1998/05/27 by Paolo Aschieri, Aschieri, Paolo
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #hep-th #math.QA
paper · pdf · doi:10.48550/arxiv.math/9805119
Ph.D. thesis, 181 pages. Added ref. [72] and [73]. Added ref. in [66] and [76]
openalex publication_date 1998/05/27 · arxiv created 1998/06/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We first give a pedagogical introduction to the differential calculus on q-groups and analize the relation between differential calculus and q-Lie algebra. Equivalent definitions of bicovariant differential calculus are studied and their geometrical interpretation is explained. Vectorfields, contraction operator and Lie derivative are defined and their properties discussed. After a review of the geometry of the (multiparametric) linear q-group GLq,r(N) we construct the inhomogeneous q-group IGLq,r(N) as a projection from GLq,r(N+1), i.e. as a quotient of GLq,r(N+1) with respect to a suitable Hopf algebra ideal. The semidirect product structure of IGLq,r(N) given by the GLq,r(N) q-subgroup times translations is easily analized. A bicovariant calculus on IGLq,r(N) is explicitly obtained as a projection from the one on GLq,r(N+1). The universal enveloping algebra of IGLq,r(N) and its R-matrix formulation are constructed along the same lines. We proceed similarly in the orthogonal and symplectic case. We find the inhomogeneous multiparametric q-groups of the Bn,Cn,Dn series via a projection from Bn+1, Cn+1,Dn+1. We give an R-matrix formulation and discuss real forms. We study their universal enveloping algebras and differential calculi. In particular we obtain the bicovariant calculus on a dilatation-free minimal deformation of the Poincaré group ISOq(3,1). The projection procedure is also used to construct differential calculi on multiparametric q-orthogonal planes in any dimension N. Real forms are studied and in particular we obtain a q-Minkowski space and its q-deformed phase-space with hermitian operators xa and pa.