Coherent state in quantum physics, by jeanpierre gazeau. The racahwigner algebra also has similarities, but there is additional structure due to the deformation parameter q. The qdeformation of w 2, 2 lie algebra is well defined based on a realization of this lie algebra by using the famous bosonic and fermionic oscillators in physics. The quantum groups are mathematical structures, lately developed as deformation of lie algebras. Operators as a realization of the quantum superalgebra bq0. Finally, the 1dimensional central extension of the qdeformed w 2, 2 lie algebra is studied, which turns out to be coincided with the conventional w 2, 2 lie algebra in the q 1 limit. Arnlind, et al hepth0602290 abhhs can be rewriten as a new algebra which contains qdeformed commutators. Infinite dimensional lie algebras have played a crucial role in physics. The point now is that the lij are the components of a.
In this paper we will explicitly construct the representation of the qdeformed algebra for generic. A combination of these two techniques will lead to a new and natural q deformation of matrix. The qdeformed algebras have a similar representation theory to lie algebras because of their common structure rosso, 1988. The rmatrices are special elements appearing in qdeformed algebras and form the basis of many of the applications of qdeformed algebras. Now, we are able to introduce the notion of a quantum lie algebra as it was given in 3739. The lie subalgebras for qdeformed heisenberg algebra h q when q is not a root of unity has been considered in 3 where especially the lie subalgebra generated by generators a and b has been. R poisson algebra are particular cases of poisson lie book groups. We show that the same can be done for their qdeformed. Based on the structure of the qdeformed virasorowitt algebra, we derive a nontrivial qdeformed virasorowitt nalgebra which is nothing but a. The full set of poisson lie pl structures on the 3d book group are presented. We first derive a calculus entirely based on the algebra and then formulate laws of physics based on this calculus.
We show that the two different q deformed versions of the sl2. Plancks constant and calling it a formal quantisation qdeformation actually. On qdeformed infinitedimensional nalgebra sciencedirect. Several clifford algebras that are covariant under the action of a lie algebra can be deformed in a way consistent with the deformation of into a quantum group or into a triangular hopf algebra, i. Some introductory notes on quantum groups, quantum algebras. A quantum lie algebra can be regarded as a subspace of a qdeformed enveloping algebra uqg being invariant under the adjoint action of uqg. Part of the research reports in physics book series resreports.
This was the subject of a paper published in 1992 lienert and butler, 1992a. We recall that any associative algebra such as hq has a natural lie. Pdf classification theorem on irreducible representations of the q. The qdeformed creation and annihilation operators as a. Fuzzy torus and qdeformed lie algebra internet archive.
The qdeformed heisenberg algebra is a mathematical abstraction of a. Algebraic curves for commuting elements in the qdeformed. Pdf a qdeformed heisenberg algebra as a normed space. The string solution exists for generic values of q qn \neq 1. Highest voted liealgebra questions page 11 physics. It will be shown that the defining relations for fuzzy torus and deformed squashed sphere proposed by j.
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