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Lie Bialgebra Structures for Centrally Extended Two- Dimensional Galilei Algebra and their Lie-Poisson Counterparts

All bialgebra structures for centrally extended Galilei algebra are classified. The corresponding Lie-Poisson structures on centrally extended Galilei group are found.


Introduction
Much interest has been attracted in last years to the problem of deformations of space-time symmetry groups [1], [2], [3], [4], [5], [6], [7]. In particular, in the recent paper [8] all inequivalent bialgebra structures on two-dimensional Galilei algebra were classified and the corresponding Lie-Poisson structures on the group were found.
From the physical point of view what is really interesting is the central extension of the Galilei algebra. This is because only the genuine projective representations of the Galilei group are relevant in nonrelativistic quantum theory [9]. In the present paper we classify all nonequivalent bialgebra Lie-Poisson structures for centrally extended two-dimensional Galilei algebra/group. In the two-dimensional case there exists two-parameter family of central extensions, the parameters being the mass of the particle and the constant force acting on it. We restrict ourselves to the case of free particles, i.e. only the mass parameter is kept nonvanishing.
The content of the paper is as follows. First, we find the general form of 1cocycle on centrally extended twodimesional Galilei algebra. Then the action of most general automorphism transformation on such 1-cocycle is considered and its orbits are classified which allows to find all nonequivalent bialgebra structures. The corresponding Lie-Poisson structures on Galilei group are then found. The whole procedure follows quite closely the one presented in Ref. [10] for E(2) group and in Ref. [8] for two-dimensional Galilei group. As a result we find 26 nonequivalent bialgebra structures (some of them being still one parameter families), 8 of them being the coboundary ones.
2 Two-dimensional Galilei group and algebra with central extension and their automorphisms The two-dimensional Galilei group is a Lie group of transformations of the space-time with one space dimensions. An arbitrary group element g is of the form g = (τ, v, a); here τ is time translation, a respectively v are space translation respectively Galilean boost. The multiplication law reads: The resulting Lie algebra takes the form: The central extension is obtained by replacing the second commutation rule by Therefore, we arrive finally at the following algebra Let us define the centrally extended Galilei group by the following global expotential parametrization of group elements Let us write g = (m, τ, v, a) Then we have the following multiplication law Lie algebra with central extension can be realized in terms of rightinvariant fields to be calculated according to the standard rules from the composition law (9) Let us now describe all automorphisms of the algebra (6). The group of automorphisms consists of the following transformations where : and, obviously, α 1 = 0, γ 3 = 0.

The Bialgebra structures on two-dimensional centrally extended Galilei algebra
Our aim here is to give a complete classification of Lie bialgebra structures for the algebra (6) up to automorphisms. Let us remind the definition of bialgebra. It is a pair (L, δ), where L is a Lie algebra while δ is a skewsymmetric cocommutator δ : We can find all bialgebra structures on our algebra. The general form of δ obeying (i) is a, b, c, d, e, f, g, h and j being arbitrary real parameters. From the condition (ii) we obtain: Eqs. (13) and (14) define all bialgebra structures on two-dimensional Galilei algebra (6). However, we are interested in classification of nonequivalent bialgebra structures. To this end we find the transformation rules for the parameters under the automorphisms (11). They read We are now in position to classify all orbits of automorphism group in the space of bialgebra structures. A simple but long and painful analysis leads to the complete list of Lie bialgebra structures summarized in Table 1. We have checked explictly that all the above bialgebra structures are consistent and inequivalent. It remains to find coboundary structures (listed also in Table1).

Conclusions
We have classified all inequivalent bialgebra structures on the centrally extended two-dimensional Galilei algebra and found the corresponding Lie-Poisson structures on the group. The resulting classification appears to be quite rich and contains 26 inequivalent cases, eigth of them being the coboundary ones. This is in contrast with semisimple case as well as the case of fourdimensional Poincare group where there are only coboundary structures.

Acknowledgment
The Author acknowledges Prof. P. Kosiński for a careful reading of the manuscript and many helpful suggestion. Special thanks are also due to Prof. S. Giller, Dr. C. Gonera, Prof. P. Maślanka and MSc. E. Kowalczyk for valuable discussion.