The Hidden Geometry of the Quantum Euclidean Space
We briefly describe how to introduce the basic notions of noncommutative differential geometry on the 3-dim quantum space covariant under the quantum group of rotations $SO_q(3)$.
math.QA↗
arXiv subjects
John Madore
Publications and source records attributed to John Madore.
Leibniz Rules and Reality Conditions
An analysis is made of reality conditions within the context of noncommutative geometry. We show that if a covariant derivative satisfies a given left Leibniz rule then a right Leibniz rule is equivalent to the reality condition. We show also that the matrix which determines the reality condition must satisfy the Yang-Baxter condition if the extension of the covariant derivative to tensor products is to satisfy the reality condition. This is equivalent to the braid condition for the matrix which determines the right Leibniz rule.
math.QA↗
Shadow of noncommutativity
We analyse the structure of the $κ=0$ limit of a family of algebras $A_κ$ describing noncommutative versions of space-time, with $κ$ a parameter of noncommutativity. Assuming the Poincaré covariance of the $κ=0$ limit, we show that, besides the algebra of functions on Minkowski space, $A_0$ must contain a nontrivial extra factor $A^I_0$ which is Lorentz covariant and which does not commute with the functions whenever it is not commutative. We give a general description of the possibilities and analyse some representative examples.
q-alg↗
Linear connections on the quantum plane
A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric.
hep-th↗