# Generalization of connection based on the concept of graded q-differential algebra/Gradueeritud q-diferentsiaalalgebrale tuginev seostuse uldistus.

1. INTRODUCTION

It is well known that the concepts of connection and its curvature are basic elements of the theory of fibre bundles and play an important role not only in modern differential geometry, but also in modern theoretical physics, namely in the gauge field theory. The development of a theory of connections has been closely related to the development of theoretical physics. The advent of supersymmetric field theories in the 1970s gave rise to interest towards [Z.sub.2]-graded structures which became known in theoretical physics under the name of superstructures. This direction of development has led to the concept of superconnection which appeared in [12]. The emergence of noncommutative geometry in the 1980s was a powerful spur to the development of the theory of connections on modules [5,6,9,11]. A basic concept used in the theory of connections on modules is the notion of graded differential algebra. This notion has been generalized to the notion of graded q-differential algebra, where q is a primitive Nth root of unity (see papers [7,8,10]).

In Section 2 and Section 3 we give a short overview of N-structures, such as N-differential module, cochain N-complex, generalized cohomologies of an N-complex, and graded q-differential algebra. In Section 4 we introduce the notion of connection form in a graded q-differential algebra and covariant N-differential, which can be viewed as analogues of the connection form in a graded differential algebra described in [13]. In order to study the structure of a connection form in a graded q-differential algebra, we introduce an algebra of polynomials in the variables [??], [a.sub.1], [a.sub.2], ... and prove the power expansion formula for an nth power of the operator [[??].sub.a] = [??] + [a.sub.1]. Applying this formula, we show that the Nth power of the covariant N-differential is the operator of multiplication by an element [F.sup.(N).sub.A], which we then define as the curvature N-form of a connection form A. We also study the concept of [[OMEGA].sub.q]-connection on module, where [[OMEGA].sub.q] is a graded q-differential algebra, introduced in [2-4], and define the notions, such as dual [[OMEGA].sub.q]-connection and [[OMEGA].sub.q]-connection consistent with the Hermitian structure of a module.

2. N-COMPLEX

Let K be a commutative ring with a unit and E be a left K-module. The module E, endowed with an endomorphism d satisfying [d.sup.2] = 0, is referred to as a differential module and an endomorphism d as its differential. If K is a field, then the differential module E will be referred to as a differential vector space. From the property of the differential [d.sup.2] = 0 it follows that Im d [subset] Ker d and one can measure the non-exactness of the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by the quotient module H(E) = Ker d/Im d, which is referred to as the homology of the differential module E. Let E, F be differential modules, respectively with differentials d, d'. A homomorphism of differential modules is a homomorphism of modules [phi] : E [right arrow] F satisfying [phi] o d = d' o [phi]. Obviously [phi] (Im d) [subset] Im d', [phi] (Ker d) [subset] Ker d', and [phi] induces the homomorphism [[phi].sub.*] : H (E) [right arrow] H (F) in homology. Given an exact sequence of differential modules

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

one can construct a homomorphism [partial derivative] : H(G) [right arrow] H(E) such that the triangle

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

is exact [8].

A cochain complex is a Z-graded differential module E = [[direct sum].sub.i[member of]Z] [E.sup.i] whose differential d has degree 1, which means d : [E.sup.n] [right arrow] [E.sup.n+1]. The homology H(E) of a cochain complex inherits a Z-graded structure of the cochain complex E. Hence H(E) = [[direct sum].sub.i[member of]Z] [H.sup.i](E), where [H.sup.i](E) = Ker d [intersection] [E.sup.i], and H(E) is usually referred to as a cohomology of the cochain complex E. Given an exact sequence of cochain complexes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

one can construct by means of (1) the following exact sequence:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ...

Let N [greater than or equal to] 2 be a positive integer. The left K-module E is said to be an N-differential module with N-differential d if d is an endomorphism of E satisfying [d.sup.N] = 0. Obviously, an N-differential module can be viewed as a generalization of the concept of differential module to any integer N [greater than or equal to] 2. If K is a field, an N-differential module will be referred to as an N-differential vector space.

For each integer m with 1 [less than or equal to] m [less than or equal to] N - 1 we can define the submodules [Z.sub.m](E) = Ker ([d.sup.m]) and [B.sub.m](E) = Im ([d.sup.N-m]). It follows from the equation [d.sup.N] = 0 that [B.sub.m](E) [subset] [Z.sub.m](E) and the quotient modules [H.sub.m](E) := [Z.sub.m](E)/[B.sub.m](E) are called the (generalized) homology of the N-differential module E. As in the case of the homology of a differential module, one can prove a proposition analogous to (1), which asserts that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an exact sequence of N-differential modules, then there exist homomorphisms [partial derivative] : [H.sub.m](G) [right arrow] [H.sub.N-m](E) for m [member of] {1, 2, ..., N - 1} such that the following hexagons of homomorphisms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are exact for n [member of] {1, 2, ..., N - 1} [8]. A cochain N-complex of modules or simply an N-complex is a Z-graded N-differential module E = [[direct sum].sub.k[member of]Z][E.sup.k] with a homogeneous N-differential d of degree 1. If E is an N-complex, then its cohomologies [H.sub.m](E) are Z-graded modules, i.e. [H.sub.m](E) = [[direct sum].sub.n[member of]Z][H.sup.n.sub.m](E), where

[H.sup.n.sub.m] (E) = Ker ([d.sup.m] : [E.sup.n] [right arrow] [E.sup.n+m])/[d.sup.N-m]([E.sup.n+m-N]).

It should be noted that many notions related to N-complexes depend only on the underlying [Z.sub.N]-graduation. For this purpose we define a [Z.sub.N]-complex to be a [Z.sub.N]-graded N-differential module with N-differential d of degree 1.

3. GRADED q-DIFFERENTIAL ALGEBRA

Let [OMEGA] = [[direct sum].sub.n[member of]Z][[OMEGA].sup.n] be a unital associative graded C-algebra. The subspace of elements of grading zero [[OMEGA].sup.0] [subset] [OMEGA] is the subalgebra of [OMEGA], which we denote by U, i.e. U = [[OMEGA].sup.0]. Any subspace [[OMEGA].sup.k] [subset] [OMEGA] of elements of grading k [member of] Z is the U-bimodule. A graded differential algebra is a unital associative graded C-algebra equipped with a linear mapping d of degree 1 such that the sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

d([omega] x [theta]) = d[omega] x [theta] + [(-1).sup.k] [omega] x d[theta],

where [omega] [member of] [[OMEGA].sup.k], [theta] [member of] [OMEGA]. Let us mention that if [OMEGA] is a graded differential algebra, then Ker d is the graded unital subalgebra of [OMEGA], whereas Im d is the graded two-sided ideal of Ker d, so the cohomology H([OMEGA]) is the unital associative graded algebra. Obviously [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a first-order differential calculus over the algebra U. In what follows we shall call [OMEGA] a differential calculus over the algebra U.

