Differential calculus on compact matrix pseudogroups (quantum groups).

*(English)*Zbl 0751.58042This is a sequel to an earlier paper by the author [ibid. 111, No. 4, 613-665 (1987; Zbl 0627.58034)]. There, he introduced and developed the finite-dimensional representation theory of a particular generalization of the concept of a compact Lie group, which has the desirable property of admitting nontrivial deformations. (A more abstract schema has been proposed to the same end by V. G. Drinfel’d [Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 798-820 (1987; Zbl 0667.16003)]. Here the author discusses the foundations of (noncommutative) differential geometry for these objects.

A “differential calculus” over an algebra \(\mathcal A\) (generalizing the algebra of smooth functions) consists, in principle, of an \(\mathcal A\)- bimodule \(\Gamma\) (generalizing the module of 1-forms), and a derivation \(d:{\mathcal A}\to \Gamma\) (corresponding to the exterior derivative). For the author’s pseudogroups, the algebra \(\mathcal A\) was part of the definition; however, the choice of differential calculus over \(\mathcal A\) is not canonical, and he remarks that nonstandard calculi may be constructed even on classical compact groups. Given a differential calculus over \(\mathcal A\) which satisfies appropriate covariance conditions, it is possible — with effort and ingenuity — to construct suitable, not always transparent, analogues of tensors and forms, of the exterior derivative in all degrees, of left-invariant vector fields and their brackets, of the Maurer-Cartan equation, and of the Jacobi identity. The author comments that for these purposes the “compactness” of his pseudogroups, meaning certain algebraic identities of orthogonality in the definition, does not appear essential.

A “differential calculus” over an algebra \(\mathcal A\) (generalizing the algebra of smooth functions) consists, in principle, of an \(\mathcal A\)- bimodule \(\Gamma\) (generalizing the module of 1-forms), and a derivation \(d:{\mathcal A}\to \Gamma\) (corresponding to the exterior derivative). For the author’s pseudogroups, the algebra \(\mathcal A\) was part of the definition; however, the choice of differential calculus over \(\mathcal A\) is not canonical, and he remarks that nonstandard calculi may be constructed even on classical compact groups. Given a differential calculus over \(\mathcal A\) which satisfies appropriate covariance conditions, it is possible — with effort and ingenuity — to construct suitable, not always transparent, analogues of tensors and forms, of the exterior derivative in all degrees, of left-invariant vector fields and their brackets, of the Maurer-Cartan equation, and of the Jacobi identity. The author comments that for these purposes the “compactness” of his pseudogroups, meaning certain algebraic identities of orthogonality in the definition, does not appear essential.

##### MSC:

58H05 | Pseudogroups and differentiable groupoids |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

58A15 | Exterior differential systems (Cartan theory) |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

58B25 | Group structures and generalizations on infinite-dimensional manifolds |

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\textit{S. L. Woronowicz}, Commun. Math. Phys. 122, No. 1, 125--170 (1989; Zbl 0751.58042)

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##### References:

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[2] | Connes, A.: Non-commutative differential geometry. Institut des Hautes Etudes Scientifiques. Extrait des Publications MathÃ©matiques No. 62 (1986) · Zbl 0592.46056 |

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[7] | Woronowicz, S.L.: Pseudospaces, pseudogroups and Pontriagin duality. Proceedings of the International Conference on Mathematics and Physics, Lausanne 1979. Lecture Notes in Physics, Vol. 116. Berlin, Heidelberg, New York: Springer 1980 · Zbl 0513.46046 |

[8] | Woronowicz, S.L.: TwistedSU(2) group. An example of a noncommutative differential calculus. Publ. Res. Inst. Math. Sci., Kyoto University23, 117-181 (1987) · Zbl 0676.46050 · doi:10.2977/prims/1195176848 |

[9] | Woronowicz, S.L.: Group structure on noncommutative spaces. Fields and Geometry 1986. Proceedings of the XXIInd Winter School and Workshop of Theoretical Physics, Karpacz, Poland, pp. 478-496. Singapore: World Scientific |

[10] | Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys.111, 613-665 (1987) · Zbl 0627.58034 · doi:10.1007/BF01219077 |

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