Inhomogeneous quantum groups and their quantized universal enveloping algebras

Lie-algebraic structure from inhomogeneous Hopf algebras

“…Remark (2,1): Equations ͑25͒ and ͑26͒ imply for ϭ0, RϭR ϭR Ϫ1 , and the R-matrix is subject to a strong constraint R 2 ϭI I used in Ref. 8.…”

“…where the left adjoint coaction action Ad:B→B B is defined as Ad(a)ϭa (1) S(a (3) ) a (2) , and satisfies…”

“…Here a (1) , a (2) , and a (3) are elements of B such that a (1) a (2) a (3) ϭ(id ⌬)⌬(a). Theorem "3,1…:…”

“…[1][2][3][4][5][6] Among them are those which are based on the q-deformed Lie algebra of the Lorentz group augmented consistently by translation generators as q-deformed four-vector derivatives in Minkowski space. 1,2 Other approaches are based on the contraction of q-deformed anti-de Sitter algebra U͑O͑3,2͒͒ 4,5 or on the projective method where a bicovariant differential calculus on inhomogeneous quantum groups such as IGL q,r (N) or ISO q,r (N) is obtained as a quotient of those of GL q,r (Nϩ1) or SO q,r (Nϩ1) with respect to a suitable Hopf ideal. 6 In this paper, one presents a quite different approach based on a bicovariant differential calculus on an inhomogeneous Hopf algebra studied in our previous paper.…”

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

“…In complete analogy with the ISO q,r (N) case we also prove that U q,r (iso(N)) is a bicovariant bimodule over U q,r (so(N)) and give a basis of right invariant elements that freely generate U q,r (iso(N)). The universal enveloping algebras of the inhomogeneous quantum groups IGL q,r (N), first studied with a different approach in [7], can be derived in a similar way.…”