Enrico Celeghini

Three‐dimensional quantum groups from contractions of SU(2)q

Contractions of Lie algebras and of their representations are generalized to define new quantum groups. An explicit and complete exposition is made for the one‐dimensional Heisenberg H(1)q and the two‐dimensional Euclidean quantum group E(2)q obtained by contracting SU(2)q.

The quantum Heisenberg group H(1)q

The structure of the quantum Heisenberg group is studied in the two different frameworks of the Lie algebra deformations and of the quantum matrix pseudogroups. The R‐matrix connecting the two approaches, together with its classical limit r, are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown.

The three‐dimensional Euclidean quantum group E(3)q and its R‐matrix

A contraction procedure starting from SO(4)q is used to determine the quantum analog E(3)q of the three‐dimensional Euclidean group and the structure of its representations. A detailed analysis of the contraction of the R‐matrix is then performed and its explicit expression has been found. The classical limit of R is shown to produce an integrable dynamical system. By means of the R‐matrix the pseudogroup of the noncommutative representative functions is considered. It will finally be shown that a further contraction made on E(3)q produces the two‐dimensional Galilei quantum group and this, in turn, can be used to give a new realization of E(3)q and E(2,1)q.

QUANTUM GROUPS, COHERENT STATES, SQUEEZING AND LATTICE QUANTUM-MECHANICS

By resorting to the Fock--Bargmann representation, we incorporate the quantum Weyl--Heisenberg algebra,

THERMO FIELD DYNAMICS AND QUANTUM ALGEBRAS

q

Exponential mapping for non-semisimple quantum groups

-WH, into the theory of entire analytic functions. The

An R-matrix approach to the quantization of the Euclidean group E(2)

q

SO q (n+1,n−1) as a real form of SO q (2n,ℂ)

--WH algebra operators are realized in terms of finite difference operators in the

Quantum Galilei group as symmetry of magnons.

z

Quantum (1+1) extended Galilei algebras: from Lie bialgebras to quantum R-matrices and integrable systems

plane. In order to exhibit the relevance of our study, several applications to different cases of physical interest are discussed: squeezed states and the relation between coherent states and theta functions on one side, lattice quantum mechanics and Bloch functions on the other, are shown to find a deeper mathematical understanding in terms of