\(q\)-Lorentz group and braided coaddition on \(q\)-Minkowski space
Recent interest in \(q\)-deformed Lorentz groups and Minkowski spaces is motivated by the idea of regularizing singularities in quantum field theories by making space-time slightly noncommutative. In the \(q\)-regularization approach noncommutativity is controlled by a parameter \(q\), symmetries are preserved as \(q\)-symmetries, using \(q\)-deformed Lie algebras. The author proposes a new version of \(q\)-Minkowski space, which has both a coaddition law and an \(SL_q (2, \mathbb{C})\)-spinor decomposition. It was noted by S. Majid that it is possible to equip a quantum plane with a coaddition law which turns it into a braided Hopf algebra living in the category of comodules over the corresponding quantum group. There is a general construction called transmutation by means of which one can convert a quantum group into a braided Hopf algebra in the corresponding (co)module category. In particular, braided \(2\times 2\) matrices as the transmutation of the well-known \(2\times 2\) quantum matrices are a natural candidate for the algebra of \(q\)-Minkowski space. Braided \(2\times 2\) matrices have the same coalgebra structure as quantum matrices, which corresponds to the product of ordinary matrices. But there is no \(q\)-deformed analogue of the additive group structure of Minkowski space. In the present paper a new braided coalgebra structure on the braided Hermitian \(2\times 2\) matrices is introduced which plays the rôle of braided coaddition. The new braided \(q\)-Minkowski space is a braided Hopf algebra in the category of comodules over the function algebra on the \(q\)-Lorentz group.
scientific article; zbMATH DE number 778960
scientific article