Making use of the notion of N-complex described in the previous section, one can generalize the concept of graded differential algebra [8,10]. Let K be the field of complex numbers C and q be a primitive Nth root of unity, where N [greater than or equal to] 2. A graded q-differential algebra is a unital associative Z-graded ([Z.sub.n]-graded) C-algebra [[OMEGA].sub.q] = [[direct sum].sub.k][[OMEGA].sup.k.sub.q] endowed with a linear mapping d of degree one such that the sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

d([omega] x [theta]) = d[omega] x [theta] + [q.sup.k] [omega] x d[theta],

where [omega] [member of] [[OMEGA].sup.k.sub.q], [theta] [member of] [[OMEGA].sub.q]. As in the case of a graded differential algebra, the subspace of elements of grading zero U = [[OMEGA].sup.0.sub.q] is the subalgebra of the graded q-differential algebra [[OMEGA].sub.q]. By analogy with the terminology used in the case of a graded differential algebra, we shall call [[OMEGA].sub.q] a q-differential calculus over the algebra U. Let us mention that d : U [right arrow] [[OMEGA].sup.1.sub.q] is a first-order differential calculus over the algebra U.

Let us remind that a q-graded centre of an associative unital graded C-algebra A = [[direct sum].sub.k[member of]Z][A.sup.k] is the graded subspace Z(A) = [[direct sum].sub.k[member of]Z] [Z.sup.k] (A) of A generated by the homogeneous elements v [member of] [A.sup.k], where k [member of] Z, satisfying vw = [q.sup.km] wv for any w [member of] [A.sup.m], m [member of] Z. The graded q-centre Z (A) is the graded subalgebra of A. A graded q-derivation of degree k [member of] Z of A is a homogeneous linear mapping D : [A.sup.m] [right arrow] [A.sup.m+k] satisfying the graded q-Leibniz rule D(vw) = D(v) w + [q.sup.km] vD(w), where v [member of] [A.sup.m]. If v [member of] [A.sup.k] is a homogeneous element, then v determines the graded q-derivation of degree k by means of a graded q-commutator as follows: [ad.sub.q](v)w = [[v, w].sub.q] = vw - [q.sup.km] wv, where w [member of] [A.sup.m], and [ad.sub.q] (v) is called an inner graded q-derivation of degree k of A.

It is proved in [1] that if A is an associative unital graded C-algebra and v is an element of grading one of this algebra satisfying [v.sup.N] [member of] Z(A), where N [greater than or equal to] 2, then the inner graded q-derivation [d.sub.v] = [ad.sub.q](v) : [A.sup.k] [right arrow] [A.sup.k+1] is the N-differential of an algebra A and A is the graded q-differential algebra with respect to d. Making use of this theorem, we can endow a generalized Clifford algebra with a structure of graded q-differential algebra. Indeed, let us remind that a generalized Clifford algebra [[C.sup.N.sub.p] is an associative unital C-algebra generated by [[gamma].sub.1], [[gamma].sub.2], ..., [[gamma].sub.p], which are subjected to the relations

[[gamma].sub.i][[gamma].sub.j] = [q.sup.sg(j-i)] [[gamma].sub.j][[gamma].sub.i], [[gamma].sup.N.sub.i] = 1, i, j = 1, 2, ..., p,

where 1 is the identity element of [[C.sup.N.sub.p]. If we assign grading zero to the identity element 1 and grading one to each generator [[gamma].sub.i], then [[C.sup.N.sub.p] becomes the [Z.sub.N]-graded algebra. It is easy to verify that the Nth power of any linear combination of generators v = [[summation].sub.[mu]] [[lambda].sub.[mu]] [[gamma].sub.[mu]], [[lambda].sub.[mu]] [member of] C belongs to the graded q-centre of [C.sup.N.sub.p], i.e. [v.sup.N] [member of] Z([C.sup.N.sub.p]). Hence, [d.sub.v] = [ad.sub.q](v) is the N-differential of [[C.sup.N.sub.p] and [C.sup.N.sub.p] is the graded q- differential algebra.

4. GENERALIZATION OF CONNECTION

In this section we propose a generalization of the concept of connection form and connection on a module by means of the notion of graded q-differential algebra [2-4]. We begin with an algebra of polynomials on two variables, which we will use later to prove propositions describing the structure of the curvature of a connection form.

Let B[[??], a] be the algebra of polynomials in [??], [a.sub.1], [a.sub.2], ... with coefficients in C and the variables [??], [a.sub.1], [a.sub.2], ... be subjected to the relations

[??][a.sub.i] = q [a.sub.i][??] + [a.sub.i+1], i = 1,2, ..., (2)

where q is a complex number. Let B[[??]] be the subalgebra of B[[??], a], generated by the variable [??], and B[a] be the subalgebra of B[[??],a], freely generated by the variables [a.sub.1], [a.sub.2], ..., [a.sub.n], .... We equip the algebra of polynomials B[[??], a] with the N-graduation, by assigning grading one to the generator d and grading i to the generator [a.sub.i]. It should be mentioned that this N-graded structure of [??][[??], a] induces the N-graded structures on the subalgebras B[[??]], B[a]. We will call B[[??], a] the N-reduced algebra of polynomials and denote it by [B.sub.N][[??], a] if q is a primitive Nth root of unity and [??] obeys the additional relation [[??].sup.N] = 0. It can be shown by means of (2) that if q is a primitive Nth root of unity and [[??].sup.N] = 0, then [a.sub.n] = 0 for any integer n [greater than or equal to] N. Indeed, making use of the commutation relations (2), we can express any variable [a.sub.n], n [greater than or equal to] 2, in terms of [??], [a.sub.1] as the polynomial

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

If n = N in (3), then the first and last terms in (3) vanish because of [[??].sup.N] = 0; other terms are zero because of the vanishing of q-binomial coefficients, provided q is a primitive Nth root of unity. Hence, [a.sub.N] = 0, and the N-reduced algebra of polynomials [B.sub.N][[??], a] is an algebra over C generated by [??], [a.sub.1], [a.sub.2], ..., [a.sub.N-1], which are subjected to the commutation relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Let us consider the algebra B[[??], a] and its subalgebra B[a] generated by [a.sub.1], [a.sub.2], ..., [a.sub.N-1]. We assign a linear operator [??] : B[a] [right arrow] B[a] to the generator [??] by putting

[??](1) = 0, [??]([a.sub.i]) = [a.sub.i+1], [??]([a.sub.i][a.sub.j]) = [a.sub.i+1][a.sub.j] + [q.sup.i][a.sub.i][a.sub.j+1].

Evidently [??] is the graded q-differential on the subalgebra B[a], i.e. [??] satisfies the graded q-Leibniz rule with respect to N-graded structure of B[a]. It is easy to see that in the case of the N-reduced algebra of polynomials [B.sub.N][[??], a], the Nth power of [??] : [B.sub.N][a] [right arrow] [B.sub.N][a] is zero, i.e. [[??].sup.N] = 0. Hence, [B.sub.N][a] is the graded q-differential algebra and [??] is its N-differential.

