connection form image is an algebra of polynomials over image generated by finite set of variables image which are subjected to the relations.

image (38)

Proof: It is suffice to show that in the case of the algebra of N-connection form we have an = 0 for any integer n ≥ N. Indeed making use of the commutation relations (37) we can express any variable an, n ≥ 2 in the terms of image as polynomial

image (39)

Now assuming n = N we see that the first and last terms in (39) vanish because of image, 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 an = 0. In what follows we will denote by image the subalgebra of image generated by a1,a2,…, aN-1.

Let us consider the algebra of connection form image and its subalgebra image generated by the set of variables {ai}i≥ 1. Let image be a polynomial in variables a1,a2,…,an,….. The commutation relations of the algebra of connection form (37) show that for any polynomial imagethe product image can be represented in the form

image

where image the uniquely determined polynomials. We define the linear operator image associated to the variable image by the formula

image

where ξ any polynomial of image .

Theorem 4.4: The linear operator imageis the graded q-differential of the algebra image , i.e. ˆd satisfies the graded q-Leibniz rule with respect to N-graded structure of image. Particularly

image

In the case of the algebra of N-connection form image the graded q-differential image is the N-differential, i.e. image, and the algebra image is the q-GDA.

Now for any integer n ≥ 1 we define the polynomials image and the operator image of degree one by

image (40)

For the first values of k the straightforward computation of polynomials imageby means of recurrent relation image gives

image (41)

image (42)

image

image (43)

image

image

image

image (44)

We associate the linear operator imageto the polynomial image replacing the variable image in image by the associated graded q-differential image . Evidently image and image . Our aim now is to find a power expansion for polynomials image with respect to variables image. 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 image’s to the right.

Lemma 4.5: Each polynomial image can be expanded with respect to variables of the algebra of connection form image as follows

image

where image. In the case of the algebra of N-connection form image the operator image induced by the polynomial image is the operator of multiplication by image.

Let Ωq be a image -graded q-differential algebra with N-differential d, where q is a Nth primitive root of unity, and image be the subalgebra of elements of degree zero.

Definition 4.6: We will call an element of degree one imagea N-connection form in a q-GDA Ωq. The linear operator of degree one image 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 image for 1 ≤ n ≤ N − 1 if we successively apply it to a N-connection form A we get the sequence of elements A, dA, d2A, . . . , dN−1A, where. Let us denote by Ωq [A] the graded subalgebra of Ωq generated by these elements A, dA, d2A, . . . , dN−1 A. For any integer n = 1, 2, . . . , N we define the polynomial image by the formula image .

Now we will use the algebra of N-connection form imageto study the structure of a kth power of the covariant N-differential dA. Indeed it is easy to see that we can identify a N-differential d in Ωq with the N-differential imagein image , a N-connection form A in Ωq with the variable a1 in image and image with image . Consequently from Theorem 4.5 we obtain

Theorem 4.7: For any integer 1 ≤ n ≤ N the nth power of the covariant N-differential dA can be expanded as follows

image

where image . Particularly if n = N then the Nth power of the covariant N-differential dA is the operator of multiplication by the element image of degree zero.

The Theorem 4.7 allows us to define the curvature of a N-connection form A as follows

Definition 4.8: The N-curvature form of a N-connection form A is the element of degree zero image

It is easy to see that in the particular case of a graded differential algebra (N = 2, q = −1) with differential d satisfying d2 = 0 the above definition yields a connection form A and its curvature image 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 Ωq the N- curvature form image satisfies the Bianchi identity

image (45)

Proof: Indeed we have

image

From (41)–(44) we obtain the expressions for N-curvature form

image (46)

image (47)

image (48)

image

image

image

image

image (49)

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 [15-17]. 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 image be an unital associative image-algebra and Ωq be an N-differential calculus over image, i.e. Ωq is a q-GDA with N-differential d and image . Let ε be a left image -module. Considering algebra Ωq as the image-bimodule we take the tensor product image of modules which clearly has the structure of left image -module. Taking into account that an algebra Ωq can be viewed as the direct sum of image-bimodules Ωi q we can split the left image -module image into the direct sum of the left image -modules image [21], i.e. image , which means that image inherits the graded structure of algebra Ωq , and image is the graded left image -module. It is worth noting that the left image submoduleimage of elements of degree zero is isomorphic to a left image -module ε, where isomorphism image can be defined by image , where e is the identity element of algebra A. Since an algebra Ωq can be considered as the image-bimodule the left image -module image can be also viewed as the left Ωq -module [21] and we will use this structure to describe a concept of N- connection.

