Universal enveloping algebra and differential calculi on inhomogeneous orthogonal q-groups

Abstract

We review the construction of the multiparametric quantum group ISO q,r (N ) as a projection from SO q,r (N + 2) and show that it is a bicovariant bimodule over SO q,r (N ). The universal enveloping algebra U q,r (iso(N )), characterized as the Hopf algebra of regular functionals on ISO q,r (N ), is found as a Hopf subalgebra of U q,r (so(N + 2)) and is shown to be a bicovariant bimodule over U q,r (so (N )).An R-matrix formulation of U q,r (iso(N )) is given and we prove the pairing U q,r (iso(N )) ↔ ISO q,r (N ). We analyze the subspaces of U q,r (iso(N )) that define bicovariant differential calculi on ISO q,r (N ). Subj. ind. class. 17B37 81R50 16W30.q-alg/9705023

On the geometry of the quantum poincaré group

“…One can also define a q-antisymmetric epsilon tensor and a quantum determinant det q,r T [7,12]; to obtain the special orthogonal quantum group SO q,r (N) we impose also the relation det q,r T = I.…”

“…Note 3 : U q,r (so(N + 2)) is completely characterized by the relations (4.4), (4.5), (4.6), (4.7), and L −A A = F (L +A A ). The functional dependence F of the L −A A in terms of the L +A A is studied in [12]; in the uniparametric case (q AB → r) it becomes the simple expression:…”

“…be the subalgebra of U q,r (so(N + 2)) generated by Since ∆(IU) ⊆ IU ⊗ IU and κ ′ (IU) ⊆ IU (as can be immediately seen at the generators level) we have that IU is a Hopf subalgebra of U q,r (so(N + 2)). Moreover one can also give the following R-marix formulation [12]:…”

BICOVARIANT CALCULUS ON TWISTED ISO(N), QUANTUM POINCARÉ GROUP AND QUANTUM MINKOWSKI SPACE

“…terms annihilating H where we have indicated the terms that do and do not annihilate the Hopf ideal H. We see that only the functionals χ • b , χ • • and χ • • do annihilate H, and therefore belong to U(IS q,r (N)). The resulting bicovariant differential calculus [18] contains dilatations and translations, but does not contain the tangent vectors of S q,r (N), i.e. the functionals χ a b .…”

“…Finally, the bicovariant calculus that includes the dilatation generator χ • • is discussed in ref. [18]. 8 The multiparametric orthogonal quantum plane as a q-coset space…”