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On free differentials on associative algebras. (English) Zbl 0833.58006
González, Santos (ed.), Non-associative algebra and its applications. 3rd international conference, Oviedo, Spain, July 12th-17th, 1993. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 303, 46-53 (1994).
The paper is one of the series of works of the same authors related with differentials on associative algebras and with the author’s concept of optimal calculi. A differential $$d: R\to {}_R M_R$$ (satisfying the usual undeformed Leibniz rule) is called free if the differential of any element $$v$$ has a unique representation of the form $$dv= dx^i\cdot v_i$$, where $$x^1,\dots, x^n$$ are generators of the unitary algebra $$R$$. Any free differential defines a commutation rule $$v dx^i= dx^k\cdot A(v)^i_k$$, where $$A: v\mapsto A(v)^i_k$$ is an algebra homomorphism $$R\to R_{n\times n}$$. It is shown that for a given commutation rule a free algebra generated by $$x^1,\dots, x^n$$ has a unique related free differential. The authors study ideals of free algebras consistent with this differential. A maximal such ideal is described in the homogeneous case and the corresponding factor-algebra is called optimal. A number of examples of optimal algebras for different commutation rules is considered. A classification theorem for calculi with a commutative optimal algebra is announced. Relations with results of Wess and Zumino about covariant differential calculus on the quantum hyperplane are discussed.
For the entire collection see [Zbl 0801.00022].

##### MSC:
 46L85 Noncommutative topology 46L87 Noncommutative differential geometry 16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 16W25 Derivations, actions of Lie algebras