The tensor product image is also the vector space over image where this vector space is the tensor product of the vector spaces Ωq and ε. It is evident that F is a graded vector space, i.e. image, where image . Let us denote the image by image . The structure of the graded vector space of F induces the structure of a graded vector space on image , and we shall denote the subspace of homogeneous linear operators of degree i by image .

Definition 5.1: A N-connection on the left Ωq-module image is a linear operator image of degree one satisfying the condition

image(50)

whereimage , imageand |ω| is the degree of the homogeneous element of algebra Ωq .

Similarly one can define a N-connection on right modules. If image is a right image-module, a N-connection on image is a linear map image of degree one such that image for any image and homogeneous element image.

Making use of the previously introduced notations we can write image . 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 image -module. The set of all homomorphisms of ε into A has the structure of the dual module of the left image -module ε and is denoted by ε*. It is evident that ε* is a right image -module.

Definition 5.2: A linear map image defined as follows

image

whereimage , image and imageis a N-connection on ε, is said to be the dual connection of image.

It is easy to verify that image has a structure of N-connection on the right module image. Indeed, for any imagewe have

image

image

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 image generated by image is equal to zero, i.e. image . The right A -module ε is called a Hermitian module if ε is endowed with a sesquilinear map image which satises

image

image

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

image

whereimage

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 Ωq-module image .

Proof. It suffices to verify that for any homogeneous element ω ∈ Ωq an endomorphism image satisfies image . Let us expand the k-th power of ∇q as follows

image(51)

where imageare the q-binomial coefficients. Since d is the N-differential of a graded q-differential algebra Ωq we have dNω = 0 . Taking into account that image m for 1 ≤ m ≤ N − 1 we see that in the case of k = N the expansion (51) takes the following

image(52)

and this proves that image is the endomorphism of the left image-module image .

This proposition allows us to define the curvature of N-connection as follows

Definition 5.5: The endomorphism image of degree N of the left image- module image is said to be the curvature of a N-connection image.

The graded vector space image 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 image as an algebra multiplication. We can extend a N- connection image to the linear operator on the vector space image by means of the graded q-commutator as follows

image(53)

where A is a homogeneous linear operator. Obviously image is the linear operator of degree one on the vector space image, i.e. image, and image satisfies the graded q-Leibniz rule with respect to the algebra structure of image . It follows from the definition of the curvature of a N-connection that F can be viewed as the linear operator of degree N on the vector space F , i.e. image. Consequently one can act on F byimage, and it holds that

Proposition 5.6: For any N-connection ∇∇q the curvature F of this connec- tion satisfies the Bianchi identity ∇q(F)=0.

Proof : We have

image

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 image is a free A-module, where V is a image-vector space, then image is N-connection on ε and this connection is flat, i.e. its curvature vanishes.

Proof : Indeed, imageand

image

Where f,g ∈ A, v ∈V. Because of dN = 0 and q is the primitive N th root of unity, we get

image i. e. the curvature of

such a N-connection vanishes.

Theorem 5.8 : Every projective module admits a N-connection.

Proof : Let image be a projective module. From the theory of modules it is known that a module image is projective if and only if there exists a module image such that image is a free module. It is well known that free left image-module ε can be represented as the tensor product image, where V is a image -vector space. A linear map image is a N-connection on a projective module image, where imageis a N-connection on a left image-module ε, π is the projection on the first sum and in the direct sum image and image , where image Taking into account Proposition 5.7 we get

image

image

where image

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 image be a basis for a left module ε. For any element image we have image .

