Quantum differential operators on the quantum plane

Remarks on Quantum Transmutation

“…Next comes questions of uniqueness and boundary conditions of Cauchy type and surprisingly little seems to have been done in creating a general theory of existence and uniqueness for q-difference equations, or more generally creating a genuinely algebraic theory of noncommutative difference equations (see however [26,37] for some starts and cf. also [30,49]). Before looking at these we want to suggest another approach based on [61] (cf.…”

Noetherian Algebras of Quantum Differential Operators

“…Let Γ = Z 2 with standard basis {e 1 , e 2 }. Here we summarise the results from [9] on the quantum differential operators for the bicharacter β : Γ × Γ → k * be given by β(e i , e j ) =…”

“…The definition of these quantum differential operators requires the base k-algebra R to be Γ-graded for some abelian group Γ and depends on the choice of a bicharacter β : Γ × Γ → k * . The first author and T. C. McCune have contributed to this theory through their identification of the quantum differential operators for appropriate bicharacters, of the polynomial algebra k[x] in one variable [8] and the coordinate algebra of the quantum plane [9]. A second approach is to study algebras that are either generated by particular operators of an intuitively quantum differential nature or are generalizations of familiar algebras of differential operators.…”

“…In Section 4, we consider quantum differential operators over the coordinate algebras of quantum n-space and quantum tori. We recall the results on the quantum plane from [9] and then study the case of the coordinate algebra of quantum n-space, concentrating on the case n ≥ 3 where there are distinct differences to the cases n = 1, 2. The short Section 5 deals with the quantum experior algebra for which the algebra of quantum differential operators is the entire algebra of linear maps on the quantum exterior algebra.…”

Remarks on Quantum Differential Operators

“…It was suggested in [9] for example that such treatments could be extended to give general results about existence and uniqueness of solutions of q-partial differential equations (qPDE) and we will address this question in more detail below. Other approaches to a general treatment of qPDE appear in [48,53] for example. To see in more detail how the Ogievetsky-Zumino technique applies in the q-plane consider REMARK 3.2.…”