Now, for any integer n [greater than or equal to] 1 we define the polynomials [p.sup.(n)] [member of] B[[??],a], [f.sup.(n).sub.a] [member of] B[a] and the operator [[??].sub.a] : B[a] [right arrow] B[a] of grading one by

[p.sup.(n)] = [([??] + [a.sub.1]).sup.n], [f.sup.(k+1).sub.a] = [[??].sup.k.sub.a]([a.sub.1]), [[??].sub.a](p) = [??](p)+ [a.sub.1] p, [for all]p [member of] B[a]. (4)

For the first values of k the straightforward computation of polynomials [f.sup.(k).sub.a] by means of the recurrent relation [f.sup.(k+1).sub.a] = [[??].sub.a]([f.sup.(k).sub.a])) gives

[f.sup.(2).sub.a] = [a.sub.2] + [a.sup.2.sub.1], (5)

[f.sup.(3).sub.a] = [a.sub.3] + [a.sub.2][a.sub.1] + [[2].sub.q][a.sub.1] [a.sub.2] + [a.sup.3.sub.1], (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

We assign the operator [[??].sup.(n)] : B[a] [right arrow] B[a] to the polynomial [p.sup.(n)], replacing the variable [??] in [p.sup.(n)] by the operator [??]. Evidently, [[??].sup.(n)] = [[??].sup.n.sub.a] and [f.sup.(n+1).sub.a] = [p.sup.(n)]([a.sub.1]). Our aim now is to find a power expansion for polynomials [p.sup.(n)] with respect to variables [??], [a.sub.1], [a.sub.2], ..., [a.sub.n]. It is obvious that making use of the commutation relations (2), we can rearrange the factors in each summand of this expansion by removing all [??]'s to the right.

Theorem 4.1. Each polynomial [p.sup.(n)] can be expanded with respect to variables of the algebra B[[??],a] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [f.sup.(n).sub.a] = [[??].sup.(n-1)] ([a.sub.1]). In the case of N-reduced algebra of polynomials [B.sub.N] [[??], a], the operator [[??].sup.(N)] = [[??].sup.N.sub.a] : [[??].sub.N][a] [right arrow] [B.sub.N][a], induced by the polynomial [p.sup.(N)], is the operator of multiplication by [f.sup.(N).sub.a].

Let [[OMEGA].sub.q] be a [Z.sub.N]-graded q-differential algebra with N-differential d, where q is an Nth primitive root of unity, and U = [[OMEGA].sup.0.sub.q] be the subalgebra of elements of grading zero. We will call an element of grading one A [member of] [[OMEGA].sup.1.sub.q] a connection form in a graded q-differential algebra [[OMEGA].sup.1.sub.q]. Since d is an N- differential, which means that [d.sup.n] [not equal to] 0 for 1 [less than or equal to] n [less than or equal to] N - 1, if we successively apply it to a connection form A, we get the sequence of elements A, dA, [d.sup.2] A, ..., [d.sup.N-1] A, where [d.sup.n] A [member of] [[OMEGA].sup.n+1.sub.q]. Let us denote by [[OMEGA].sub.q][A] the graded subalgebra of [[OMEGA].sub.q] generated by A, dA, [d.sup.2] A, [d.sup.N-1] A. The linear operator of degree one [d.sub.A] = d + A : [[OMEGA].sup.i] [right arrow] [Q.sup.i+1] will be called a covariant N-differential induced by a connection form A. For any integer n = 1,2, ..., N we define the polynomial [F.sup.(n).sub.A] [member of] [[OMEGA].sub.q][A] by the formula [F.sup.(n).sub.A] = [d.sup.n- 1.sub.A](A).

Now we apply the N-reduced algebra of polynomials [B.sub.N] [[??], a], constructed above to study the structure of a k th power of the covariant N-differential [d.sub.A]. Indeed, it is easy to see that we can identify an N- differential d in [[OMEGA].sub.q] with the variable [??] in [B.sub.N] [[??], a], a connection form A in [[OMEGA].sub.q] with the variable [a.sub.1] in [B.sub.N] [[??], a] and [d.sup.n] A [member of] [[OMEGA].sub.q] with [a.sub.n+1] [member of] [B.sub.N] [[??],a]. Then the commutation relations (4) between [??],[a.sub.i] are equivalent to the graded q-Leibniz rule for an N-differential d, and [f.sup.(n).sub.a] can be identified with [F.sup.(n).sub.A]. Consequently, from Theorem 4.1 we obtain

Proposition 4.2. For any integer 1 [less than or equal to] n [less than or equal to] N the nth power of the covariant N-differential [d.sub.A] can be expanded as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [F.sup.(n).sub.A] = [([d.sub.A]).sup.n-1](A). Particularly, if n = N, then the Nth power of the covariant N- differential [d.sub.A] is the operator of multiplication by the element [F.sup.(N).sub.A] of degree zero.

Proposition 4.2 allows us to define the curvature of a connection form A as follows.

Definition 4.3. The curvature N-form of a connection form A is the element of grading zero [F.sup.(N).sub.A] [member of] U.

It is easy to see that in the particular case of a graded differential algebra (N = 2, q = -1) with differential d satisfying [d.sup.2] = 0 the above definition yields a connection form A and its curvature [F.sup.(2).sub.A] = dA + [A.sup.2] as elements of a graded differential algebra, respectively, of grading one and two [13].

Proposition 4.4. For any connection form A in a graded q-differential algebra [[OMEGA].sub.q] the curvature N-form satisfies the Bianchi identity

d[F.sup.(N).sub.A] + [[A, [F.sup.(N).sub.A])].sub.q] = 0.

Proof. Indeed, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From (5)-(8) we obtain the expressions for curvature form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let U be a unital associative C-algebra and [OMEGA] be a differential calculus over U, i.e. [OMEGA] is a graded differential algebra [OMEGA] = [[direct sum].sub.k][[OMEGA].sup.k] with [[OMEGA].sup.0] = U and differential d. Let E be a left module over algebra U. It is evident that E has the structure of C-vector space induced by a left U-module structure if one defines [alpha][xi] = ([alpha]e) [xi], where [alpha] [member of] C, [xi] [member of] E, e is the identity element of algebra U. Let us remind [9] that an [OMEGA]-connection on module E is a linear map [nabla] : E [right arrow] [[OMEGA].sup.1] [[cross product].sub.U] E satisfying the condition

[nabla]([omega][xi]) = d[omega][[cross product].sub.U][xi] + [omega][nabla]([xi]),

where [omega] [member of] U, [xi] [member of] E. Since [[OMEGA].sup.k] can be viewed as the (U, U)-bimodule, the tensor product [[OMEGA].sup.1] [[cross product].sub.U] E has the structure of the left U-module. Let us denote F = [OMEGA] [[cross product].sub.U] E. Obviously, F is the left [OMEGA]- module and also a graded left U-module, i.e. F = [[direct sum].sub.U] [F.sup.k], where [F.sup.k] = [[OMEGA].sup.k] [[cross product].sub.U] E. One can extend an [OMEGA]-connection [nabla] to any [[OMEGA].sup.k] [[cross product].sub.U] E by means of the formula

[nabla]([omega] [[cross product].sub.U] E) = d[omega] [[cross product].sub.U] [xi] + [(-1).sup.k] [omega][nabla]([xi]),

where [omega] [member of] [[OMEGA].sup.k], [xi] [member of] E.

Let [[OMEGA].sub.q] be a graded q-differential algebra with N-differential d. In order to generalize the notion of [OMEGA]-connection, we define as in [2-4] an [[OMEGA].sub.q]-connection [[nabla].sub.q] on the left [[OMEGA].sub.q]- module [[OMEGA].sub.q] [[cross product].sub.U] E as a linear operator of degree one satisfying the condition

[[nabla].sub.q]([omega] [[cross product].sub.U] [xi])= d[omega] [[cross product].sub.U][xi] + [q.sup.[absolute value of [omega]]] [omega] [[nabla].sub.q]([xi]),

where [omega] [member of] [[OMEGA].sup.k.sub.q], [xi] [member of] E, and [absolute value of [omega]] is the grading of the homogeneous element of algebra [[OMEGA].sub.q]. Analogously, if G is a right U-module, we define an [[OMEGA].sub.q]-connection on G as a linear map [[nabla].sub.q] : G [right arrow] G [[cross product].sub.U] [[OMEGA].sup.1.sub.q] such that [[nabla].sub.q]([xi]f) = [xi] [[cross product].sub.U] d f + [[nabla].sub.q]([xi])f for any [xi] [member of] G, f [member of] U.

