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**Viktor ABRAMOV ^{*} and Olga LIIVAPUU^{*}**

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

- *Corresponding Author:
- Viktor ABRAMOV
[email protected]

E-mail:

Olga LIIVAPUU

**E-mail:**[email protected]

**Received Date**: December 20,2007; **Revised Date:** March 13, 2008

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

We study a concept of a q-connection on a left module, where q is a primitive Nth root of unity. This concept is based on a notion of a graded q-differential algebra whose differential d satisfies dN = 0. We propose a notion of a graded q-differential algebra with involution and making use of this notion we introduce and study a concept of a q-connection consistent with a Hermitian structure of a left module. Assuming module to be a finitely generated free module we define the components of q-connection and show that these components with respect to different basises are related by gauge transformation. We also derive the relation for components of a q-connection consistent with Hermitian structure of a module.

Let q be a primitive Nth root of unity, where concept of a q-connection and graded q-connection on a left module [1,2,3,4] is based on a notion of a graded q-differential algebra A [5,6,7]. The differential d of a graded q-differential algebra A satisfies the graded q-Leibniz rule and d^{N} = 0. If N = 2, q = −1 then the graded q-Leibniz rule takes the form of graded
Leibniz rule and d^{2} = 0. Hence a graded q-differential algebra can be viewed as a generalization a graded differential algebra. If ε is a left module over the subalgebra of
elements of grading zero and then a q-connection on the left A-module is a
linear operator D of grading one satisfying the graded q-Leibniz rule. It can be shown that the N th power of a q-connection D is the endomorphism of the left A-module and this allows to define the curvature of q-connection as It can be proved that the curvature F of
q-connection satisfies the Bianchi identity. In this paper we continue to study the concept of a
q-connection started in [2,3,4] and propose a notion of a q-connection on the left module F
consistent with a Hermitian structure of the module F. A Hermitian structure on the module
F requires an involution on a graded q-differential algebra A, and we introduce a notion of a
graded q-differential algebra with involution proving that the differential d is consistent with
an involution. Assuming the left A-module to be a finitely generated free left module
we define the components of a q-connection with respect to a basis for the module and show
that the components of a q-connection with respect to different basises are related by gauge
transformation. Assuming that D is a q-connection consistent with a Hermitian structure of F
we derive the relation for the components of D. Finally we find the expressions for components
of the curvature in terms of the components of a q-connection.

The aim of this section is to remind a concept of a graded q-differential algebra, where q is
a primitive Nth root of unity This algebra is a basic component in our algebraic
approach to q-generalization of connection, and it may be viewed as an analog of algebra of
differential forms with exterior differential satisfying d^{N} = 0. It should be noted that within the
framework of this analogy the subalgebra of elements of grading zero plays a role of an algebra
of functions on a base manifold. In order to have an algebraic model of differential forms with
values in a vector bundle we introduce a left module over the subalgebra of elements of grading
zero of a graded q-differential algebra. Assuming that this module is a finitely generated free
module we describe an algebraic analog of transition from one local trivialization of a vector
bundle to another.

Let q be a primitive Nth root of unity and be an associative unital graded algebra over the complex numbers. Let us denote the identity element of this algebra by e and the grading of a homogeneous element ω of An algebra A is said to be a graded qdifferential algebra if it is endowed with a linear mapping d of degree one, i.e satisfying the graded q-Leibniz rule

where and the N-nilpotency condition d^{N} = 0. It is easy to see that the subspace A^{0} of elements of grading zero is the subalgebra of an algebra A. We will denote this subalgebra by Obviously is the associative unital algebra over C with the identity element e. Given an associative unital algebra A we call a graded q-differential algebra A an N-differential calculus over an algebra Let us mention that taking N = 2, q = −1 in the
definition of a graded q-differential algebra we get a graded differential algebra (with differential
d satisfying d^{2} = 0). Thus a graded q-differential algebra can be considered as a generalization
of a graded differential algebra for any integer N > 2. It follows from the graded structure of an
algebra A that each subspace of homogeneous elements of grading i can be considered
as the bimodule over the algebra Thus we have the following sequence of bimodules

The part d : of this sequence is the first order differential calculus over the algebra

We define a graded q-differential algebra with involution as a graded q-differential algebra A which is equipped with a mapping of grading zero satisfying

where It is easy to show that the involution is consistent with the graded q-Leibniz rule. We have

On the other hand,

From the above formulae and

it follows that the involution ¤ is consistent with the graded q-Leibniz rule. Let ε be a left -module. Considering a graded q-differential algebra A as the bimodule we take the tensor product of modules which clearly has the structure of left -module. Let us
denote this left -module by F, i.e. Obviously F inherits the graded structure of A. Indeed for every i we have the left submodule of the left module F. It is easy to see that the left module F is the direct sum of its submodules It
is worth noting that the left submodule F^{0} of elements of grading zero is isomorphic to a left module ε, i. e. where the isomorphism can be defined by The left module F can be also considered as the left A-module and in the next section we will use this structure to describe a concept of q-connection. Let us mention that multiplication by elements of A^{i}, where i 6= 0, does not preserve the graded structure of the module F.

Since A is a graded algebra the tensor product of vector spaces is the graded vector space over C, where Hence we have the graded algebra of linear operators of the graded vector space F, which we denote by where is the subspace of homogeneous linear operators of grading i. If is a homogeneous linear operator then we can extend it to the linear operator on the whole graded algebra of linear operators by means of the graded q-commutator as follows

where B is a homogeneous linear operator and A · B is the product of two linear operators.

