Geometrical Tools for Quantum Euclidean Spaces

Abstract

We apply one of the formalisms of noncommutative geometry to R N q , the quantum space covariant under the quantum group SO q (N ). Over R N q there are two SO q (N )-covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of R 3 q . As in the case N = 3, one has to slightly enlarge the algebra R N q ; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N = 3 there is a conformal ambiguity in the natural metrics on the differential calculi over R N q . While in our previous article the frame was found 'by hand', here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of R N q with U q so(N ) into R N q , an interesting result in itself.

A dynamical 2-dimensional fuzzy space

“…In particular, since 6) we conclude that the frame necessarily commutes with all the elements of the algebra A . The 1-form θ defined as θ = −p a θ a can be considered as an analog of the Dirac operator in ordinary geometry.…”

Modified braid equations for SOq(3) and noncommutative spaces

“…To render the geometry more transparent one should construct the "frame basis" in terms of operators commuting with the algebra. ( See [5], [6] and sources cited there.) Thus equipped, one can study possible attractive consequences concerning the metric of implementingR c as in (3.53).…”

“…But for completeness and convenience they are presented below explicitly. where the matrix P ( permuting the rows (2, 4), (3,7), (6,8) ) leavesP (0) invariant. In Hieterinta's classification [8] of 4 × 4 R matrices appear examples without free parameters.…”

Realization of the N (odd)-Dimensional Quantum Euclidean Space by Differential Operators

“…When N is odd the geometrical structure is similar to R 3 q case. When, on the other hand, N is even, one should add an additional component to the algebra of R N q [6]. Also in Ref.…”