Graded contractions and bicrossproduct structure of deformed inhomogeneous algebras

Abstract

A family of deformed Hopf algebras corresponding to the classical maximal isometry algebras of zero-curvature N -dimensional spaces (the inhomogeneous algebras iso(p, q), p + q = N, as well as some of their contractions) are shown to have a bicrossproduct structure. This is done for both the algebra and, in a low-dimensional example, for the (dual) group aspects of the deformation. * PACS numbers: 02.20, 02.20Sv, 02.40 † St. John's College Overseas Visiting Scholar. ‡ On sabbatical (J.A.) leave and on leave of absence (J.C.P.B

Central extensions of the quasi-orthogonal Lie algebras

“…In particular, a whole family of deformations of inhomogeneous Lie algebras [30], or working to first order, of the corresponding bialgebras [31], has been found. The semidirect structure of the 'classical' CK ω 1 = 0 inhomogeneous Lie algebras becomes [32] a bicrossproduct [33] structure for their CK deformed counterparts. Whether or not this extends to the deformations of other semidirect structures associated to the vanishing of any ω a requires further study.…”

Dynamical systems and quantum bicrossproduct algebras

“…In [9] the algebra U z (T N ) was considered as a noncommutative deformation of the Lie algebra of the group of translations of R N . However, here we can profit the commutativity of U z (T N ) for interpreting it as the algebra of functions over a group, in such a way that we have the following bicrossproduct decomposition…”

“…Indeed, it belongs to the center of the algebra U z (iso ω 2 ,ω 3 ,... ,ω N (N )) and is the Casimir C z given in [9], but now it appears in a natural way.…”

“…Among the elements of this family we find the Poincaré and the Galilei algebras in (N −1, 1) dimensions and the Euclidean algebra in N dimensions. The bicrossproduct structure that share these quantum algebras [9] allows to present a unified study of the properties mentioned above.…”

On the bicrossproduct structures for the family of algebras

“…In [21] it was shown that all these deformed algebras are endowed with a bicrossproduct structure that corresponds to the undeformed semidirect one (2.3) in which the abelian algebra is given by the single row with generators J 0i (see Fig. 2.1 for a = 1).…”