The tensor product F = [[OMEGA].sup.k.sub.q] [[cross product].sub.C] E of vector spaces is the graded C-vector space. Let us denote the vector space of linear operators on F by L(F). The graded structure of the vector space F induces a graduation on the vector space L(F) = [[direct sum].sub.k] [L.sup.k] (F). If F [right arrow] F is a homogeneous linear operator, we can extend it to the linear operator [L.sub.A] = L(F) [right arrow] L(F) on the graded algebra of linear operators L(F) by means of the graded q-commutator as follows:

[L.sub.A] (B) = [[A, B].sub.q] = A x B - [q.sup.[absolute value of A][absolute value of B]] B x A,

where B is a homogeneous linear operator and A x B is the product of two linear operators. It can be shown that the Nth power of any [[OMEGA].sub.q]-connection [[nabla].sub.q] is the endomorphism of degree N of the left [[OMEGA].sub.q]-module F. The proof is based on the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [omega] [member of] [[OMEGA].sub.q], [xi] [member of] E. This allows us to define the curvature of an [[OMEGA].sub.q]-connection [[nabla].sub.q] as the endomorphism F = [[nabla].sup.N.sub.q] of degree N of the left [OMEGA]-module F.

Let E be a left U-module. The set of all homomorphisms of E into U has the structure of the dual module of the left U-module E, and is denoted by [E.sup.*]. It is easy to see that [E.sup.*] is a right A-module. If [[nabla].sub.q] is an [[OMEGA].sub.q]-connection on E, then a linear map [[nabla].sup.*.sub.q] : [E.sup.*] [right arrow] [E.sup.*] [[cross product].sub.U] [[OMEGA].sup.1.sub.q] defined as

[[nabla].sup.*.sbu.q]([eta])([xi]) = d([eta]([xi])) - [eta]([[nabla].sub.q]([xi])),

where [omega] [member of] E, [eta] [member of] [E.sup.*], is an [[OMEGA].sub.q]-connection on the right module [E.sup.*]. Indeed, for any f [member of] U, [eta] [member of] [E.sup.*], [xi] [member of] E, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In order to define a Hermitian structure on a right U-module E, we assume U to be a graded q-differential algebra with involution * such that the largest linear subset contained in the convex cone C [member of] U generated by [a.sup.*] a is equal to zero, i.e. C [intersection] (-C) = 0. The right U-module E is called a Hermitian module if E is endowed with a sesquilinear map h : E x E [right arrow] U which satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We have used the convention for a sesquilinear map to take the second argument to be linear. If E is a Hermitian right U-module, an [[OMEGA].sub.q]-connection [[nabla].sub.q] on E is said to be consistent with a Hermitian structure of E if it satisfies

dh([xi], [xi]')= h([[nabla].sub.q]([xi]), [xi]') + h([xi], [[nabla].sub.q]([xi]')),

where [xi], [xi]' [member of] E.

In analogy with the theory of [OMEGA]-connection [9] we can prove that there is an [[OMEGA].sub.q]-connection on every projective module. For this we need the following proposition.

Proposition 4.5. If E = U [cross product] V is a free U-module, where V is a C-vector space, then [[nabla].sub.q] = d [cross product] [I.sub.V] is an [[OMEGA].sub.q]-connection on E and this connection is flat, i.e. its curvature vanishes.

Proof. Indeed, [[nabla].sub.q] : U [cross product] V [right arrow] [[OMEGA].sup.1.sub.q] [cross product] (U [cross product] V) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where f, g [member of] U, v [member of] V. As [d.sup.N] = 0 and q is the primitive Nth root of unity, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

i.e. the curvature of such an [[OMEGA].sub.q]-connection vanishes.

Theorem 4.6. Every projective module admits an [[OMEGA].sub.q]-connection.

Proof. Let M be a projective module. From the theory of modules it is known that a module M is projective if and only if there exists a module N such that E = M [direct sum] N is a free module. It is well known that a free left U-module E can be represented as the tensor product U [cross product] V, where V is a C-vector space. A linear map [[nabla].sub.q] = [pi] o (d [cross product] [I.sub.V]) : M [right arrow] [[OMEGA].sup.1.sub.q] [[cross product].sub.U] M is an [[OMEGA].sub.q]-connection on a projective module M, where d [cross product] [I.sub.V] is an [[OMEGA].sub.q]-connection on a left U-module E, [pi] is the projection on the first summand in the direct sum M [direct sum] N, and [pi]([omega] [[cross product].sub.U] (g [cross product] v)) = [omega] [[cross product].sub.U] [pi](g [cross product] v) = [omega] [[cross product].sub.U] m, where [omega] [member of] [[OMEGA].sup.1.sub.q], g [member of] U, v [member of] V, m [member of] M. Taking into account Proposition 4.5, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where f [member of] U, m [member of] M.

doi: 10.3176/proc.2010.4.02

ACKNOWLEDGEMENT

We gratefully acknowledge the financial support of the Estonian Science Foundation under the research grant ETF 7427.

REFERENCES

[1.] Abramov, V. On a graded q-differential algebra. J. Nonlinear Math. Phys., 2006, 13 Supplement, 1-8.

[2.] Abramov, V. Generalization of superconnection in noncommutative geometry. Proc. Estonian Acad. Sci. Phys. Math., 2006, 55, 3-15.

[3.] Abramov, V. Graded q-differential algebra approach to q-connection. In Generalized Lie Theory in Mathematics, Physics and Beyond (Silvestrov, S. and Paal, E., eds). Springer, 2009, 71-79.

[4.] Abramov, V. and Liivapuu, O. Geometric approach to BRST-symmetry and [Z.sub.N]-generalization of superconnection. J. Nonlinear Math. Phys., 2006, 13 Supplement, 9-20.

[5.] Connes, A. [C.sup.*] algebres et geometrie differentielle. C. R. Acad. Sci. Paris, 1980, 290, Serie A, 599-604.

[6.] Cuntz, J. and Quillen, D. Algebra extensions and nonsingularity. J. Amer. Math. Soc., 1995, 8, 251-289.

[7.] Dubois-Violette, M. [d.sup.N] = 0: generalized homology. K-Theory, 1998, 14, 371-404.

[8.] Dubois-Violette, M. Lectures on differentials, generalized differentials and on some examples related to theoretical physics. arXiv: math.gA/0005256, 2000.

[9.] Dubois-Violette, M. Lectures on graded differential algebras and noncommutative geometry. In Noncommutative Differential Geometry and Its Applications to Physics: Proceedings of the Workshop (Maeda, Y., Moriyoshi, H., Omori, H., Sternheimer, D., Tate, T., and Watamura, S., eds). Math. Phys. Stud., 2001, 23, 245-306.

