2011
DOI: 10.1007/978-3-642-23568-9_24
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On Two-Generated Non-commutative Algebras Subject to the Affine Relation

Abstract: Abstract. We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + αx + βy + γ for q ∈ K * and α, β, γ ∈ K. We prove, that among such algebras there are precisely five isomorphism classes. The representatives of these classes, which are ubiquitous operator algebras, are called model algebras. We derive explicit multiplication formulas for y m · x n in terms of standard monomials x i y j for many algebras of the considered type. Such formulas are used in … Show more

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Cited by 4 publications
(2 citation statements)
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“…With regard to notation, [a + b] n stands for writing out the right hand side of the binomial formula in its commutative version. Example 6 ( [21]). Suppose that K is a field of characteristic zero.…”
Section: G-algebrasmentioning
confidence: 99%
“…With regard to notation, [a + b] n stands for writing out the right hand side of the binomial formula in its commutative version. Example 6 ( [21]). Suppose that K is a field of characteristic zero.…”
Section: G-algebrasmentioning
confidence: 99%
“…Moreover, for non-commutative rings, one can prove that it is always possible to find a finite left Gröbner basis for every ideal of a G-algebra. These bases are usually constructed using the generalized Buchberger's algorithm (see [37,38]).…”
Section: Pξ ¡1mentioning
confidence: 99%