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

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

doi:10.1142/S0129055X00000095

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

Robert Coquereaux
(
- Marseille, CPT
)
,
A.O. Garcia
(
- Centro Atomico Bariloche and
- Cuyo U.
)
,
R. Trinchero
(
- Centro Atomico Bariloche and
- Cuyo U.
)

2000

59 pages

Published in:

Rev.Math.Phys. 12 (2000) 227-285

DOI:

10.1142/S0129055X00000095

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Abstract: (WSP)

We consider the algebra of N×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, {\mathbb C}))

, q being an Nth 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.

quantum group: SL(2,C)
algebra: differential
quantum group: representation
geometry: noncommutative
operator: differential
gauge field theory: action
duality
differential forms
space-time
group theory: Lorentz

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