Semi-commutative Galois Extension and Reduced Quantum Plane
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 integ
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Abstract 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 q-differential 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 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 quantum plane
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Galois extension
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Reduced
V. Abramov (B) · Md. Raknuzzaman Institute of Mathematics, University of Tartu, Liivi 2–602, Tartu 50409, Estonia e-mail: [email protected] Md. Raknuzzaman e-mail: [email protected] © Springer International Publishing Switzerland 2016 S. Silvestrov and M. Ranˇci´c (eds.), Engineering Mathematics II, Springer Proceedings in Mathematics & Statistics 179, DOI 10.1007/978-3-319-42105-6_2
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V. Abramov and Md. Raknuzzaman
1 Introduction Let us briefly remind a definition of noncommutative Galois extension [12–15]. Suppose A˜ is an associative unital C-algebra, A ⊂ A˜ is its subalgebra, and there is an element τ ∈ A˜ which satisfies τ ∈ / A , τ N = 1, where N ≥ 2 is an integer and 1 is the identity element of A˜. A noncommutative Galois extension of A by means of τ is the smallest subalgebra A [τ ] ⊂ A˜ such that A ⊂ A [τ ], and τ ∈ A [τ ]. It should be pointed out that a concept of noncommutative Galois extension can be applied not only to associative unital algebra with a binary multiplication law but as well as to the algebra with a ternary multiplication law, for instant to a ternary analog of Grassmann and Clifford algebra [6, 14, 15], and this approach can be used in particle physics to construct an elegant algebraic model for quarks. A graded q-differential
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