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Coordinate calculi on associative algebras. (English) Zbl 0879.16023
Lukierski, Jerzy (ed.) et al., Quantum groups: formalism and applications. Proceedings of the XXX Karpacz winter school of theoretical physics, Karpacz, Poland, February 14–26, 1994. Warszawa: PWN-Polish Scientific Publishers. 231-241 (1995).
The paper reviews some previous results due to the authors, partially jointly with Z. Oziewicz, and supplies them with comments and examples. The subject is the noncommutative first-order differential calculus \(d\colon R\to M\) over a field \(\mathbb{F}\) where \(R=\mathbb{F}\langle x^1,\dots,x^n\rangle/I_R\). One assumes that the \(R\)-bimodule \(M\) is freely generated from the right by \(dx^1,\dots,dx^n\) and speaks then about a coordinate differential calculus. Consequently the commutation relations have the form \(v dx^i=dx^k\cdot A(v)^i_k\) where \(A\colon R\to R_{n\times n}\) is a homomorphism. Then attention is paid to the inverse procedure: given an ideal \(I\neq\widehat R\) in the free algebra \(\widehat R=\mathbb{F}\langle x^1,\dots,x^n\rangle\) and a homomorphism \(A\colon\widehat R\to\widehat R_{n\times n}\) one solves the problem of consistency conditions between \(I\) and \(A\) in order to construct a coordinate differential calculus. Moreover, for a given \(A\) there exists a largest \(A\)-consistent ideal \(I(A)\subset\widehat R\) containing only polynomials with zero constant terms; \(\widehat R/I(A)\) is called an optimal algebra. The obtained results are further applied to quadratic algebras. The final section contains an announcement of a construction of a higher order calculus (graded differential algebra).
For the entire collection see [Zbl 0856.00018].

16W25 Derivations, actions of Lie algebras
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W50 Graded rings and modules (associative rings and algebras)
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