Deformed and extended Galilei group Hopf algebras

Abstract

Theκ-deformed extended Galilei Hopf group algebra, Funκ(G (m) ), is introduced. It provides an explicit example of a deformed group with cocycle bicrossproduct structure, and is shown to be the contraction limit of a pseudoextension of the κ-Poincaré group algebra. The possibility of obtaining another deformed extended Galilei groups is discussed, including one obtained from a non-standard Poincaré deformation.

Expansions of Algebras and Superalgebras and Some Applications

“…The contraction process has also been considered for 'quantum' algebras (see e.g., [29]) and used, in particular, to obtain the κ-Poincaré [30] and κ-Galilei algebras [31].…”

Representations of quantum bicrossproduct algebras

“…The non-standard quantum Galilei algebra U z (g (1, 1)) is isomorphic to the quantum Heisenberg algebra H q (1) [29,30] and to the deformed Heisenberg-Weyl algebra U ρ (HW ) [31]. It can be obtained by contraction [31] of a non-standard deformation of the Poincaré algebra [27] (the null-plane quantum Poincaré).…”

“…It can be obtained by contraction [31] of a non-standard deformation of the Poincaré algebra [27] (the null-plane quantum Poincaré).…”

On the bicrossproduct structures for the family of algebras

“…Then, the algebra H dual to H is given by ∆R ij = R ik ⊗ R kj , ∆x i = 1 ⊗ x i + x k ⊗ R ki , ∆y i = 1 ⊗ y i + y k ⊗ R As we may see the bicrossproduct structure (with cocycle in this case) allows us to recover Fun λ (HW) in an easy way from the enveloping (dual) algebra U λ (HW). 6 The antisymmetric form of the cocycle is a matter of convention; different forms of the cocycle are related by a coboundary change (see [24] for an explicit example). 7 This algebra is a true rotation algebra for ω i = 1 i = 1, .…”