Differential calculus and connections on a quantum plane at a cubic root of unity

A. O. Garcia; R. Coquereaux; R. Trinchero

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Differential calculus and connections on a quantum plane at a cubic root of unity

R. Coquereaux, A. O. Garcia, R. Trinchero

Mathematical Physics · Submitted 12 Jul 1998

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.

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arXiv IDmath-ph/9807012 (version 2)

Pith IDIOMC27BQ, Pith's permanent record of this paper and its review

CitationA. O. Garcia, R. Coquereaux, R. Trinchero. "Differential calculus and connections on a quantum plane at a cubic root of unity." arXiv:math-ph/9807012 (1998). https://pith.science/paper/IOMC27BQ

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.24 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.83 · compares with prior results 0.06 · mentions code or data availability 0.01.

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We consider the algebra of N x N matrices as a reduced quantum plane on which a finite-dimensional quantum group H acts. This quantum group is a quotient of U_q(sl(2,C)), q being an N-th root of unity. Most of the time we shall take N=3; in that case \dim(H) = 27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess-Zumino complex. The quantum group H also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of H. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider the algebra of N x N matrices as a reduced quantum plane on which a finite-dimensional quantum group H acts.	contribution 0.64	0.60	0.06	asserted 1.00
s1	This quantum group is a quotient of U_q(sl(2,C)), q being an N-th root of unity.	background 0.80	0.26	0.04	asserted 1.00
s2	Most of the time we shall take N=3; in that case \dim(H) = 27.	background 0.62	0.32	0.03	asserted 0.86
s3	We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess-Zumino complex.	contribution 0.64	0.69	0.06	asserted 0.99
s4	The quantum group H also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of H.	contribution 0.94	0.87	0.11	asserted 0.99
s5	We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra.	contribution 0.97	0.81	0.08	asserted 0.99
s6	By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.	contribution 0.57	0.76	0.26	hedged possibility 0.72

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 1.00.

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Original abstract

We consider the algebra of N x N matrices as a reduced quantum plane on which a finite-dimensional quantum group H acts. This quantum group is a quotient of U_q(sl(2,C)), q being an N-th root of unity. Most of the time we shall take N=3; in that case \dim(H) = 27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess-Zumino complex. The quantum group H also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of H. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.

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