Noncommutative Galois Extension and Graded q-Differential Algebra

Abstract

Abstract. We show that a semi-commutative Galois extension of a unital associative algebra can be endowed with the structure of a graded q-differential algebra. We study the first and higher order noncommutative differential calculus of semi-commutative Galois extension induced by the graded q-differential algebra. As an example we consider the quaternions which can be viewed as the semi-commutative Galois extension of complex numbers.Mathematics Subject Classification (2010). Primary 46L87; Secondary 81R60.

On Perfect Hypercomplex Algebras

“…A class of HNs endowed with addition and product forms a special vector space over R, called a hypercomplex algebra (HA). HAs have various applications including signal and image processing [20], dealing with differential operators [1,2], designing neural networks [19], etc.…”