Universal Enveloping Algebra and Differential Calculi on Orthogonal q-groups

Abstract

We review the construction of the multiparametric quantum group $IS{O}_{q,r}(N)$ as a projection from $S{O}_{q,r}(N+2)$ and show that it is a bicovariant bimodule over $S{O}_{q,r}(N)$. The universal enveloping algebra $U_{q,r}(iso(N))$, characterized as the Hopf algebra of regular functionals on $IS{O}_{q,r}(N)$, 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 $U_{q,r}(iso(N))\leftrightarrow IS{O}_{q,r}(N)$. We analyze the subspaces of $U_{q,r}(iso(N))$ that define bicovariant differential calculi on $IS{O}_{q,r}(N)$.

LaTeX, 25 pages

References in corpus (10)

• The quantum structure of spacetime at the Planck scale and quantum fields
• Differential Calculus on Iso_q (N), Quantum Poincare' Algebra and q - Gravity
• Bicovariant Quantum Algebras and Quantum Lie Algebras
• Vector Fields on Quantum Groups
• The Lagrangian of q-Poincare' Gravity
• Bicovariant Calculus on Twisted ISO(N), Quantum Poincare' Group and Quantum Minkowski Space
• R-Matrix Formulation of the Quantum Inhomogeneous Groups Iso_qr(N) and Isp_qr(N)
• Inhomogeneous quantum groups IGL_{q,r}(N): Universal enveloping algebra and differential calculus
• Deformations of Multiparameter Quantum gl(N)
• Differential calculus on quantum groups: Constructive procedure

Cited by in corpus (3)

• Noncommutative instantons from twisted conformal symmetries
• Real forms of quantum orthogonal groups, q-Lorentz groups in any dimension
• Noncommutative Bundles and Instantons in Tehran