[10.] Dubois-Violette, M. and Kerner, R. Universal q-differential calculus and q-analog of homological algebra. Acta Math. Univ. Comenian, 1996, 65, 175-188.

[11.] Dubois-Violette, M. and Masson, T. On the first order operators in bimodules. Lett. Math. Phys., 1996, 37, 467- 474.

[12.] Mathai, V. and Quillen, D. Superconnections, Thom classes and equivariant differential forms. Topology, 1986, 25, 85-110.

[13.] Quillen, D. G. Chern-Simons forms and cyclic cohomology. In The Interface of Mathematics and Particle Physics (Quillen, D. G., Tsou, S. T., and Segal, G. B., eds). Clarendon Press, Oxford, 1990, 117-134.

Viktor Abramov * and Olga Liivapuu

Institute of Mathematics, University of Tartu, Liivi 2, 50409 Tartu, Estonia; olgai@ut.ee

Received 29 September 2009, accepted 9 October 2009

* Corresponding author, viktor.abramov@ut.ee

It is well known that the concepts of connection and its curvature are basic elements of the theory of fibre bundles and play an important role not only in modern differential geometry, but also in modern theoretical physics, namely in the gauge field theory. The development of a theory of connections has been closely related to the development of theoretical physics. The advent of supersymmetric field theories in the 1970s gave rise to interest towards [Z.sub.2]-graded structures which became known in theoretical physics under the name of superstructures. This direction of development has led to the concept of superconnection which appeared in [12]. The emergence of noncommutative geometry in the 1980s was a powerful spur to the development of the theory of connections on modules [5,6,9,11]. A basic concept used in the theory of connections on modules is the notion of graded differential algebra. This notion has been generalized to the notion of graded q-differential algebra, where q is a primitive Nth root of unity (see papers [7,8,10]).

In Section 2 and Section 3 we give a short overview of N-structures, such as N-differential module, cochain N-complex, generalized cohomologies of an N-complex, and graded q-differential algebra. In Section 4 we introduce the notion of connection form in a graded q-differential algebra and covariant N-differential, which can be viewed as analogues of the connection form in a graded differential algebra described in [13]. In order to study the structure of a connection form in a graded q-differential algebra, we introduce an algebra of polynomials in the variables [??], [a.sub.1], [a.sub.2], ... and prove the power expansion formula for an nth power of the operator [[??].sub.a] = [??] + [a.sub.1]. Applying this formula, we show that the Nth power of the covariant N-differential is the operator of multiplication by an element [F.sup.(N).sub.A], which we then define as the curvature N-form of a connection form A. We also study the concept of [[OMEGA].sub.q]-connection on module, where [[OMEGA].sub.q] is a graded q-differential algebra, introduced in [2-4], and define the notions, such as dual [[OMEGA].sub.q]-connection and [[OMEGA].sub.q]-connection consistent with the Hermitian structure of a module.

2. N-COMPLEX

Let K be a commutative ring with a unit and E be a left K-module. The module E, endowed with an endomorphism d satisfying [d.sup.2] = 0, is referred to as a differential module and an endomorphism d as its differential. If K is a field, then the differential module E will be referred to as a differential vector space. From the property of the differential [d.sup.2] = 0 it follows that Im d [subset] Ker d and one can measure the non-exactness of the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by the quotient module H(E) = Ker d/Im d, which is referred to as the homology of the differential module E. Let E, F be differential modules, respectively with differentials d, d'. A homomorphism of differential modules is a homomorphism of modules [phi] : E [right arrow] F satisfying [phi] o d = d' o [phi]. Obviously [phi] (Im d) [subset] Im d', [phi] (Ker d) [subset] Ker d', and [phi] induces the homomorphism [[phi].sub.*] : H (E) [right arrow] H (F) in homology. Given an exact sequence of differential modules

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

one can construct a homomorphism [partial derivative] : H(G) [right arrow] H(E) such that the triangle

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

is exact [8].

A cochain complex is a Z-graded differential module E = [[direct sum].sub.i[member of]Z] [E.sup.i] whose differential d has degree 1, which means d : [E.sup.n] [right arrow] [E.sup.n+1]. The homology H(E) of a cochain complex inherits a Z-graded structure of the cochain complex E. Hence H(E) = [[direct sum].sub.i[member of]Z] [H.sup.i](E), where [H.sup.i](E) = Ker d [intersection] [E.sup.i], and H(E) is usually referred to as a cohomology of the cochain complex E. Given an exact sequence of cochain complexes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

one can construct by means of (1) the following exact sequence:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ...

Let N [greater than or equal to] 2 be a positive integer. The left K-module E is said to be an N-differential module with N-differential d if d is an endomorphism of E satisfying [d.sup.N] = 0. Obviously, an N-differential module can be viewed as a generalization of the concept of differential module to any integer N [greater than or equal to] 2. If K is a field, an N-differential module will be referred to as an N-differential vector space.

For each integer m with 1 [less than or equal to] m [less than or equal to] N - 1 we can define the submodules [Z.sub.m](E) = Ker ([d.sup.m]) and [B.sub.m](E) = Im ([d.sup.N-m]). It follows from the equation [d.sup.N] = 0 that [B.sub.m](E) [subset] [Z.sub.m](E) and the quotient modules [H.sub.m](E) := [Z.sub.m](E)/[B.sub.m](E) are called the (generalized) homology of the N-differential module E. As in the case of the homology of a differential module, one can prove a proposition analogous to (1), which asserts that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an exact sequence of N-differential modules, then there exist homomorphisms [partial derivative] : [H.sub.m](G) [right arrow] [H.sub.N-m](E) for m [member of] {1, 2, ..., N - 1} such that the following hexagons of homomorphisms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are exact for n [member of] {1, 2, ..., N - 1} [8]. A cochain N-complex of modules or simply an N-complex is a Z-graded N-differential module E = [[direct sum].sub.k[member of]Z][E.sup.k] with a homogeneous N-differential d of degree 1. If E is an N-complex, then its cohomologies [H.sub.m](E) are Z-graded modules, i.e. [H.sub.m](E) = [[direct sum].sub.n[member of]Z][H.sup.n.sub.m](E), where

[H.sup.n.sub.m] (E) = Ker ([d.sup.m] : [E.sup.n] [right arrow] [E.sup.n+m])/[d.sup.N-m]([E.sup.n+m-N]).

It should be noted that many notions related to N-complexes depend only on the underlying [Z.sub.N]-graduation. For this purpose we define a [Z.sub.N]-complex to be a [Z.sub.N]-graded N-differential module with N-differential d of degree 1.

3. GRADED q-DIFFERENTIAL ALGEBRA

Let [OMEGA] = [[direct sum].sub.n[member of]Z][[OMEGA].sup.n] be a unital associative graded C-algebra. The subspace of elements of grading zero [[OMEGA].sup.0] [subset] [OMEGA] is the subalgebra of [OMEGA], which we denote by U, i.e. U = [[OMEGA].sup.0]. Any subspace [[OMEGA].sup.k] [subset] [OMEGA] of elements of grading k [member of] Z is the U-bimodule. A graded differential algebra is a unital associative graded C-algebra equipped with a linear mapping d of degree 1 such that the sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

d([omega] x [theta]) = d[omega] x [theta] + [(-1).sup.k] [omega] x d[theta],

where [omega] [member of] [[OMEGA].sup.k], [theta] [member of] [OMEGA]. Let us mention that if [OMEGA] is a graded differential algebra, then Ker d is the graded unital subalgebra of [OMEGA], whereas Im d is the graded two-sided ideal of Ker d, so the cohomology H([OMEGA]) is the unital associative graded algebra. Obviously [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a first-order differential calculus over the algebra U. In what follows we shall call [OMEGA] a differential calculus over the algebra U.

