Abstract
Quantum spaces with su(2) noncommutativity can be modelled by using a family of SO(3)-equivariant differential *-representations. The quantization maps are determined from the combination of the Wigner theorem for SU(2) with the polar decomposition of the quantized plane waves. A tracial star-product, equivalent to the Kontsevich product for the Poisson manifold dual to su(2) is obtained from a subfamily of differential *-representations. Noncommutative (scalar) field theories free from UV/IR mixing and whose commutative limit coincides with the usual ϕ4 theory on ℛ3 are presented. A generalization of the construction to semi-simple possibly non simply connected Lie groups based on their central extensions by suitable abelian Lie groups is discussed.
Highlights
In the recent years, deformations of R3 for which the algebra of coordinates forms a su(2) Lie algebra have received some interest
Quantum spaces with su(2) noncommutativity can be modelled by using a family of SO(3)-equivariant differential ∗-representations
A generalization of the construction to semi-simple possibly non connected Lie groups based on their central extensions by suitable abelian Lie groups is discussed
Summary
Deformations of R3 for which the algebra of coordinates forms a su(2) Lie algebra have received some interest. Speaking, the star-product which will be determined in a while is essentially unique up to unitary equivalence This result can be understood [16] within the more general framework of central extensions of Lie groups which in addition offers a convenient way for generalizations of the present work. The above conclusion extends to the central extension of any semi-simple and connected Lie group by R/D so that the extension of the present construction to spaces with noncommutativity based on the corresponding Lie algebra should produce uniqueness of the star-product (up to equivalence). When G is semi-simple but not necessarily connected, its inequivalent central extensions where HCˆ by R/D denotes are the classified (up to additional technical requirements) by Cech cohomology. One expect that the extension of the present construction of SL(2, R) gives rise to inequivalent classes of star-products
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