Engineering Mathematics II pp 13-31 | Cite as

# Semi-commutative Galois Extension and Reduced Quantum Plane

## Abstract

In this paper we show that a semi-commutative Galois extension of associative unital algebra by means of an element \(\tau \), which satisfies \(\tau ^N={\mathbbm {1}}\) (\({\mathbbm {1}}\) is the identity element of an algebra and \(N\ge 2\) is an integer) induces a structure of graded *q*-differential algebra, where *q* is a primitive *N*th root of unity. A graded *q*-differential algebra with differential *d*, which satisfies \(d^N=0, N\ge 2\), can be viewed as a generalization of graded differential algebra. The subalgebra of elements of degree zero and the subspace of elements of degree one of a graded *q*-differential algebra together with a differential *d* can be considered as a first order noncommutative differential calculus. In this paper we assume that we are given a semi-commutative Galois extension of associative unital algebra, then we show how one can construct the graded *q*-differential algebra and when this algebra is constructed we study its first order noncommutative differential calculus. We also study the subspaces of graded *q*-differential algebra of degree greater than one which we call the higher order noncommutative differential calculus induced by a semi-commutative Galois extension of associative unital algebra. We also study the subspaces of graded *q*-differential algebra of degree greater than one which we call the higher order noncommutative differential calculus induced by a semi-commutative Galois extension of associative unital algebra. Finally we show that a reduced quantum plane can be viewed as a semi-commutative Galois extension of a fractional one-dimensional space and we apply the noncommutative differential calculus developed in the previous sections to a reduced quantum plane.

## Keywords

Noncommutative differential calculus Galois extension Reduced quantum plane## Notes

### Acknowledgement

The authors is gratefully acknowledge the Estonian Science Foundation for financial support of this work under the Research Grant No. ETF9328. This research was also supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. The second author also acknowledges his gratitude to the Doctoral School in Mathematics and Statistics for financial support of his doctoral studies at the Institute of Mathematics, University of Tartu. The authors are also grateful for partial support from Linda Peeters Foundation for cooperation between Sweden and Estonia provided by Swedish Mathematical Society.

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