Quantum Galilei Research Articles

On simple modules and automorphisms of the quantum Galilei group

This paper is devoted to ring theoretic properties and representations of the quantum Galilei group F q ( G ) and its Hopf dual U q ( g ) . We classify the primitive ideals and simple modules of F q ( G ) and U q ( g ) over an algebraically closed field of arbitrary characteristic. We determine their groups of automorphisms over a field of characteristic zero.

Induced Representation of the (1 + 1)-Quantum Extended Galilei Algebra on the Bargmann Space-Time

We present a q-deformed local representations of the (1 + 1) extended Galilei group acting on the space of wave-functions defined in the Bargmann space-time. This paper is a development of the work made by F. Bonechi et al on the induced representations of (1 + 1) quantum Galilei.

Representations of quantum bicrossproduct algebras

We present a method to construct induced representations of quantum algebras having the structure of bicrossproduct. We apply this procedure to some quantum kinematical algebras in (1+1)--dimensions with this kind of structure: null-plane quantum Poincare algebra, non-standard quantum Galilei algebra and quantum kappa Galilei algebra.

Induced representations of quantum (1+1) Galilei algebras

We develop a systematic method to construct induced representations of quantum algebras. The procedure makes use of two Hopf algebras with a nondegenerate pairing and a pair of dual bases for them. We apply the method on three different quantum deformations of the Galilei algebra in (1+1) dimensions. We obtain several families of induced representations including some results already known.

Two-dimensional centrally extended quantum Galilei groups and their algebras

All deformations of two dimensional centrally extended Galilei group are classified. The corresponding quantum Lie algebras are found.

Unitarity of induced representations from coisotropic quantum subgroups

We study unitarity of the induced representations from coisotropic quantum subgroups which were introduced by Ciccoli in math.QA/9804138. We define a real structure on coisotropic subgroups which determines an involution on the homogeneous space. We give general invariance properties for functionals on the homogeneous space which are sufficient to build a unitary representation starting from the induced one. We present the case of the one-dimensional quantum Galilei group where we have to use in all generality our definition of quasi-invariant functional.

Induced representations of the one dimensional quantum Galilei group

We study the representations of the quantum Galilei group by a suitable generalization of the Kirillov method on spaces of noncommutative functions. On these spaces, we determine a quasi-invariant measure with respect to the action of the quantum group by which we discuss unitary and irreducible representations. The latter are equivalent to representations on l2, i.e. on the space of square summable functions on a one-dimensional lattice.

THE QUANTUM GALILEI GROUP

The quantum Galilei group Gκ is defined. The bicross-product structure of Gκ and the corresponding Lie algebra is revealed. The projective representations for two-dimensional quantum Galilei group are constructed.

Quantum Galilei group as symmetry of magnons.

Inhomogeneous quantum groups are shown to be an effective algebraic tool in the study of integrable systems and to provide solutions equivalent to the Bethe ansatz. The method is illustrated on the 1D Heisenberg ferromagnet whose symmetry is shown to be the quantum Galilei group Gamma_q(1) here introduced. Both the single magnon and the s=1/2 bound states of n-magnons are completely described by the algebra.

Heisenberg XXZ model and quantum Galilei group

The 1D Heisenberg spin chain with anisotropy of the XXZ type is analyzed in terms of the symmetry given by the quantum Galilei group Gamma_q(1). We show that the magnon excitations and the s=1/2, n-magnon bound states are determined by the algebra. Thus the Gamma_q(1) symmetry provides a description that naturally induces the Bethe Ansatz. The recurrence relations determined by Gamma_q(1) permit to express the energy of the n-magnon bound states in a closed form in terms of Tchebischeff polynomials.