Q deformed lie algebra book

Part of the research reports in physics book series resreports. 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. In section 1 we consider the simplest version of a q deformed heisenberg algebra as an example of a noncommutative structure. Infinite dimensional lie algebras have played a crucial role in physics. In this paper we extend the eliminant construction of burchnall and chaundy for commuting differential operators in the heisenberg algebra to the qde. Examples of representations of quantum groups mathoverflow. Some introductory notes on quantum groups, quantum algebras.

It will be shown that the defining relations for fuzzy torus and deformed squashed sphere proposed by j. The qdeformed heisenberg algebra is a mathematical abstraction of a. Arnlind, et al hepth0602290 abhhs can be rewriten as a new algebra which contains qdeformed commutators. The rmatrices are special elements appearing in qdeformed algebras and form the basis of many of the applications of qdeformed algebras. A combination of these two techniques will lead to a new and natural qdeformation of matrix. 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. 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. Operators as a realization of the quantum superalgebra bq0. Algebraic curves for commuting elements in the qdeformed. The racahwigner algebra also has similarities, but there is additional structure due to the deformation parameter q. 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.

A combination of these two techniques will lead to a new and natural q deformation of matrix. On qdeformed infinitedimensional nalgebra sciencedirect. R poisson algebra are particular cases of poisson lie book groups. Pdf classification theorem on irreducible representations of the q. The string solution exists for generic values of q qn \neq 1. Pdf a qdeformed heisenberg algebra as a normed space. This was the subject of a paper published in 1992 lienert and butler, 1992a. The quantum groups are mathematical structures, lately developed as deformation of lie algebras.

We recall that any associative algebra such as hq has a natural lie. Coherent state in quantum physics, by jeanpierre gazeau. The full set of poisson lie pl structures on the 3d book group are presented. We show that the two different q deformed versions of the sl2. Highest voted liealgebra questions page 11 physics. Based on the structure of the qdeformed virasorowitt algebra, we derive a nontrivial qdeformed virasorowitt nalgebra which is nothing but a. Fuzzy torus and qdeformed lie algebra internet archive. The qdeformed algebras have a similar representation theory to lie algebras because of their common structure rosso, 1988. Plancks constant and calling it a formal quantisation qdeformation actually. A quantum lie algebra can be regarded as a subspace of a qdeformed enveloping algebra uqg being invariant under the adjoint action of uqg. The point now is that the lij are the components of a. We show that the same can be done for their qdeformed. Now, we are able to introduce the notion of a quantum lie algebra as it was given in 3739.