Making use of the notion of N-complex described in the previous section, one can generalize the concept of graded differential algebra [8,10]. Let K be the field of complex numbers C and q be a primitive Nth root of unity, where N [greater than or equal to] 2. A graded q-differential algebra is a unital associative Z-graded ([Z.sub.n]-graded) C-algebra [[OMEGA].sub.q] = [[direct sum].sub.k][[OMEGA].sup.k.sub.q] endowed with a linear mapping d of degree one such that the sequence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

d([omega] x [theta]) = d[omega] x [theta] + [q.sup.k] [omega] x d[theta],

where [omega] [member of] [[OMEGA].sup.k.sub.q], [theta] [member of] [[OMEGA].sub.q]. As in the case of a graded differential algebra, the subspace of elements of grading zero U = [[OMEGA].sup.0.sub.q] is the subalgebra of the graded q-differential algebra [[OMEGA].sub.q]. By analogy with the terminology used in the case of a graded differential algebra, we shall call [[OMEGA].sub.q] a q-differential calculus over the algebra U. Let us mention that d : U [right arrow] [[OMEGA].sup.1.sub.q] is a first-order differential calculus over the algebra U.

Let us remind that a q-graded centre of an associative unital graded C-algebra A = [[direct sum].sub.k[member of]Z][A.sup.k] is the graded subspace Z(A) = [[direct sum].sub.k[member of]Z] [Z.sup.k] (A) of A generated by the homogeneous elements v [member of] [A.sup.k], where k [member of] Z, satisfying vw = [q.sup.km] wv for any w [member of] [A.sup.m], m [member of] Z. The graded q-centre Z (A) is the graded subalgebra of A. A graded q-derivation of degree k [member of] Z of A is a homogeneous linear mapping D : [A.sup.m] [right arrow] [A.sup.m+k] satisfying the graded q-Leibniz rule D(vw) = D(v) w + [q.sup.km] vD(w), where v [member of] [A.sup.m]. If v [member of] [A.sup.k] is a homogeneous element, then v determines the graded q-derivation of degree k by means of a graded q-commutator as follows: [ad.sub.q](v)w = [[v, w].sub.q] = vw - [q.sup.km] wv, where w [member of] [A.sup.m], and [ad.sub.q] (v) is called an inner graded q-derivation of degree k of A.

It is proved in [1] that if A is an associative unital graded C-algebra and v is an element of grading one of this algebra satisfying [v.sup.N] [member of] Z(A), where N [greater than or equal to] 2, then the inner graded q-derivation [d.sub.v] = [ad.sub.q](v) : [A.sup.k] [right arrow] [A.sup.k+1] is the N-differential of an algebra A and A is the graded q-differential algebra with respect to d. Making use of this theorem, we can endow a generalized Clifford algebra with a structure of graded q-differential algebra. Indeed, let us remind that a generalized Clifford algebra [[C.sup.N.sub.p] is an associative unital C-algebra generated by [[gamma].sub.1], [[gamma].sub.2], ..., [[gamma].sub.p], which are subjected to the relations

[[gamma].sub.i][[gamma].sub.j] = [q.sup.sg(j-i)] [[gamma].sub.j][[gamma].sub.i], [[gamma].sup.N.sub.i] = 1, i, j = 1, 2, ..., p,

where 1 is the identity element of [[C.sup.N.sub.p]. If we assign grading zero to the identity element 1 and grading one to each generator [[gamma].sub.i], then [[C.sup.N.sub.p] becomes the [Z.sub.N]-graded algebra. It is easy to verify that the Nth power of any linear combination of generators v = [[summation].sub.[mu]] [[lambda].sub.[mu]] [[gamma].sub.[mu]], [[lambda].sub.[mu]] [member of] C belongs to the graded q-centre of [C.sup.N.sub.p], i.e. [v.sup.N] [member of] Z([C.sup.N.sub.p]). Hence, [d.sub.v] = [ad.sub.q](v) is the N-differential of [[C.sup.N.sub.p] and [C.sup.N.sub.p] is the graded q- differential algebra.

4. GENERALIZATION OF CONNECTION

In this section we propose a generalization of the concept of connection form and connection on a module by means of the notion of graded q-differential algebra [2-4]. We begin with an algebra of polynomials on two variables, which we will use later to prove propositions describing the structure of the curvature of a connection form.

Let B[[??], a] be the algebra of polynomials in [??], [a.sub.1], [a.sub.2], ... with coefficients in C and the variables [??], [a.sub.1], [a.sub.2], ... be subjected to the relations

[??][a.sub.i] = q [a.sub.i][??] + [a.sub.i+1], i = 1,2, ..., (2)

where q is a complex number. Let B[[??]] be the subalgebra of B[[??], a], generated by the variable [??], and B[a] be the subalgebra of B[[??],a], freely generated by the variables [a.sub.1], [a.sub.2], ..., [a.sub.n], .... We equip the algebra of polynomials B[[??], a] with the N-graduation, by assigning grading one to the generator d and grading i to the generator [a.sub.i]. It should be mentioned that this N-graded structure of [??][[??], a] induces the N-graded structures on the subalgebras B[[??]], B[a]. We will call B[[??], a] the N-reduced algebra of polynomials and denote it by [B.sub.N][[??], a] if q is a primitive Nth root of unity and [??] obeys the additional relation [[??].sup.N] = 0. It can be shown by means of (2) that if q is a primitive Nth root of unity and [[??].sup.N] = 0, then [a.sub.n] = 0 for any integer n [greater than or equal to] N. Indeed, making use of the commutation relations (2), we can express any variable [a.sub.n], n [greater than or equal to] 2, in terms of [??], [a.sub.1] as the polynomial

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

If n = N in (3), then the first and last terms in (3) vanish because of [[??].sup.N] = 0; other terms are zero because of the vanishing of q-binomial coefficients, provided q is a primitive Nth root of unity. Hence, [a.sub.N] = 0, and the N-reduced algebra of polynomials [B.sub.N][[??], a] is an algebra over C generated by [??], [a.sub.1], [a.sub.2], ..., [a.sub.N-1], which are subjected to the commutation relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Let us consider the algebra B[[??], a] and its subalgebra B[a] generated by [a.sub.1], [a.sub.2], ..., [a.sub.N-1]. We assign a linear operator [??] : B[a] [right arrow] B[a] to the generator [??] by putting

[??](1) = 0, [??]([a.sub.i]) = [a.sub.i+1], [??]([a.sub.i][a.sub.j]) = [a.sub.i+1][a.sub.j] + [q.sup.i][a.sub.i][a.sub.j+1].

Evidently [??] is the graded q-differential on the subalgebra B[a], i.e. [??] satisfies the graded q-Leibniz rule with respect to N-graded structure of B[a]. It is easy to see that in the case of the N-reduced algebra of polynomials [B.sub.N][[??], a], the Nth power of [??] : [B.sub.N][a] [right arrow] [B.sub.N][a] is zero, i.e. [[??].sup.N] = 0. Hence, [B.sub.N][a] is the graded q-differential algebra and [??] is its N-differential.