As mentioned above image and the basis for a module ε induces the basis image, where image , for the left image-module image . For any image we have image . Taking into account that image and image is the left Ωq –module we can multiply the elements of the basis image by elements of Ωq. It is easy to see that if image q then for any μ we have image . Consequently we can express any element of the image as a linear combination of image with coefficients from image . Indeed let image be an element of image . Then

image

image

where image

Let image . Obviously image , and making use of (50) we can express the element image as follows

image(54)

Let image be the vector space of the image-matrices whose entries are the elements of a q-GDA image. If each entry of a matriximage is an element of a homogeneous subspace image then Θ will be refered to as a homogeneous matrix of degree i and shall denote the vector space of such matrices by image . Obviously image. The vector space image of image -matrices becomes the associative unital graded algebra if we define the product of two matrices image as follows

image(55)

If image are homogeneous matrices then we define the graded q-commutator by image. We extend the N –differential d of a q-GDA Ωq to the algebra image as follows image

Since any element of a left image-module image can be expressed in terms of the basis image with coefficients from image we have

image(56)

where image. In analogy with the classical theory of connections in differential geometry of fibre bundles we introduce a notion of a matrix of N-connection.

Definition 5.9: A r × r-matrix image , whose entries image are the elements of image i.e. image , is said to be a matrix of a N-connection image with respect to the basis image of the left image-module image .

Using (54) and (56) we obtain

image(57)

In order to express the curvature F of a N-connection image in the terms of the entries of the matrix Θ of a N-connection image we should express the kth power of a N-connection image, where 1 ≤ k ≤ N , in the terms of the entries of the matrix Θ. It can be calculated that the kth power of image has the following form

image

image(58)

where image are polynomials on the entries image of the matrix Θ of a N-connection image and their differentials. We can calculate the polynomials image by means of the following recursion formula

image(59)

or in the matrix form

image(60)

where we begin with the polynomial image , and e is the identity element of A ⊂ Ωq. For example the first four polynomials in the expansion (58) obtained with the help of the recursion formula (59) have the form

image(61)

image(62)

image(63)

image

image

image(64)

From (58) it follows that if k = N then the first term image in this expansion vanishes because of the N-nilpotency of the N-differential d, and the next terms corresponding to the l values from 1 to N − 1 also vanish because of the well known property of q-binomial coefficients imageprovided q is a primitive Nth root of unity. Hence if k = N then the formula (58) takes on the form

image(65)

In order to simplify the notations and assuming that N is fixed we shall denote image.

Definition 5.10 : A (r × r) -matrix image , whose entries are the elements of degree N of a q-GDA Ωq, is said to be the curvature matrix of a N- connection ∇q .

Obviously image . In new notations the formula (65) can be written as follows image , and it clearly demonstrates that image is the endomorphism of degree N of the left Ωq -module image .

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 Ωq is a GDA with differential d satisfying d2 = 0. This is a classical case, and if we assume that Ωq is the algebra of differential forms on a smooth manifold M with exterior differential d and exterior multiplication image is the module of smooth sections of a vector bundle π : EM over M , ∇q is a connection on E, e is a local frame of a vector bundle E then Θ is the matrix of 1-forms of a connection ∇q and the formula (62) gives the expression for the curvature 2-form

image

in which we immediately recognize the classical expression for the curvature.

If N = 3 then image is the cubic root of unity satisfying the relations q3=1, 1+q+q2=0. This is a first noN-classical case of a q-connection, and the formula (63) gives the following expression for the curvature of a N-connection

image

image

image

It is useful to write the above expression for the curvature in a matrix form

image(66)

If N = 4 then q = i is the fourth root of unity satisfying relations1+q2 = 0, q2= -1. The expression (64) for curvature in this case takes on the form

image

image(67)

This expression can be put into a matrix form as follows

image(68)

It should be mentioned that in the case of N = 4 it holds that

image(69)

and the graded q-commutator degenerates in the case of dΘ, i.e. [dΘ, dΘ]q= dΘ dΘ - q4dΘ dΘ = 0.

From the proposition 5.6 it follows that the curvature of a N – connection satisfies the Bianchi identity. Straight forward computation shows that in both cases of (66) and (68) this identity in terms of the matrices of the curvature and a connection takes on the form

image(70)

Acknowledgement

V. Abramov and O. Liivapuu gratefully acknowledge the financial support of their research by the Estonian Science Foundation under the research grant ETF 7427.

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