Noncommutative Galois Extensions and Ternary Clifford Analysis

“…In this paper we will study a particular case of a noncommutative Galois extension which is called a semi-commutative Galois extension [15]. A noncommutative Galois extension is referred to as a semi-commutative Galois extension if for any element x ∈ A there exists an element x ′ ∈ A such that x τ = τ x ′ .…”

Section: Definitionmentioning

confidence: 99%

“…Let us briefly remind a definition of noncommutative Galois extension [12,13,14,15]. Suppose Ã is an associative unital C-algebra, A ⊂ Ã is its subalgebra, and there is an element τ ∈ Ã which satisfies τ / ∈ A , τ N = ½, where N ≥ 2 is an integer and ½ is the identity element of Ã .…”

Section: Introductionmentioning

confidence: 99%

Semi-commutative Galois Extension and Reduced Quantum Plane

Abramov

Raknuzzaman

2016

Springer Proceedings in Mathematics &Amp; Statistics

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In this paper we show that a semi-commutative Galois extension of associative unital algebra by means of an element τ, which satisfies τ N = 1 (1 is the identity element of an algebra and N ≥ 2 is an integer) induces a structure of graded q-differential algebra, where q is a primitive Nth root of unity. A graded qdifferential algebra with differential d, which satisfies d N = 0, N ≥ 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 semicommutative 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.

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“…Our aim in this section is to show that we can apply this theorem to a noncommutative Galois extension to construct a graded q-differential algebra with N -differential satisfying d N = 0. First of all we remind a notion of a noncommutative Galois extension [10,11,12,13].…”

Section: Graded Q-differential Algebra Structure Of Noncommutative Gamentioning

confidence: 99%

Noncommutative Galois Extension and Graded q-Differential Algebra

Abramov

2015

Adv. Appl. Clifford Algebras

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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.

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Noncommutative Galois Extensions and Ternary Clifford Analysis

Cited by 9 publications

References 5 publications

Binary and ternary structures in physics III. Galois-type theory for binary and ternary structures

Binary and ternary structures in physics III. Galois-type theory for binary and ternary structures

Semi-commutative Galois Extension and Reduced Quantum Plane

Noncommutative Galois Extension and Graded q-Differential Algebra

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