Now, for any integer n [greater than or equal to] 1 we define the polynomials [p.sup.(n)] [member of] B[[??],a], [f.sup.(n).sub.a] [member of] B[a] and the operator [[??].sub.a] : B[a] [right arrow] B[a] of grading one by

[p.sup.(n)] = [([??] + [a.sub.1]).sup.n], [f.sup.(k+1).sub.a] = [[??].sup.k.sub.a]([a.sub.1]), [[??].sub.a](p) = [??](p)+ [a.sub.1] p, [for all]p [member of] B[a]. (4)

For the first values of k the straightforward computation of polynomials [f.sup.(k).sub.a] by means of the recurrent relation [f.sup.(k+1).sub.a] = [[??].sub.a]([f.sup.(k).sub.a])) gives

[f.sup.(2).sub.a] = [a.sub.2] + [a.sup.2.sub.1], (5)

[f.sup.(3).sub.a] = [a.sub.3] + [a.sub.2][a.sub.1] + [[2].sub.q][a.sub.1] [a.sub.2] + [a.sup.3.sub.1], (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

We assign the operator [[??].sup.(n)] : B[a] [right arrow] B[a] to the polynomial [p.sup.(n)], replacing the variable [??] in [p.sup.(n)] by the operator [??]. Evidently, [[??].sup.(n)] = [[??].sup.n.sub.a] and [f.sup.(n+1).sub.a] = [p.sup.(n)]([a.sub.1]). Our aim now is to find a power expansion for polynomials [p.sup.(n)] with respect to variables [??], [a.sub.1], [a.sub.2], ..., [a.sub.n]. It is obvious that making use of the commutation relations (2), we can rearrange the factors in each summand of this expansion by removing all [??]'s to the right.

Theorem 4.1. Each polynomial [p.sup.(n)] can be expanded with respect to variables of the algebra B[[??],a] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [f.sup.(n).sub.a] = [[??].sup.(n-1)] ([a.sub.1]). In the case of N-reduced algebra of polynomials [B.sub.N] [[??], a], the operator [[??].sup.(N)] = [[??].sup.N.sub.a] : [[??].sub.N][a] [right arrow] [B.sub.N][a], induced by the polynomial [p.sup.(N)], is the operator of multiplication by [f.sup.(N).sub.a].

Let [[OMEGA].sub.q] be a [Z.sub.N]-graded q-differential algebra with N-differential d, where q is an Nth primitive root of unity, and U = [[OMEGA].sup.0.sub.q] be the subalgebra of elements of grading zero. We will call an element of grading one A [member of] [[OMEGA].sup.1.sub.q] a connection form in a graded q-differential algebra [[OMEGA].sup.1.sub.q]. Since d is an N- differential, which means that [d.sup.n] [not equal to] 0 for 1 [less than or equal to] n [less than or equal to] N - 1, if we successively apply it to a connection form A, we get the sequence of elements A, dA, [d.sup.2] A, ..., [d.sup.N-1] A, where [d.sup.n] A [member of] [[OMEGA].sup.n+1.sub.q]. Let us denote by [[OMEGA].sub.q][A] the graded subalgebra of [[OMEGA].sub.q] generated by A, dA, [d.sup.2] A, [d.sup.N-1] A. The linear operator of degree one [d.sub.A] = d + A : [[OMEGA].sup.i] [right arrow] [Q.sup.i+1] will be called a covariant N-differential induced by a connection form A. For any integer n = 1,2, ..., N we define the polynomial [F.sup.(n).sub.A] [member of] [[OMEGA].sub.q][A] by the formula [F.sup.(n).sub.A] = [d.sup.n- 1.sub.A](A).

Now we apply the N-reduced algebra of polynomials [B.sub.N] [[??], a], constructed above to study the structure of a k th power of the covariant N-differential [d.sub.A]. Indeed, it is easy to see that we can identify an N- differential d in [[OMEGA].sub.q] with the variable [??] in [B.sub.N] [[??], a], a connection form A in [[OMEGA].sub.q] with the variable [a.sub.1] in [B.sub.N] [[??], a] and [d.sup.n] A [member of] [[OMEGA].sub.q] with [a.sub.n+1] [member of] [B.sub.N] [[??],a]. Then the commutation relations (4) between [??],[a.sub.i] are equivalent to the graded q-Leibniz rule for an N-differential d, and [f.sup.(n).sub.a] can be identified with [F.sup.(n).sub.A]. Consequently, from Theorem 4.1 we obtain

Proposition 4.2. For any integer 1 [less than or equal to] n [less than or equal to] N the nth power of the covariant N-differential [d.sub.A] can be expanded as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [F.sup.(n).sub.A] = [([d.sub.A]).sup.n-1](A). Particularly, if n = N, then the Nth power of the covariant N- differential [d.sub.A] is the operator of multiplication by the element [F.sup.(N).sub.A] of degree zero.

Proposition 4.2 allows us to define the curvature of a connection form A as follows.

Definition 4.3. The curvature N-form of a connection form A is the element of grading zero [F.sup.(N).sub.A] [member of] U.

It is easy to see that in the particular case of a graded differential algebra (N = 2, q = -1) with differential d satisfying [d.sup.2] = 0 the above definition yields a connection form A and its curvature [F.sup.(2).sub.A] = dA + [A.sup.2] as elements of a graded differential algebra, respectively, of grading one and two [13].

Proposition 4.4. For any connection form A in a graded q-differential algebra [[OMEGA].sub.q] the curvature N-form satisfies the Bianchi identity

d[F.sup.(N).sub.A] + [[A, [F.sup.(N).sub.A])].sub.q] = 0.

Proof. Indeed, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From (5)-(8) we obtain the expressions for curvature form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let U be a unital associative C-algebra and [OMEGA] be a differential calculus over U, i.e. [OMEGA] is a graded differential algebra [OMEGA] = [[direct sum].sub.k][[OMEGA].sup.k] with [[OMEGA].sup.0] = U and differential d. Let E be a left module over algebra U. It is evident that E has the structure of C-vector space induced by a left U-module structure if one defines [alpha][xi] = ([alpha]e) [xi], where [alpha] [member of] C, [xi] [member of] E, e is the identity element of algebra U. Let us remind [9] that an [OMEGA]-connection on module E is a linear map [nabla] : E [right arrow] [[OMEGA].sup.1] [[cross product].sub.U] E satisfying the condition

[nabla]([omega][xi]) = d[omega][[cross product].sub.U][xi] + [omega][nabla]([xi]),

where [omega] [member of] U, [xi] [member of] E. Since [[OMEGA].sup.k] can be viewed as the (U, U)-bimodule, the tensor product [[OMEGA].sup.1] [[cross product].sub.U] E has the structure of the left U-module. Let us denote F = [OMEGA] [[cross product].sub.U] E. Obviously, F is the left [OMEGA]- module and also a graded left U-module, i.e. F = [[direct sum].sub.U] [F.sup.k], where [F.sup.k] = [[OMEGA].sup.k] [[cross product].sub.U] E. One can extend an [OMEGA]-connection [nabla] to any [[OMEGA].sup.k] [[cross product].sub.U] E by means of the formula

[nabla]([omega] [[cross product].sub.U] E) = d[omega] [[cross product].sub.U] [xi] + [(-1).sup.k] [omega][nabla]([xi]),

where [omega] [member of] [[OMEGA].sup.k], [xi] [member of] E.

