R-Matrix Formulation of the Quantum Inhomogeneous Groups Iso_qr(N) and Isp_qr(N)

TL;DR

This paper develops an R-matrix based formulation for inhomogeneous quantum groups of types B, C, D, including the quantum Poincaré group, by establishing their algebraic relations and Hopf structures through projections from higher-rank groups.

Contribution

It introduces a new R-matrix approach to define inhomogeneous quantum groups of types B, C, D, and constructs their Hopf algebra structures via projections from larger quantum groups.

Findings

Derived quantum commutation relations for inhomogeneous q-groups.

Constructed consistent Hopf structures for these groups.

Identified quantum Poincaré group as a special case.

Abstract

The quantum commutations $RTT=TTR$ and the orthogonal (symplectic) conditions for the inhomogeneous multiparametric $q$-groups of the $B_n,C_n,D_n$ type are found in terms of the $R$-matrix of $B_{n+1},C_{n+1},D_{n+1}$. A consistent Hopf structure on these inhomogeneous $q$-groups is constructed by means of a projection from $B_{n+1},C_{n+1},D_{n+1}$. Real forms are discussed: in particular we obtain the $q$-groups $ISO_{q,r}(n+1,n-1)$, including the quantum Poincar\'e group.