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01-Applied Mathematics & Information Sciences
An International Journal
               
 
 
 
 
 
 
 
 
 
 
 

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Volumes > Volume 9 > No. 1L

 
   

Graded q-Differential Algebra Approach to Chern-Simons Form

PP: 29-38
Author(s)
Viktor Abramov, Olga Liivapuu,
Abstract
In the present paper we develop noncommutative approach to a connection which is based on a notion of graded qdifferential algebra, where q is a primitive Nth root of unity. We define the curvature of connection form and prove Bianchi identity. We construct a graded q-differential algebra to calculate the curvature of connection for any integer N  2. Making use of Bianchi identity we introduce the Chern character form of connection form and show that this form is closed. We study the case N = 3 which is the first non-trivial generalization because in the case N = 2 we have a classical theory. We calculate the curvature of connection form and show that it can be expressed in terms of graded q-commutators, where q is a primitive cubic root of unity. This allows us to prove an infinitesimal homotopy formula, and making use of this formula we introduce the Chern-Simons form.

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