Let [[OMEGA].sub.q] be a graded q-differential algebra with N-differential d. In order to generalize the notion of [OMEGA]-connection, we define as in [2-4] an [[OMEGA].sub.q]-connection [[nabla].sub.q] on the left [[OMEGA].sub.q]- module [[OMEGA].sub.q] [[cross product].sub.U] E as a linear operator of degree one satisfying the condition

[[nabla].sub.q]([omega] [[cross product].sub.U] [xi])= d[omega] [[cross product].sub.U][xi] + [q.sup.[absolute value of [omega]]] [omega] [[nabla].sub.q]([xi]),

where [omega] [member of] [[OMEGA].sup.k.sub.q], [xi] [member of] E, and [absolute value of [omega]] is the grading of the homogeneous element of algebra [[OMEGA].sub.q]. Analogously, if G is a right U-module, we define an [[OMEGA].sub.q]-connection on G as a linear map [[nabla].sub.q] : G [right arrow] G [[cross product].sub.U] [[OMEGA].sup.1.sub.q] such that [[nabla].sub.q]([xi]f) = [xi] [[cross product].sub.U] d f + [[nabla].sub.q]([xi])f for any [xi] [member of] G, f [member of] U.

The tensor product F = [[OMEGA].sup.k.sub.q] [[cross product].sub.C] E of vector spaces is the graded C-vector space. Let us denote the vector space of linear operators on F by L(F). The graded structure of the vector space F induces a graduation on the vector space L(F) = [[direct sum].sub.k] [L.sup.k] (F). If F [right arrow] F is a homogeneous linear operator, we can extend it to the linear operator [L.sub.A] = L(F) [right arrow] L(F) on the graded algebra of linear operators L(F) by means of the graded q-commutator as follows:

[L.sub.A] (B) = [[A, B].sub.q] = A x B - [q.sup.[absolute value of A][absolute value of B]] B x A,

where B is a homogeneous linear operator and A x B is the product of two linear operators. It can be shown that the Nth power of any [[OMEGA].sub.q]-connection [[nabla].sub.q] is the endomorphism of degree N of the left [[OMEGA].sub.q]-module F. The proof is based on the formula

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [omega] [member of] [[OMEGA].sub.q], [xi] [member of] E. This allows us to define the curvature of an [[OMEGA].sub.q]-connection [[nabla].sub.q] as the endomorphism F = [[nabla].sup.N.sub.q] of degree N of the left [OMEGA]-module F.

Let E be a left U-module. The set of all homomorphisms of E into U has the structure of the dual module of the left U-module E, and is denoted by [E.sup.*]. It is easy to see that [E.sup.*] is a right A-module. If [[nabla].sub.q] is an [[OMEGA].sub.q]-connection on E, then a linear map [[nabla].sup.*.sub.q] : [E.sup.*] [right arrow] [E.sup.*] [[cross product].sub.U] [[OMEGA].sup.1.sub.q] defined as

[[nabla].sup.*.sbu.q]([eta])([xi]) = d([eta]([xi])) - [eta]([[nabla].sub.q]([xi])),

where [omega] [member of] E, [eta] [member of] [E.sup.*], is an [[OMEGA].sub.q]-connection on the right module [E.sup.*]. Indeed, for any f [member of] U, [eta] [member of] [E.sup.*], [xi] [member of] E, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In order to define a Hermitian structure on a right U-module E, we assume U to be a graded q-differential algebra with involution * such that the largest linear subset contained in the convex cone C [member of] U generated by [a.sup.*] a is equal to zero, i.e. C [intersection] (-C) = 0. The right U-module E is called a Hermitian module if E is endowed with a sesquilinear map h : E x E [right arrow] U which satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We have used the convention for a sesquilinear map to take the second argument to be linear. If E is a Hermitian right U-module, an [[OMEGA].sub.q]-connection [[nabla].sub.q] on E is said to be consistent with a Hermitian structure of E if it satisfies

dh([xi], [xi]')= h([[nabla].sub.q]([xi]), [xi]') + h([xi], [[nabla].sub.q]([xi]')),

where [xi], [xi]' [member of] E.

In analogy with the theory of [OMEGA]-connection [9] we can prove that there is an [[OMEGA].sub.q]-connection on every projective module. For this we need the following proposition.

Proposition 4.5. If E = U [cross product] V is a free U-module, where V is a C-vector space, then [[nabla].sub.q] = d [cross product] [I.sub.V] is an [[OMEGA].sub.q]-connection on E and this connection is flat, i.e. its curvature vanishes.

Proof. Indeed, [[nabla].sub.q] : U [cross product] V [right arrow] [[OMEGA].sup.1.sub.q] [cross product] (U [cross product] V) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where f, g [member of] U, v [member of] V. As [d.sup.N] = 0 and q is the primitive Nth root of unity, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

i.e. the curvature of such an [[OMEGA].sub.q]-connection vanishes.

Theorem 4.6. Every projective module admits an [[OMEGA].sub.q]-connection.

Proof. Let M be a projective module. From the theory of modules it is known that a module M is projective if and only if there exists a module N such that E = M [direct sum] N is a free module. It is well known that a free left U-module E can be represented as the tensor product U [cross product] V, where V is a C-vector space. A linear map [[nabla].sub.q] = [pi] o (d [cross product] [I.sub.V]) : M [right arrow] [[OMEGA].sup.1.sub.q] [[cross product].sub.U] M is an [[OMEGA].sub.q]-connection on a projective module M, where d [cross product] [I.sub.V] is an [[OMEGA].sub.q]-connection on a left U-module E, [pi] is the projection on the first summand in the direct sum M [direct sum] N, and [pi]([omega] [[cross product].sub.U] (g [cross product] v)) = [omega] [[cross product].sub.U] [pi](g [cross product] v) = [omega] [[cross product].sub.U] m, where [omega] [member of] [[OMEGA].sup.1.sub.q], g [member of] U, v [member of] V, m [member of] M. Taking into account Proposition 4.5, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where f [member of] U, m [member of] M.

doi: 10.3176/proc.2010.4.02

ACKNOWLEDGEMENT

We gratefully acknowledge the financial support of the Estonian Science Foundation under the research grant ETF 7427.

REFERENCES

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[10.] Dubois-Violette, M. and Kerner, R. Universal q-differential calculus and q-analog of homological algebra. Acta Math. Univ. Comenian, 1996, 65, 175-188.

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[13.] Quillen, D. G. Chern-Simons forms and cyclic cohomology. In The Interface of Mathematics and Particle Physics (Quillen, D. G., Tsou, S. T., and Segal, G. B., eds). Clarendon Press, Oxford, 1990, 117-134.

Viktor Abramov * and Olga Liivapuu

Institute of Mathematics, University of Tartu, Liivi 2, 50409 Tartu, Estonia; olgai@ut.ee

Received 29 September 2009, accepted 9 October 2009

* Corresponding author, viktor.abramov@ut.ee

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Title Annotation: | MATHEMATICS |
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Author: | Abramov, Viktor; Liivapuu, Olga |

Publication: | Proceedings of the Estonian Academy of Sciences |

Article Type: | Report |

Geographic Code: | 4EXES |

Date: | Dec 1, 2010 |

Words: | 5266 |

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