In order to have an algebraic analog of the local structure of a vector bundle in this approach
we assume ε to be a finitely generated free left module. Let be a basis for a left module ε. This basis induces the basis where for the left -module F^{0}. Taking into account that and F is the left A-module we can multiply the elements of the basis f by elements of A. It is easy to see that if then for any μ we have Consequently we can express any element of F^{i} as a linear combination of f_{μ} with coefficients from A^{i}. Indeed let be an element of Then

where

Denote by the vector space of r × r-matrices whose entries are the elements of an algebra A. This vector space is a graded vector space with graded structure induced by the graded structure of a graded q-differential algebra A. Hence where is the subspace of homogeneous matrices of grading i, i.e. if then The vector space matrices becomes the associative unital graded algebra if we define the product of two matrices In the next section we shall use the graded q-commutator of homogeneous matrices which is defined by

We extend the differential d of a graded q-differential algebra A to the algebra as usual:

let be another basis for the left module F^{0} with the same number of elements (this will always be the case if is a division algebra or if is commutative). Then where is the transition matrix from the basis f to the basis f^{0}. It is well known [8] that in the case of finitely generated free module transition matrix is an invertible matrix, and we denote the inverse matrix of G by

In order to define a Hermitian structure on the left A-module F we assume A to be a graded q-differential algebra with involution ¤. We will call the left module F a Hermitian (left) module if F^{0} is endowed with a bilinear form which satisfies where and It is easy to extend a Hermitian form h to the whole left A-module F if we put

where and Consequently it holds The matrix of this Hermitian form with respect to a basis f is denoted by

**q-connection on module F**

In this section we describe a concept of q-connection [2,3,4] on the left A-module F, curvature of q-connection and Bianchi identity. Assuming that graded q-differential algebra A is an algebra with involution and F is the Hermitian module over this algebra we define a q-connection consistent with a Hermitian structure of F. Then assuming the submodule to be a finitely generated free module we introduce the matrices of q-connection and its curvature.

A q-connection on the left A-module F is a linear operator of degree one satisfying the condition

where and d is the differential of a graded q-differential algebra A. If the left A-module F is the Hermitian left module with Hermitian form h a q-connection D on F is said to be consistent with a Hermitian structure of F if it satisfies

(3.1)

where

It can be shown that the N-th power of any q-connection D is the endomorphism of degree N of the left A-module F. This allows us to define the curvature of a q-connection D as the endomorphism F = DN of degree N of the left A-module F. The curvature F of any q-connection D on F satisfies the Bianchi identity LD(F) = 0 [3], where is the extention of D to the algebra of linear operators of F.

Let F^{0} be a finitely generated free module with a basis and where Obviously The coefficients of a q-connection D with respect to a basis f are defined by The matrix is called the matrix of q-connection D with respect to f. Using (3.1) we obtain

(3.2)

where be another basis for the left A-module F^{0}, and where is a transition matrix. If we denote by the coefficients of D with respect to basis are the entries of the inverse matrix then and this clearly shows that the components of D with respect to different basises of module F^{0} are related by the gauge transformation. Let A be a graded q-differential
algebra with involution ¤ : be a Hermitian module with a Hermitian form h, and D
be a q-connection on F consistent with a Hermitian structure of F. Then the components of D obey the relation

Our next aim is to express the components of the curvature F of a q-connection D in terms of the coefficients of a q-connection D. We define the components of curvature F with respect to a basis f by and denote the matrix of curvature by Straightforward computation gives for different k = 1, 2, . . . ,N the polynomial

where are q-binomial coefficients, and From this polynomial we get the recursion formula for the components of curvature

This recursion formula gives the following expressions for the first three values of k:

(3.3)

Let us consider the expressions for curvature in two cases when N = 2 and N = 3. If
N = 2, q = −1 then a graded q-differential algebra A is a differential superalgebra (Z_{2}-graded),and we have for the components of curvature Assuming A to be a
super-commutative algebra we can put the expression for components of curvature into the form or by means of matrices in which we recognize the classical
expression for the curvature.

If N = 3 then is the cubic root of unity satisfying the relations q^{3} = 1, 1+q+q^{2} =0. This is the first non-classical case of a q-connection, and we have for components

Finally we derive the form of Bianchi identity in terms of the components of a q-connection
and its curvature. The curvature F of a q-connection satisfies the Bianchi identity L_{D}(F) = are the components of a q-connection D and its curvature F with respect
to a basis f for the module F then the Bianchi identity takes on the form

The authors gratefully acknowledge the financial support of their research by the Estonian Science Foundation under the grant ETF 7427.

- Kapranov MM, On the q-analog of homological algebra. Preprintar Xiv:math QA/9611005.
- Abramov V (2008) q-connection and its matrix. In”Generalized Lie Theory in Mathematics, Physics and Beyond“.Abramov V, Paal E, Silvestrov S, Stolin A Eds. Springer-Verlag, Berlin–New York.
- Abramov V (2006) Generalization of superconnection in non-commutative geometry. Proc Estonian Acad Sci Phys Math 55: 3–15.
- Abramov V, Liivapuu O (2006) Geometric approach to BRST-symmetry and ZN-generalization ofsuperconnection. J Nonlinear Math Phys13Supplement pp: 9–20.
- Dubois-Violette M, Kerner R (1996) Universal q-differential calculus and q-analog of homological al-gebra. Acta Math Univ Comenian 65: 175-188.
- Dubois-Violette M (2001) Lectures on graded differential algebras and noncommutative geometry. In”Non-commutative differential geometry and its applications to physics. Proceedings of the workshop heldin Shonan, May 31–June 4, 1999“Maeda Y, Moriyoshi H, Omori H, Sternheimer D, Tate T, Watamura S, Eds. Kluwer Academic Publishers, Math Phys Stud 23: 245-306.
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- Godement R (1969) Algebra Hermann, Paris; Kershow Publishing Company Ltd, London.

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